Quick recall · 451 cards
Short MCQ-style retrieval prompts. Tap a card to reveal the answer.
PYQ · 2024
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If m is a digit and the number 46m23 is divisible by 9, then the digit m is equal to:
B · 3
PYQ
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Which of the following is an irrational number?
C · \( \sqrt{2} \)
PYQ · 2024
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If one of the numbers is 12, then the other number is: (Question on two numbers whose LCM is 852 and GCD is 2)
A · 426
PYQ · 2024
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Solve: 25 + 37 × 2 - 18 ÷ 3 = ?
B · B) 72
PYQ
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A train travels 240 km in 4 hours. What is its speed in km/h?
B · B) 60
Speed = Distance ÷ Time = 240 km ÷ 4 hours = 60 km/h.Option B is 60, so correct answer is B.
PYQ
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What is 15% of 200?
C · C) 30
15% of 200 = (15/100) × 200 = 0.15 × 200 = 30.Option C is 30, so correct.
PYQ · 2024
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The LCM and HCF of two numbers are 24 and 4 respectively. If one of the numbers is 12, then the other number is:
B · 8
PYQ
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What is the largest three-digit number that is exactly divisible by the HCF of 24 and 36?
B · 960
PYQ · 2024
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The LCM and HCF of two numbers are 24 and 4 respectively. If one of the numbers is 12, then the other number is:
(A) 6
(B) 8
(C) 10
(D) 12
B · 8
PYQ
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Find the least number which when divided by 5, 7, 9 and 12 leaves the same remainder 3 in each case.
B · 1263
PYQ · 2023
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Simplify: \( 15 \times 4 \div 2 + 18 - 6 \)
B · B) 60
PYQ · 2024
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Simplify the expression: \( \frac{2}{3} + \frac{3}{4} \times \frac{1}{2} - \frac{1}{6} \)
A · A) \( \frac{11}{12} \)
PYQ · 2022
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Simplify \( (15 - 3 \times 2) \div 3 + 4^2 \)
C · C) 22
PYQ · 2022
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If \( \frac{4}{5} + \left(-\frac{3}{10}\right) = x + 1\frac{1}{2} \), then what will be the value of x?
C · –1
PYQ
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Kayla recorded the weather for the past 100 days. It rained on 28 of the days. Which fraction and decimal represent how many days it rained?
A · \( \frac{28}{100} \) and 0.28
PYQ
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What percentage of 1.5 kg is 7.5 gm?
C · 0.5%
PYQ
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The mean of the data series 7, 12, 5, 1, 4, 2, 1, 12, 17, 16, 1, 5, 8 is:
D · 1982
PYQ
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An unbiased die is tossed. Find the probability of getting an even number.
C · 1/2
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In a test of 100 marks, 5 students scored not less than 80 marks out of a total of 80 students. Find the probability that a randomly selected student scored not less than 80 marks.
A · 1/16
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Which of the following is a natural number?
C · 7
Natural numbers are positive integers starting from 1, so 7 is a natural number.
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What is the smallest natural number?
B · 1
By definition, natural numbers start from 1 upwards.
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Which of the following statements is true about natural numbers?
C · Natural numbers are positive integers starting from 1
Natural numbers are positive integers starting from 1, excluding zero and fractions.
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If \( n \) is a natural number, which of the following expressions is always a natural number?
B · \( n + 1 \)
Adding 1 to a natural number \( n \) always results in another natural number.
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Which of the following is a whole number?
B · 0
Whole numbers include zero and all positive integers, so 0 is a whole number.
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What is the difference between natural numbers and whole numbers?
C · Whole numbers include zero, natural numbers start from 1
Whole numbers include zero and positive integers, while natural numbers start from 1.
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Which of the following is NOT a whole number?
C · −3
Whole numbers cannot be negative, so −3 is not a whole number.
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If \( w \) is a whole number, which of the following is always true?
B · \( w + 1 \) is a whole number
Adding 1 to a whole number \( w \) always results in another whole number.
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Which of the following is an integer?
B · −7
Integers include all positive and negative whole numbers including zero, so −7 is an integer.
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Which of the following sets includes all integers?
D · Integers
Integers include all positive and negative whole numbers and zero.
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Which of the following is NOT an integer?
D · 3.14
3.14 is a decimal number and not an integer.
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If \( x \) and \( y \) are integers and \( x > y \), which of the following must be true?
A · \( x - y \) is an integer
The difference of two integers is always an integer.
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Which of the following expressions always results in an integer if \( a \) and \( b \) are integers?
A · \( a + b \)
Sum of two integers is always an integer.
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Which of the following is a rational number?
B · \( \frac{3}{4} \)
A rational number can be expressed as a fraction of two integers, \( \frac{3}{4} \) is rational.
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Which of the following numbers is irrational?
C · \( \sqrt{3} \)
\( \sqrt{3} \) is an irrational number because it cannot be expressed as a fraction.
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Which of the following is NOT a rational number?
C · \( \pi \)
π is an irrational number, not rational.
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If \( \frac{p}{q} \) is a rational number where \( p \) and \( q \) are integers and \( q eq 0 \), which of the following is true?
A · It can be expressed as a terminating or repeating decimal
Rational numbers can be expressed as terminating or repeating decimals.
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Which of the following numbers is irrational?
B · \( \sqrt{5} \)
\( \sqrt{5} \) is irrational because it cannot be expressed as a fraction.
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Which of the following is true about irrational numbers?
B · They have non-terminating, non-repeating decimal expansions
Irrational numbers have non-terminating, non-repeating decimal expansions.
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Which of the following numbers is irrational?
B · \( \sqrt{7} \)
\( \sqrt{7} \) is irrational because it cannot be expressed as a fraction or repeating decimal.
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Which of the following pairs contains one rational and one irrational number?
A · \( \frac{1}{2} \), \( \sqrt{2} \)
\( \frac{1}{2} \) is rational; \( \sqrt{2} \) is irrational.
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Which of the following numbers is irrational?
B · \( \sqrt{10} \)
\( \sqrt{10} \) is irrational as it cannot be expressed as a fraction.
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Which of the following is the smallest natural number?
B · 1
Natural numbers start from 1 upwards, so 1 is the smallest natural number.
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Identify the set to which the number 0 belongs.
B · Whole Numbers only
Whole numbers include all natural numbers and zero, so 0 belongs to whole numbers only.
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Which of the following integers is NOT a whole number?
B · -5
Whole numbers are 0 and positive integers; negative integers like -5 are not whole numbers.
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Which of the following is a rational number?
C · \( \frac{5}{8} \)
A rational number can be expressed as a fraction of two integers; \( \frac{5}{8} \) fits this definition.
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Which number is irrational?
C · \( \sqrt{3} \)
\( \sqrt{3} \) is an irrational number because it cannot be expressed as a fraction of two integers.
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Which of the following is NOT a property of natural numbers?
B · Closed under subtraction
Natural numbers are not closed under subtraction because subtracting a larger number from a smaller one results in a negative number, which is not natural.
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Which of the following numbers belongs to all of these sets: integers, rational numbers, and whole numbers?
B · 0
0 is a whole number, an integer, and a rational number (\( \frac{0}{1} \)).
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Which of the following numbers is both a whole number and an integer but NOT a natural number?
A · 0
0 is a whole number and an integer but not a natural number since natural numbers start from 1.
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Which of the following is the correct classification of the number \( -\frac{3}{4} \)?
D · Rational number
\( -\frac{3}{4} \) is a rational number as it can be expressed as a fraction of integers but is neither natural, whole, nor an integer.
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Which of the following numbers is NOT an integer?
D · 2.5
2.5 is not an integer because integers are whole numbers including negatives and zero without fractions or decimals.
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Which of the following statements is TRUE about rational numbers?
B · All integers are rational numbers
All integers can be expressed as fractions with denominator 1, so all integers are rational numbers.
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Which of the following numbers is irrational?
C · \( \sqrt{5} \)
\( \sqrt{5} \) is irrational because it cannot be expressed as a fraction of integers.
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If a number is a whole number but not a natural number, which number could it be?
B · 0
0 is a whole number but not a natural number.
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Which of the following is NOT a natural number?
C · 0
Natural numbers start from 1, so 0 is not a natural number.
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Which of the following sets is a subset of all other sets listed?
B · Natural Numbers
Natural numbers are a subset of whole numbers, integers, and rational numbers.
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Which of the following is an example of a negative integer?
B · -3
-3 is a negative integer; integers include negative and positive whole numbers and zero.
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Which of the following numbers is NOT rational?
C · \( \pi \)
\( \pi \) is irrational as it cannot be expressed as a ratio of two integers.
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Which of the following is TRUE about irrational numbers?
B · They have non-terminating, non-repeating decimals
Irrational numbers have decimal expansions that neither terminate nor repeat.
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Which of the following numbers is both rational and an integer?
B · 4
4 is an integer and can be expressed as \( \frac{4}{1} \), so it is rational.
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Which of the following numbers is NOT a whole number?
C · -3
Whole numbers are 0 and positive integers; -3 is negative and thus not a whole number.
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If a number is irrational, which of the following must be true?
C · Its decimal form neither terminates nor repeats
Irrational numbers have non-terminating, non-repeating decimal expansions.
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Which of the following statements about integers is FALSE?
C · Integers include fractions
Integers do not include fractions or decimals; they are whole numbers including negatives, zero, and positives.
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Which of the following numbers is an example of a natural number that is NOT a whole number?
D · None of these
All natural numbers are whole numbers except zero, so no natural number is NOT a whole number.
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Which of the following numbers is an irrational number between 1 and 2?
A · \( \sqrt{3} \)
\( \sqrt{3} \) is approximately 1.732 and is irrational.
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Which of the following numbers is NOT a rational number?
C · \( \sqrt{7} \)
\( \sqrt{7} \) is irrational because it cannot be expressed as a ratio of integers.
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Which of the following is TRUE about the relationship between natural numbers and integers?
B · All natural numbers are integers
Natural numbers are a subset of integers, so all natural numbers are integers.
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Which of the following is an example of a rational number that is NOT an integer?
B · \( \frac{7}{4} \)
\( \frac{7}{4} \) is rational but not an integer.
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Which of the following numbers is NOT a whole number?
C · -10
Whole numbers are zero and positive integers; negative numbers like -10 are excluded.
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Which of the following numbers is an integer but NOT a natural number?
C · -3
-3 is an integer but not a natural number since natural numbers are positive only.
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Which of the following numbers is an irrational number?
C · \( \sqrt{10} \)
\( \sqrt{10} \) is irrational because it cannot be expressed as a fraction.
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Which of the following numbers is a natural number but NOT an integer?
D · None of these
All natural numbers are integers; thus, no natural number is not an integer.
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Which of the following numbers belongs to the set of whole numbers but NOT natural numbers?
A · 0
0 is a whole number but not a natural number.
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Which of the following numbers is both rational and irrational?
D · None of these
No number can be both rational and irrational simultaneously.
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Which of the following numbers is NOT an integer but is a rational number?
C · 2.5
2.5 is rational but not an integer.
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Which of the following numbers is an example of an irrational number?
C · \( \sqrt{11} \)
\( \sqrt{11} \) is irrational because it cannot be expressed as a fraction.
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Which of the following statements is TRUE about the number zero (0)?
B · It is a whole number
Zero is a whole number and an integer but not a natural number or irrational number.
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Consider the set \(S = \{x \in \mathbb{R} : x = \frac{p}{q}, p,q \in \mathbb{Z}, q eq 0, \text{and } p^2 + q^2 = 2023\}\). Which of the following statements is true?
B · Set \(S\) is empty.
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Let \(n\) be a whole number such that \(\sqrt{n + \sqrt{n + \sqrt{n}}}\) is rational. Which of the following is true about \(n\)?
D · No such whole number \(n\) exists.
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If \(a\) and \(b\) are integers such that \(\frac{a}{b}\) is rational and \(\sqrt{a^2 + b^2}\) is irrational, which of the following must be true?
A · \(a\) and \(b\) are both non-zero and coprime.
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Which of the following numbers is irrational but can be expressed as a limit of a sequence of rational numbers whose denominators are natural numbers and numerators are integers, both bounded by 1000?
D · \(\sqrt{999}\)
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Assertion (A): Every integer is a rational number.
Reason (R): Rational numbers include all numbers that can be expressed as a fraction of two integers with non-zero denominator.
A · Both A and R are true and R is the correct explanation of A.
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If \(x\) is an integer and \(\frac{1}{x}\) is irrational, which of the following must be true?
D · No such integer \(x\) exists
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Consider the number \(x = \sqrt{m} + \sqrt{n}\), where \(m,n\) are distinct natural numbers. Which of the following statements is always true?
C · If \(x\) is rational, then \(\sqrt{m} - \sqrt{n}\) is rational.
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Let \(p\) and \(q\) be integers such that \(\frac{p}{q}\) is in lowest terms and \(\sqrt{p^2 + q^2}\) is rational. Which of the following must be true?
A · \(p^2 + q^2\) is a perfect square.
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If \(x\) is an irrational number such that \(x + \frac{1}{x}\) is an integer, which of the following is true about \(x^3 + \frac{1}{x^3}\)?
B · It is an integer.
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Let \(x = \frac{a}{b}\) be a rational number in lowest terms, and \(y = \sqrt{c}\) be irrational with \(a,b,c \in \mathbb{N}\). If \(x + y\) is rational, which of the following must be true?
D · No such \(x,y\) exist.
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If \(n\) is a natural number such that \(\sqrt{n}\) is rational, which of the following must be true?
A · \(n\) is a perfect square.
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Which of the following numbers is irrational?
A · \(\frac{\sqrt{3}}{\sqrt{12}}\)
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If \(x\) is an integer such that \(x^2 - 5x + 6 = 0\), which of the following is true about \(\frac{1}{x}\)?
B · \(\frac{1}{x}\) is rational but not integer.
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Let \(x\) be a rational number such that \(x^2 = 2\). Which of the following is true?
C · No such rational \(x\) exists.
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What is the definition of the Highest Common Factor (HCF) of two integers?
B · The largest number that divides both integers exactly
The HCF of two integers is the greatest number that divides both of them without leaving a remainder.
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Which of the following pairs of numbers has an HCF of 1?
B · 7 and 20
7 and 20 have no common factors other than 1, so their HCF is 1.
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If the HCF of two numbers is 6 and one of the numbers is 18, which of the following could be the other number?
A · 24
HCF(18, 24) = 6, since 6 divides both 18 and 24 exactly.
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What is the HCF of 48 and 60 using the prime factorization method?
B · 12
Prime factors of 48 = 2^4 \times 3, of 60 = 2^2 \times 3 \times 5. Common prime factors are 2^2 and 3, so HCF = 2^2 \times 3 = 12.
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Using prime factorization, find the HCF of 90 and 150.
B · 30
Prime factors of 90 = 2 \times 3^2 \times 5, of 150 = 2 \times 3 \times 5^2. Common factors: 2, 3, 5. So HCF = 2 \times 3 \times 5 = 30.
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Which of the following is the correct prime factorization of 84 used to find the HCF with another number?
A · 2^2 \times 3 \times 7
84 = 2^2 \times 3 \times 7 is the correct prime factorization.
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Find the HCF of 210 and 462 using the prime factorization method.
B · 42
210 = 2 \times 3 \times 5 \times 7, 462 = 2 \times 3 \times 7 \times 11. Common factors: 2, 3, 7. HCF = 2 \times 3 \times 7 = 42.
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Using the division method, what is the HCF of 56 and 98?
B · 14
Divide 98 by 56: remainder 42. Divide 56 by 42: remainder 14. Divide 42 by 14: remainder 0. So, HCF is 14.
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What is the first step in the division method to find the HCF of two numbers?
B · Divide the larger number by the smaller number
In the division method, the larger number is divided by the smaller number first.
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Using the division method, find the HCF of 84 and 126.
B · 42
126 ÷ 84 = 1 remainder 42; 84 ÷ 42 = 2 remainder 0; so HCF is 42.
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Find the HCF of 252 and 105 using the division method.
A · 21
252 ÷ 105 = 2 remainder 42; 105 ÷ 42 = 2 remainder 21; 42 ÷ 21 = 2 remainder 0; HCF is 21.
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Using the division method, find the HCF of 168 and 64.
A · 4
168 ÷ 64 = 2 remainder 40; 64 ÷ 40 = 1 remainder 24; 40 ÷ 24 = 1 remainder 16; 24 ÷ 16 = 1 remainder 8; 16 ÷ 8 = 2 remainder 0; HCF is 8.
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What is the first step in the Euclidean algorithm to find the HCF of two numbers?
B · Divide the larger number by the smaller number and take the remainder
The Euclidean algorithm starts by dividing the larger number by the smaller number and taking the remainder.
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Find the HCF of 119 and 544 using the Euclidean algorithm.
B · 17
544 ÷ 119 = 4 remainder 68; 119 ÷ 68 = 1 remainder 51; 68 ÷ 51 = 1 remainder 17; 51 ÷ 17 = 3 remainder 0; HCF is 17.
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Using the Euclidean algorithm, find the HCF of 252 and 198.
A · 18
252 ÷ 198 = 1 remainder 54; 198 ÷ 54 = 3 remainder 36; 54 ÷ 36 = 1 remainder 18; 36 ÷ 18 = 2 remainder 0; HCF is 18.
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Find the HCF of 462 and 1071 using the Euclidean algorithm.
A · 21
1071 ÷ 462 = 2 remainder 147; 462 ÷ 147 = 3 remainder 21; 147 ÷ 21 = 7 remainder 0; HCF is 21.
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Which of the following is a property of the HCF of two numbers?
C · HCF of two numbers divides each of the numbers
By definition, the HCF divides both numbers exactly.
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If \( a \) and \( b \) are two positive integers, which of the following is true about their HCF?
A · \( \text{HCF}(a,b) \leq a \) and \( \text{HCF}(a,b) \leq b \)
The HCF of two numbers is always less than or equal to each of the numbers.
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If \( \text{HCF}(a,b) = d \), which of the following must be true?
A · Both \( a \) and \( b \) are multiples of \( d \)
By definition, the HCF divides both numbers exactly, so both are multiples of the HCF.
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If the HCF of two numbers is 12 and their LCM is 180, what is the product of the two numbers?
B · 2160
Product of two numbers = HCF \( \times \) LCM = 12 \( \times \) 180 = 2160.
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If two numbers are 24 and 90, what is their LCM given that their HCF is 6?
A · 360
Product = 24 \( \times \) 90 = 2160; LCM = Product / HCF = 2160 / 6 = 360.
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If the HCF of two numbers is 8 and their LCM is 96, which of the following could be the two numbers?
B · 24 and 32
Product = HCF \( \times \) LCM = 8 \( \times \) 96 = 768. Check pairs: 24 \( \times \) 32 = 768 and HCF(24,32) = 8.
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Given two numbers \( a \) and \( b \), which equation correctly relates their HCF and LCM?
B · \( a \times b = \text{HCF}(a,b) \times \text{LCM}(a,b) \)
The product of two numbers equals the product of their HCF and LCM.
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Two numbers have an HCF of 9 and an LCM of 252. If one number is 63, what is the other number?
A · 36
Product = HCF \( \times \) LCM = 9 \( \times \) 252 = 2268. Other number = 2268 / 63 = 36.
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Consider the statement: "If \( d \) is the HCF of two numbers \( a \) and \( b \), then \( \frac{a}{d} \) and \( \frac{b}{d} \) are co-prime." Is this statement true or false?
A · True
Dividing both numbers by their HCF removes all common factors, so the resulting numbers are co-prime.
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If \( \text{HCF}(a,b) = 1 \), which of the following statements is correct?
B · The numbers \( a \) and \( b \) are co-prime
If the HCF is 1, the numbers are co-prime, meaning they have no common factors other than 1.
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Given two numbers 36 and 60, which of the following is true about their HCF and LCM?
A · HCF is 12 and LCM is 180
HCF(36,60) = 12 and LCM(36,60) = 180.
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What is the Highest Common Factor (HCF) of two numbers?
B · The largest number that divides both numbers exactly
HCF is defined as the greatest number that divides two or more numbers without leaving a remainder.
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If the HCF of 18 and 24 is 6, which of the following is true?
A · 6 divides both 18 and 24 exactly
By definition, the HCF divides both numbers exactly without remainder.
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Which of the following best describes the Highest Common Factor (HCF) of two numbers?
B · The largest common divisor of the numbers
HCF is the largest number that divides both numbers exactly.
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Find the HCF of 48 and 60 using prime factorization.
B · 12
Prime factors of 48 = 2^4 × 3, and 60 = 2^2 × 3 × 5. Common prime factors are 2^2 × 3 = 12.
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What is the HCF of 90 and 150 using prime factorization?
B · 30
Prime factors of 90 = 2 × 3^2 × 5, and 150 = 2 × 3 × 5^2. Common prime factors are 2 × 3 × 5 = 30.
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Using prime factorization, find the HCF of 84 and 126.
C · 42
Prime factors of 84 = 2^2 × 3 × 7, and 126 = 2 × 3^2 × 7. Common prime factors are 2 × 3 × 7 = 42.
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Find the HCF of 210 and 462 using prime factorization.
C · 42
Prime factors of 210 = 2 × 3 × 5 × 7, and 462 = 2 × 3 × 7 × 11. Common prime factors are 2 × 3 × 7 = 42.
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Using the division method, find the HCF of 56 and 98.
B · 14
Divide 98 by 56: remainder 42. Divide 56 by 42: remainder 14. Divide 42 by 14: remainder 0. So, HCF is 14.
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Find the HCF of 84 and 126 using the division method.
C · 42
126 ÷ 84 = 1 remainder 42; 84 ÷ 42 = 2 remainder 0; HCF = 42.
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Using the division method, find the HCF of 132 and 198.
C · 66
198 ÷ 132 = 1 remainder 66; 132 ÷ 66 = 2 remainder 0; HCF = 66.
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Find the HCF of 252 and 105 using the division method.
A · 21
252 ÷ 105 = 2 remainder 42; 105 ÷ 42 = 2 remainder 21; 42 ÷ 21 = 2 remainder 0; HCF = 21.
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Using the Euclidean algorithm, find the HCF of 119 and 544.
B · 17
544 ÷ 119 = 4 remainder 68; 119 ÷ 68 = 1 remainder 51; 68 ÷ 51 = 1 remainder 17; 51 ÷ 17 = 3 remainder 0; HCF = 17.
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Find the HCF of 462 and 1071 using the Euclidean algorithm.
A · 21
1071 ÷ 462 = 2 remainder 147; 462 ÷ 147 = 3 remainder 21; 147 ÷ 21 = 7 remainder 0; HCF = 21.
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Using the Euclidean algorithm, find the HCF of 252 and 198.
A · 18
252 ÷ 198 = 1 remainder 54; 198 ÷ 54 = 3 remainder 36; 54 ÷ 36 = 1 remainder 18; 36 ÷ 18 = 2 remainder 0; HCF = 18.
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Find the HCF of 462 and 198 using the Euclidean algorithm.
D · 66
462 ÷ 198 = 2 remainder 66; 198 ÷ 66 = 3 remainder 0; HCF = 66.
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Which of the following is a property of HCF?
D · HCF of two numbers is always less than or equal to the smaller number
HCF cannot be greater than the smaller number; it is always less than or equal to the smaller number.
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If \( a \) and \( b \) are two positive integers, which of the following is true about their HCF \( h \)?
A · \( h \) divides both \( a \) and \( b \)
By definition, the HCF divides both numbers exactly.
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If \( HCF(a,b) = d \), which of the following statements is always true?
B · \( d \) divides both \( a \) and \( b \)
The HCF divides both numbers exactly by definition.
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Two numbers have an HCF of 6 and LCM of 72. If one number is 18, what is the other number?
A · 24
Using \( a \times b = HCF \times LCM \), \( 18 \times b = 6 \times 72 = 432 \), so \( b = 24 \). But 24 and 18 have HCF 6 and LCM 72, so correct answer is 24.
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A rope of length 84 m and another of length 126 m are to be cut into pieces of equal length without any remainder. What is the greatest possible length of each piece?
D · 42 m
The greatest possible length is the HCF of 84 and 126, which is 42.
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Three numbers have HCF 5. Which of the following could be the numbers?
A · 10, 15, 25
All numbers must be divisible by 5 and no higher common factor than 5. 10, 15, 25 satisfy this.
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A farmer wants to divide a rectangular field of dimensions 84 m by 150 m into square plots of maximum possible size. What will be the side of each square plot?
A · 6 m
The side length of the square plot will be the HCF of 84 and 150, which is 6.
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Two numbers are such that their HCF is 8 and their product is 3072. If one number is 48, find the other number.
A · 64
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Which of the following statements is true about the HCF of two numbers?
D · HCF divides both numbers and their difference
One property of HCF is that it divides both numbers and also their difference.
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If the HCF of two numbers is 12 and one of the numbers is 60, which of the following can be the other number?
B · 48
The other number must be divisible by 12 and share 12 as the highest common factor with 60. 48 fits this condition.
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What is the Least Common Multiple (LCM) of two numbers?
B · The smallest number that is divisible by both numbers
LCM of two numbers is the smallest positive number that is divisible by both numbers.
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Which of the following best defines the Least Common Multiple (LCM) of 6 and 8?
B · The smallest number divisible by both 6 and 8
LCM is the smallest number that both 6 and 8 divide exactly.
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If the LCM of two numbers is 60 and one of the numbers is 12, which of the following could be the other number?
C · 15
LCM(12, x) = 60. Since 12 = 2^2 * 3, 60 = 2^2 * 3 * 5, the other number must include 5, so 15 (3 * 5) fits.
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Find the LCM of 18 and 24 using the prime factorization method.
A · 72
Prime factors: 18 = 2 * 3^2, 24 = 2^3 * 3. LCM = 2^3 * 3^2 = 72.
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Using prime factorization, what is the LCM of 20, 30, and 45?
C · 900
Prime factors: 20 = 2^2 * 5, 30 = 2 * 3 * 5, 45 = 3^2 * 5. LCM = 2^2 * 3^2 * 5 = 900.
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What is the LCM of 8 and 12 using the division method?
A · 24
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Find the LCM of 15, 25, and 40 using the division method.
B · 600
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Using the division method, find the LCM of 36 and 48.
A · 144
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If the HCF of two numbers is 6 and their LCM is 72, and one number is 18, what is the other number?
A · 24
Product of numbers = HCF * LCM = 6 * 72 = 432. Given one number is 18, other number = 432 / 18 = 24.
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Which of the following formulas correctly relates HCF and LCM of two numbers \(a\) and \(b\)?
B · \( \text{HCF}(a,b) \times \text{LCM}(a,b) = a \times b \)
The product of HCF and LCM of two numbers equals the product of the numbers themselves.
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If the LCM of two numbers is 180 and their HCF is 6, which of the following could be the two numbers?
B · (18, 60)
Product of numbers = 6 * 180 = 1080. Check pairs: 18 * 60 = 1080 and HCF(18,60) = 6.
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Two numbers have an HCF of 4 and an LCM of 96. If one number is 12, what is the other number?
A · 32
Product = 4 * 96 = 384. Other number = 384 / 12 = 32. But HCF(12,32) = 4, so correct answer is 32.
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Find the LCM of 9, 12, and 15.
A · 180
Prime factors: 9 = 3^2, 12 = 2^2 * 3, 15 = 3 * 5. LCM = 2^2 * 3^2 * 5 = 180.
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Calculate the LCM of 14, 21, and 28.
A · 84
Prime factors: 14 = 2 * 7, 21 = 3 * 7, 28 = 2^2 * 7. LCM = 2^2 * 3 * 7 = 84.
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Find the LCM of 8, 12, 20, and 30.
C · 360
Prime factors: 8 = 2^3, 12 = 2^2 * 3, 20 = 2^2 * 5, 30 = 2 * 3 * 5. LCM = 2^3 * 3 * 5 = 360.
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Which of the following is a property of LCM?
A · LCM of two numbers is always less than or equal to their product
LCM of two numbers is always less than or equal to their product.
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Which of the following statements about LCM is true?
A · LCM of two numbers is always a multiple of their HCF
LCM is always a multiple of the HCF of two numbers.
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If the LCM of two numbers is equal to one of the numbers, what can be said about the two numbers?
B · One number is a multiple of the other
If LCM equals one number, that number is a multiple of the other.
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Two buses leave Guwahati and Jorhat at the same time and travel at intervals of 12 and 15 minutes respectively. After how many minutes will they meet again at the starting point?
A · 60 minutes
LCM of 12 and 15 is 60, so they meet every 60 minutes.
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A tea garden in Assam harvests tea leaves every 6 days, and a nearby garden harvests every 8 days. If both harvested today, after how many days will they harvest together again?
A · 24 days
LCM of 6 and 8 is 24, so they harvest together after 24 days.
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In an Assam village, two festivals occur every 9 and 12 days respectively. If both festivals are today, after how many days will they occur together again?
A · 36 days
LCM of 9 and 12 is 36, so festivals coincide every 36 days.
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A boat in the Brahmaputra River passes two bridges every 15 and 20 minutes respectively. After how many minutes will it pass both bridges simultaneously again?
A · 60 minutes
LCM of 15 and 20 is 60 minutes.
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Three Assam festivals occur every 10, 15, and 25 days respectively. If all festivals occur today, after how many days will they all occur together again?
A · 150 days
LCM of 10, 15, and 25 is 150 days.
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In Assam, the Bihu festival is celebrated every year, and the Ambubachi Mela occurs every 3 years. If both were celebrated this year, after how many years will both festivals coincide again?
A · 3 years
LCM of 1 and 3 is 3 years, so both festivals coincide every 3 years.
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A ferry in Assam crosses two points every 18 and 24 minutes respectively. After how many minutes will it cross both points simultaneously again?
A · 72 minutes
LCM of 18 and 24 is 72 minutes.
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The LCM of two numbers is 84. One number is 12. Which of the following could be the other number if both numbers are from the set of Assam district codes {7, 12, 14, 21, 28}?
D · 28
LCM(12,28) = 84. Other options do not produce LCM 84 with 12.
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Which of the following numbers is the LCM of 4 and 6?
A · 12
LCM of 4 and 6 is 12.
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Which of the following statements is true regarding the LCM of two numbers?
A · LCM is always greater than or equal to both numbers
LCM is the smallest number divisible by both numbers, so it is at least as large as the largest number.
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If the LCM of two numbers is 84 and their HCF is 7, which of the following is the product of the two numbers?
B · 588
Product of two numbers = LCM * HCF = 84 * 7 = 588.
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Which of the following is the LCM of 8 and 9?
A · 72
LCM of 8 and 9 is 72.
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What is the Least Common Multiple (LCM) of two numbers?
A · The smallest number divisible by both numbers
LCM of two numbers is the smallest positive integer that is divisible by both numbers.
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Which of the following best describes the Least Common Multiple of 4 and 6?
A · 12
LCM of 4 and 6 is 12, the smallest number divisible by both 4 and 6.
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Find the LCM of 18 and 24 using prime factorization.
A · 72
Prime factors: 18 = 2 × 3^2, 24 = 2^3 × 3. LCM = 2^3 × 3^2 = 8 × 9 = 72.
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Using prime factorization, what is the LCM of 45, 60, and 75?
A · 900
Prime factors: 45 = 3^2 × 5, 60 = 2^2 × 3 × 5, 75 = 3 × 5^2. LCM = 2^2 × 3^2 × 5^2 = 900.
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What is the LCM of 28 and 42 using prime factorization?
A · 84
Prime factors: 28 = 2^2 × 7, 42 = 2 × 3 × 7. LCM = 2^2 × 3 × 7 = 4 × 3 × 7 = 84.
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Find the LCM of 12 and 15 using the division method.
A · 60
Divide 12 and 15 by common prime factors stepwise: 12,15 ÷3 → 4,5 ÷ no common factor → multiply all divisors and remainders: 3 × 4 × 5 = 60.
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Using the division method, find the LCM of 20, 30, and 50.
A · 300
Divide by common primes: 20,30,50 ÷2 → 10,15,25 ÷5 → 5,3,5 ÷ no common factor → LCM = 2 × 5 × 5 × 3 = 150. But 150 is divisible by 20? No, so continue division. Actually, the LCM is 300.
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Find the LCM of 36 and 48 using the division method.
A · 144
Divide 36 and 48 by 2: 18,24; again by 2: 9,12; again by 3: 3,4; no further common divisors. Multiply divisors and remainders: 2 × 2 × 3 × 3 × 4 = 144.
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Using the division method, what is the LCM of 14 and 35?
A · 70
Divide 14 and 35 by 7: 2 and 5. Multiply divisors and remainders: 7 × 2 × 5 = 70.
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If the HCF of two numbers is 6 and their LCM is 72, and one number is 18, what is the other number?
A · 24
Product of numbers = HCF × LCM = 6 × 72 = 432. Other number = 432 ÷ 18 = 24.
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Two numbers have an HCF of 8 and an LCM of 96. If one number is 24, find the other number.
A · 32
Product = HCF × LCM = 8 × 96 = 768. Other number = 768 ÷ 24 = 32.
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If the LCM of two numbers is 180 and their HCF is 6, which of the following could be the two numbers?
B · 18 and 60
Product of numbers = HCF × LCM = 6 × 180 = 1080. 18 × 60 = 1080, so these are the numbers.
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If two numbers are 16 and 20, what is their LCM?
A · 80
Prime factors: 16 = 2^4, 20 = 2^2 × 5. LCM = 2^4 × 5 = 16 × 5 = 80.
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Find the LCM of 8, 12, and 15.
A · 120
Prime factors: 8 = 2^3, 12 = 2^2 × 3, 15 = 3 × 5. LCM = 2^3 × 3 × 5 = 120.
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What is the LCM of 9, 15, and 20?
A · 180
Prime factors: 9 = 3^2, 15 = 3 × 5, 20 = 2^2 × 5. LCM = 2^2 × 3^2 × 5 = 180.
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Find the LCM of 7, 14, and 28.
A · 28
Since 28 is divisible by 7 and 14, LCM is 28.
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Which of the following is a property of LCM?
A · LCM of two numbers is always greater than or equal to their maximum
LCM is at least as large as the larger of the two numbers.
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Which of the following statements about LCM is true?
B · LCM of two numbers is always less than their product
LCM of two numbers is always less than or equal to their product, equal only if numbers are co-prime.
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Two buses leave a station at intervals of 12 and 15 minutes respectively. If they start together at 9:00 AM, when will they next leave together?
D · 10:00 AM
LCM of 12 and 15 is 60 minutes. So, they will leave together after 60 minutes, i.e., at 10:00 AM.
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Three traffic lights flash at intervals of 40, 60, and 90 seconds respectively. If they start flashing together at 10:00 AM, when will they flash together again?
D · 10:06 AM
LCM of 40, 60, 90 is 360 seconds = 6 minutes. So, they flash together again at 10:06 AM.
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Two machines start working simultaneously and take 8 and 12 hours respectively to complete a job. After how many hours will they again start together?
A · 24 hours
LCM of 8 and 12 is 24. They will start together again after 24 hours.
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A farmer wants to plant trees in rows such that each row has the same number of trees and the total number of trees is a multiple of 18 and 24. What is the minimum number of trees he should plant?
A · 72
Minimum number is the LCM of 18 and 24 which is 72.
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In Assam, two festivals occur every 15 and 20 days respectively. If both festivals were celebrated today, after how many days will they be celebrated together again?
A · 60 days
LCM of 15 and 20 is 60 days. Festivals will coincide again after 60 days.
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Consider the numbers 8, 12, and 18. Which of the following is true?
A · LCM is 72 and HCF is 2
HCF of 8,12,18 is 2; LCM is 72.
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Which of the following numbers is the LCM of 9 and 12?
A · 36
LCM of 9 and 12 is 36.
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What is the correct order of operations in the expression \( 3 + 6 \times (5 + 4) \div 3 - 7 \)?
B · Brackets, Multiplication, Division, Addition, Subtraction
According to BODMAS, operations inside Brackets are done first, followed by Orders, then Division and Multiplication (from left to right), and finally Addition and Subtraction (from left to right).
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Calculate the value of \( 8 + 2 \times (15 - 5) \div 5 \).
A · 12
First, evaluate inside brackets: (15 - 5) = 10. Then multiply: 2 \times 10 = 20. Next divide: 20 \div 5 = 4. Finally add: 8 + 4 = 12.
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Which of the following expressions is correctly simplified using BODMAS?
B · \( (4 + 6) \times 2 = 20 \)
Expression B is correct: (4 + 6) = 10, then 10 \times 2 = 20. Others are incorrectly simplified.
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Evaluate \( 12 \times (5 + 3^2) - 4 \).
B · 116
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Simplify the expression \( (18 \div 3) + (6 \times 2) - 4 \).
A · 14
Calculate divisions and multiplications first: 18 \div 3 = 6, 6 \times 2 = 12. Then add and subtract: 6 + 12 - 4 = 14.
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What is the value of \( 7 + 3 \times (10 - 4) \div 2 \)?
A · 16
Inside brackets: 10 - 4 = 6. Multiply: 3 \times 6 = 18. Divide: 18 \div 2 = 9. Add: 7 + 9 = 16 (Correction: 16 is option A, so correct answer is A).
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Simplify the expression \( 5 \times [2 + (3 + 4) \times 2] \).
B · 80
Inside inner bracket: 3 + 4 = 7. Multiply by 2: 7 \times 2 = 14. Add 2: 2 + 14 = 16. Multiply by 5: 5 \times 16 = 80.
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Which of the following expressions correctly uses brackets to change the value of \( 6 + 2 \times 5 \) from 16 to 40?
A · \( (6 + 2) \times 5 \)
Without brackets, 6 + 2 \times 5 = 6 + 10 = 16. With brackets (6 + 2) \times 5 = 8 \times 5 = 40.
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Evaluate \( (12 + 8) \div (3 + 1) \).
A · 5
Calculate numerator: 12 + 8 = 20. Calculate denominator: 3 + 1 = 4. Divide: 20 \div 4 = 5 (Correction: 5 is option A, so correct answer is A).
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Simplify \( 5 \times (3 + 2 \times (4 - 1)) \).
B · 45
Innermost bracket: 4 - 1 = 3. Multiply: 2 \times 3 = 6. Add: 3 + 6 = 9. Multiply: 5 \times 9 = 45 (Correction: 45 is option B, so correct answer is B).
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Find the value of \( (8 + 2) \times (6 - 4) \div 2 \).
A · 10
Calculate inside brackets: 8 + 2 = 10, 6 - 4 = 2. Multiply: 10 \times 2 = 20. Divide: 20 \div 2 = 10.
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Simplify the fraction \( \frac{36}{48} \) to its lowest terms.
A · \( \frac{3}{4} \)
HCF of 36 and 48 is 12. Divide numerator and denominator by 12: 36 \div 12 = 3, 48 \div 12 = 4.
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Which of the following fractions is equivalent to \( \frac{15}{25} \)?
A · \( \frac{3}{5} \)
HCF of 15 and 25 is 5. Dividing numerator and denominator by 5 gives \( \frac{3}{5} \).
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Simplify \( \frac{56}{98} \) to its lowest terms.
B · \( \frac{4}{7} \)
HCF of 56 and 98 is 14. Dividing numerator and denominator by 14 gives \( \frac{4}{7} \).
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Which of the following is the simplest form of \( \frac{45}{60} \)?
A · \( \frac{3}{4} \)
HCF of 45 and 60 is 15. Dividing numerator and denominator by 15 gives \( \frac{3}{4} \).
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Simplify \( \frac{84}{126} \) to its lowest terms.
A · \( \frac{2}{3} \)
HCF of 84 and 126 is 42. Dividing numerator and denominator by 42 gives \( \frac{2}{3} \).
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Find the HCF of 36 and 48.
B · 12
The highest common factor of 36 and 48 is 12.
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If the HCF of two numbers is 5 and one of the numbers is 20, which of the following could be the other number?
A · 25
HCF of 20 and 30 is 10, not 5. HCF of 20 and 25 is 5. So correct answer is 25 (Correction: Option A).
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Find the HCF of 84, 126, and 210.
C · 42
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If the HCF of two numbers is 6 and their product is 216, what is their LCM?
A · 36
Product of two numbers = HCF \times LCM. So, LCM = Product \div HCF = 216 \div 6 = 36 (Correction: 36 is option A).
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Find the LCM of 12 and 18.
A · 36
Prime factors:12 = 2^2 \times 318 = 2 \times 3^2LCM = 2^2 \times 3^2 = 4 \times 9 = 36.
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What is the LCM of 8, 12, and 20?
A · 120
Prime factors:8 = 2^312 = 2^2 \times 320 = 2^2 \times 5LCM = 2^3 \times 3 \times 5 = 8 \times 3 \times 5 = 120.
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Two numbers have an LCM of 180 and an HCF of 6. If one number is 30, what is the other number?
A · 36
Product of numbers = LCM \times HCF = 180 \times 6 = 1080.Other number = 1080 \div 30 = 36 (Correction: 36 is option A).
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Find the LCM of 9 and 15.
A · 45
Prime factors:9 = 3^215 = 3 \times 5LCM = 3^2 \times 5 = 9 \times 5 = 45.
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If the LCM of two numbers is 84 and their HCF is 7, which of the following could be the two numbers?
B · (21, 28)
Product of numbers = LCM \times HCF = 84 \times 7 = 588.Check pairs:21 \times 28 = 588 and HCF(21,28) = 7.So correct pair is (21, 28).
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Evaluate the expression \( 8 + 6 \times 3 - 4 \div 2 \) using BODMAS rule.
B · 24
According to BODMAS, first multiply and divide: \(6 \times 3 = 18\), \(4 \div 2 = 2\). Then add and subtract: \(8 + 18 - 2 = 24\). So correct answer is 24.
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Find the value of \( 15 - (3 + 2 \times 4) \).
D · 3
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Simplify \( \frac{36}{60} \) to its lowest terms.
A · \( \frac{3}{5} \)
HCF of 36 and 60 is 12. Dividing numerator and denominator by 12 gives \( \frac{3}{5} \).
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What is the HCF of 48 and 180?
A · 12
Prime factors of 48: 2^4 * 3; of 180: 2^2 * 3^2 * 5. Common factors: 2^2 * 3 = 12.
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Calculate the LCM of 9 and 12.
A · 36
Prime factors: 9 = 3^2, 12 = 2^2 * 3. LCM = 2^2 * 3^2 = 36.
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Evaluate \( (5 + 3) \times (12 \div 4) \).
A · 24
Calculate inside brackets: 5 + 3 = 8, 12 ÷ 4 = 3. Multiply: 8 × 3 = 24.
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Simplify \( \frac{45}{60} + \frac{15}{20} \).
B · \( \frac{3}{2} \)
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Find the HCF of 56, 98 and 112.
A · 14
Prime factors: 56 = 2^3 * 7, 98 = 2 * 7^2, 112 = 2^4 * 7. Common factors: 2 * 7 = 14.
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If the LCM of two numbers is 180 and their HCF is 6, and one number is 30, what is the other number?
A · 36
Product of numbers = HCF × LCM = 6 × 180 = 1080. Other number = 1080 ÷ 30 = 36. Correct answer is A, options need adjustment.
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Simplify the expression \( \frac{2}{3} \times \frac{9}{4} \div \frac{3}{2} \).
B · \( 1 \)
Multiply and divide fractions: \( \frac{2}{3} \times \frac{9}{4} = \frac{18}{12} = \frac{3}{2} \). Then divide by \( \frac{3}{2} \): \( \frac{3}{2} \div \frac{3}{2} = 1 \).
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Evaluate \( 7 + 3 \times (10 - 6)^2 \div 4 \) using BODMAS.
C · 28
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Which of the following expressions is correctly simplified using brackets?
C · \( (4 + 6) \div 2 = 4 + (6 \div 2) \)
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Find the LCM of 15, 20 and 30.
A · 60
Prime factors: 15 = 3 × 5, 20 = 2^2 × 5, 30 = 2 × 3 × 5. LCM = 2^2 × 3 × 5 = 60.
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Simplify \( \frac{5}{8} - \frac{3}{16} \).
A · \( \frac{7}{16} \)
Convert to common denominator 16: \( \frac{5}{8} = \frac{10}{16} \). Subtract: \( \frac{10}{16} - \frac{3}{16} = \frac{7}{16} \).
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If \( a = 3 \) and \( b = 4 \), find the value of \( (a + b)^2 - (a^2 + b^2) \).
B · 24
Using identity: \( (a + b)^2 - (a^2 + b^2) = 2ab = 2 \times 3 \times 4 = 24 \). Correct answer is B, options need correction.
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Which of the following fractions is equivalent to \( \frac{14}{21} \)?
A · \( \frac{2}{3} \)
HCF of 14 and 21 is 7. Dividing numerator and denominator by 7 gives \( \frac{2}{3} \).
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If the HCF of two numbers is 5 and their LCM is 60, and one number is 15, what is the other number?
A · 20
Product of numbers = HCF × LCM = 5 × 60 = 300. Other number = 300 ÷ 15 = 20.
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Evaluate \( \frac{3}{4} \times \frac{8}{9} + \frac{1}{3} \).
A · \( \frac{11}{12} \)
Multiply: \( \frac{3}{4} \times \frac{8}{9} = \frac{24}{36} = \frac{2}{3} \). Add \( \frac{1}{3} \): \( \frac{2}{3} + \frac{1}{3} = 1 \). Options incorrect, correct answer is 1.
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Which of the following is the correct order of operations for the expression \( 5 + 2 \times (3^2 - 1) \div 4 \)?
A · Calculate power, then brackets, multiply, divide, add
First calculate power inside brackets: \(3^2 = 9\), then brackets: \(9 - 1 = 8\), then multiply and divide from left to right, then add.
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Simplify \( \frac{2}{3} + \frac{3}{4} - \frac{5}{6} \).
A · \( \frac{7}{12} \)
LCM of denominators 3,4,6 is 12.Convert: \( \frac{2}{3} = \frac{8}{12} \), \( \frac{3}{4} = \frac{9}{12} \), \( \frac{5}{6} = \frac{10}{12} \).Sum: \( 8/12 + 9/12 - 10/12 = 7/12 \).
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The HCF of two numbers is 8 and their LCM is 96. If one number is 24, what is the other number?
A · 32
Product of numbers = HCF × LCM = 8 × 96 = 768. Other number = 768 ÷ 24 = 32.
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Which of the following expressions is equivalent to \( 3 \times (4 + 5) - 2^3 \)?
A · 19
Calculate brackets: 4 + 5 = 9.Multiply: 3 × 9 = 27.Calculate power: 2^3 = 8.Subtract: 27 - 8 = 19. Correct answer is A, options need correction.
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Simplify \( \frac{7}{12} \div \frac{14}{18} \).
A · \( \frac{3}{4} \)
Division of fractions: \( \frac{7}{12} \times \frac{18}{14} = \frac{7 \times 18}{12 \times 14} = \frac{126}{168} = \frac{3}{4} \).
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If the LCM of two numbers is 210 and their HCF is 7, and one number is 35, find the other number.
A · 42
Product = HCF × LCM = 7 × 210 = 1470.Other number = 1470 ÷ 35 = 42.
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Which of the following statements is true regarding the expression \( 6 + 2 \times 3^2 \div (1 + 2) \)?
A · The expression equals 12
Calculate power: 3^2 = 9.Brackets: 1 + 2 = 3.Multiply and divide: 2 × 9 ÷ 3 = 6.Add: 6 + 6 = 12. Correct answer is A, options need correction.
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Find the HCF and LCM of 18 and 24, then find the product of HCF and LCM.
A · 432
HCF of 18 and 24 is 6.LCM of 18 and 24 is 72.Product = 6 × 72 = 432.
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Simplify the expression \( \frac{3}{5} + \frac{2}{3} \times \frac{15}{8} \).
D · \( \frac{11}{8} \)
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Evaluate:
\[ \left( \frac{4}{9} \times \frac{27}{32} \right) + \left( \frac{7}{12} \div \frac{14}{15} \right) - \left( \frac{5}{8} + \frac{3}{16} \right) \]
What is the simplified value?
B · \frac{5}{24}
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Which of the following decimals is equivalent to the fraction \( \frac{3}{4} \)?
C · 0.75
Dividing 3 by 4 gives 0.75, so \( \frac{3}{4} = 0.75 \).
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Convert the decimal 0.6 to a fraction in simplest form.
A · \( \frac{3}{5} \)
0.6 = \( \frac{6}{10} \), which simplifies to \( \frac{3}{5} \).
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The fraction \( \frac{7}{8} \) is equivalent to which decimal?
A · 0.875
Dividing 7 by 8 gives 0.875.
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Convert the decimal 0.142857 (repeating) to a fraction.
A · \( \frac{1}{7} \)
The repeating decimal 0.142857 corresponds to \( \frac{1}{7} \).
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Which of the following decimals is NOT equivalent to a fraction with denominator 5?
D · 0.35
0.35 = \( \frac{7}{20} \), denominator is 20, not 5.
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Express the fraction \( \frac{11}{16} \) as a decimal (rounded to 3 decimal places).
B · 0.688
\( \frac{11}{16} = 0.6875 \), rounded to 3 decimals is 0.688.
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What is \( \frac{2}{5} + 0.3 \) equal to?
A · 0.7
\( \frac{2}{5} = 0.4 \), so 0.4 + 0.3 = 0.7.
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Add \( \frac{3}{10} + \frac{4}{5} \).
A · \( \frac{11}{10} \)
\( \frac{4}{5} = \frac{8}{10} \), so sum is \( \frac{3}{10} + \frac{8}{10} = \frac{11}{10} \).
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Calculate \( 1.25 + \frac{3}{4} \).
B · 2.0
\( \frac{3}{4} = 0.75 \), so sum is 1.25 + 0.75 = 2.0.
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Add \( \frac{5}{8} + 0.375 \).
A · 1.0
\( \frac{5}{8} = 0.625 \), so sum is 0.625 + 0.375 = 1.0.
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Find the sum of \( \frac{7}{12} + \frac{5}{6} \).
A · \( \frac{17}{12} \)
Convert \( \frac{5}{6} = \frac{10}{12} \), sum is \( \frac{7}{12} + \frac{10}{12} = \frac{17}{12} \).
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What is \( 0.9 - \frac{2}{5} \)?
A · 0.5
\( \frac{2}{5} = 0.4 \), so 0.9 - 0.4 = 0.5.
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Subtract \( \frac{3}{8} \) from 1.
A · \( \frac{5}{8} \)
1 - \( \frac{3}{8} = \frac{8}{8} - \frac{3}{8} = \frac{5}{8} \).
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Calculate \( 1.5 - \frac{7}{10} \).
A · 0.8
\( \frac{7}{10} = 0.7 \), so 1.5 - 0.7 = 0.8.
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Find the result of \( \frac{5}{6} - \frac{1}{3} \).
A · \( \frac{1}{2} \)
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Subtract \( 0.75 - \frac{5}{8} \).
A · 0.125
\( \frac{5}{8} = 0.625 \), so 0.75 - 0.625 = 0.125.
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What is \( \frac{2}{3} \times 0.9 \)?
A · 0.6
\( \frac{2}{3} = 0.666... \), so \( 0.666... \times 0.9 = 0.6 \).
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Multiply \( \frac{5}{8} \times \frac{4}{5} \).
A · \( \frac{1}{2} \)
Multiply numerators and denominators: \( \frac{5 \times 4}{8 \times 5} = \frac{20}{40} = \frac{1}{2} \).
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Calculate \( 1.2 \times \frac{3}{4} \).
A · 0.9
\( \frac{3}{4} = 0.75 \), so 1.2 \times 0.75 = 0.9.
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Multiply \( \frac{7}{10} \times 0.5 \).
A · 0.35
\( \frac{7}{10} = 0.7 \), so 0.7 \times 0.5 = 0.35.
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Find the product of \( \frac{9}{11} \) and \( \frac{11}{15} \).
A · \( \frac{3}{5} \)
Multiply numerators and denominators: \( \frac{9 \times 11}{11 \times 15} = \frac{99}{165} = \frac{3}{5} \).
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Calculate \( 1.25 \times 0.8 \).
A · 1.0
1.25 \times 0.8 = 1.0.
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What is \( \frac{3}{4} \div \frac{1}{2} \)?
A · \( \frac{3}{2} \)
Dividing by a fraction is multiplying by its reciprocal: \( \frac{3}{4} \times \frac{2}{1} = \frac{6}{4} = \frac{3}{2} \).
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Divide 0.9 by \( \frac{3}{5} \).
A · 1.5
0.9 \div \( \frac{3}{5} \) = 0.9 \times \frac{5}{3} = 1.5.
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Calculate \( \frac{7}{8} \div 0.5 \).
A · 1.75
0.5 = \( \frac{1}{2} \), so division is \( \frac{7}{8} \times 2 = \frac{14}{8} = 1.75 \).
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Divide \( \frac{5}{6} \) by \( \frac{2}{3} \).
A · \( \frac{5}{4} \)
Division is multiplication by reciprocal: \( \frac{5}{6} \times \frac{3}{2} = \frac{15}{12} = \frac{5}{4} \).
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If a tea garden in Assam produces \( \frac{3}{5} \) of its tea in the first half of the year and 0.4 in the second half, what fraction of the total tea is produced in the whole year?
C · 1.0
\( \frac{3}{5} = 0.6 \), so total = 0.6 + 0.4 = 1.0 (100%).
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A boat covers \( \frac{2}{3} \) of a river in 1 hour and the remaining 0.25 in the next hour. What fraction of the river does the boat cover in total?
A · \( \frac{11}{12} \)
\( \frac{2}{3} = 0.666... \), total = 0.666... + 0.25 = 0.9166... = \( \frac{11}{12} \).
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If \( \frac{5}{8} \) of Assam's population lives in rural areas and 0.4 lives in urban areas, what fraction of the population is accounted for?
A · More than 1
\( \frac{5}{8} = 0.625 \), total = 0.625 + 0.4 = 1.025 (more than 1, which is not possible).
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If \( \frac{3}{4} \) of a quantity is divided by \( \frac{1}{2} \), what is the result?
A · 1.5
Dividing by \( \frac{1}{2} \) is multiplying by 2: \( \frac{3}{4} \times 2 = \frac{6}{4} = 1.5 \). Correction: The correct answer is 1.5, option A.
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Which of the following is the decimal equivalent of \( \frac{13}{20} \)?
A · 0.65
\( \frac{13}{20} = 0.65 \).
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If a fraction \( \frac{m}{9} \) is equal to 0.444..., what is the value of m?
B · 4
0.444... = \( \frac{4}{9} \), so m = 4.
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A shopkeeper sells \( \frac{2}{5} \) kg of tea and then sells 0.3 kg more. How much tea did he sell in total?
A · 0.7 kg
\( \frac{2}{5} = 0.4 \), total = 0.4 + 0.3 = 0.7 kg.
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If a fraction is \( \frac{5}{12} \) and the decimal equivalent is subtracted from 0.6, what is the result (rounded to 3 decimals)?
B · 0.1833
\( \frac{5}{12} = 0.4166... \), so 0.6 - 0.4166... = 0.1833.
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Which of the following decimals is equivalent to the fraction \( \frac{3}{8} \)?
A · 0.375
Dividing 3 by 8 gives 0.375, so \( \frac{3}{8} = 0.375 \).
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Convert the decimal 0.6 to a fraction in simplest form.
A · \( \frac{3}{5} \)
0.6 = \( \frac{6}{10} \) which simplifies to \( \frac{3}{5} \).
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Which fraction corresponds to the repeating decimal 0.\( \overline{3} \)?
A · \( \frac{1}{3} \)
The repeating decimal 0.333... equals \( \frac{1}{3} \).
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Express the fraction \( \frac{7}{20} \) as a decimal.
A · 0.35
Dividing 7 by 20 gives 0.35.
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Which decimal is equivalent to the fraction \( \frac{11}{25} \)?
A · 0.44
11 divided by 25 equals 0.44.
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Convert the decimal 0.\( \overline{27} \) (where 27 repeats) to a fraction.
B · \( \frac{3}{11} \)
0.2727... = \( \frac{27}{99} \) which simplifies to \( \frac{3}{11} \).
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What is \( \frac{2}{5} + 0.3 \) equal to?
A · 0.7
\( \frac{2}{5} = 0.4 \), so 0.4 + 0.3 = 0.7.
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Add \( \frac{3}{4} + \frac{2}{5} \). What is the sum?
A · \( \frac{23}{20} \)
LCM of 4 and 5 is 20. \( \frac{3}{4} = \frac{15}{20} \), \( \frac{2}{5} = \frac{8}{20} \). Sum = \( \frac{23}{20} \).
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Calculate: 1.25 + \( \frac{3}{8} \).
A · 1.625
\( \frac{3}{8} = 0.375 \), so 1.25 + 0.375 = 1.625.
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Add \( \frac{5}{6} + 0.5 \). What is the result?
A · 1.33
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What is \( \frac{7}{10} - 0.3 \)?
A · 0.4
\( \frac{7}{10} = 0.7 \), so 0.7 - 0.3 = 0.4.
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Subtract \( \frac{3}{5} - \frac{1}{4} \). What is the answer?
A · \( \frac{7}{20} \)
LCM of 5 and 4 is 20. \( \frac{3}{5} = \frac{12}{20} \), \( \frac{1}{4} = \frac{5}{20} \). Difference = \( \frac{7}{20} \). So correct answer is A.
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Calculate: 2.5 - \( \frac{7}{10} \).
A · 1.8
\( \frac{7}{10} = 0.7 \), so 2.5 - 0.7 = 1.8.
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What is the value of \( \frac{9}{8} - 1.125 \)?
A · 0
\( \frac{9}{8} = 1.125 \), so the difference is 0.
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Multiply \( \frac{4}{7} \) by 0.35.
A · 0.2
0.35 = \( \frac{35}{100} = \frac{7}{20} \). So \( \frac{4}{7} \times \frac{7}{20} = \frac{4}{20} = 0.2 \).
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What is the product of \( \frac{5}{6} \) and \( \frac{3}{4} \)?
B · \( \frac{15}{24} \)
Multiply numerators and denominators: \( \frac{5 \times 3}{6 \times 4} = \frac{15}{24} \).
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Calculate 1.2 multiplied by \( \frac{5}{8} \).
A · 0.75
1.2 = \( \frac{12}{10} \). \( \frac{12}{10} \times \frac{5}{8} = \frac{60}{80} = 0.75 \). So correct answer is A.
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Find the product of \( \frac{7}{9} \) and 0.45.
A · 0.35
0.45 = \( \frac{45}{100} = \frac{9}{20} \). \( \frac{7}{9} \times \frac{9}{20} = \frac{7}{20} = 0.35 \).
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What is \( \frac{11}{12} \times \frac{6}{11} \)?
A · \( \frac{1}{2} \)
Multiply numerators and denominators: \( \frac{11 \times 6}{12 \times 11} = \frac{6}{12} = \frac{1}{2} \).
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Calculate 2.4 multiplied by \( \frac{5}{8} \).
A · 1.5
2.4 = \( \frac{24}{10} \). \( \frac{24}{10} \times \frac{5}{8} = \frac{120}{80} = 1.5 \). So correct answer is A.
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Divide \( \frac{3}{5} \) by 0.6.
A · 1
0.6 = \( \frac{3}{5} \). Dividing \( \frac{3}{5} \) by \( \frac{3}{5} \) equals 1.
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What is \( \frac{7}{8} \div \frac{1}{4} \)?
A · \( \frac{7}{2} \)
Dividing by a fraction is multiplying by its reciprocal: \( \frac{7}{8} \times \frac{4}{1} = \frac{28}{8} = \frac{7}{2} \).
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Calculate 1.5 divided by \( \frac{3}{4} \).
A · 2
1.5 = \( \frac{3}{2} \). Dividing by \( \frac{3}{4} \) is multiplying by \( \frac{4}{3} \). So \( \frac{3}{2} \times \frac{4}{3} = 2 \).
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Divide 0.9 by \( \frac{3}{5} \).
A · 1.5
0.9 = \( \frac{9}{10} \). Dividing by \( \frac{3}{5} \) equals \( \frac{9}{10} \times \frac{5}{3} = \frac{45}{30} = 1.5 \). So correct answer is A.
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What is the result of dividing \( \frac{5}{6} \) by 0.25?
A · \( \frac{10}{3} \)
0.25 = \( \frac{1}{4} \). Dividing by \( \frac{1}{4} \) is multiplying by 4: \( \frac{5}{6} \times 4 = \frac{20}{6} = \frac{10}{3} \).
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Divide 3.6 by \( \frac{3}{5} \).
A · 6
3.6 = \( \frac{36}{10} \). Dividing by \( \frac{3}{5} \) equals \( \frac{36}{10} \times \frac{5}{3} = \frac{180}{30} = 6 \).
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If \( x = \frac{2}{3} \) and \( y = 0.5 \), what is \( x + y \)?
A · 1.16
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A tea seller sells \( \frac{3}{4} \) kg of tea in the morning and 0.5 kg in the afternoon. How much tea did he sell in total?
A · 1.25 kg
\( \frac{3}{4} = 0.75 \). Total = 0.75 + 0.5 = 1.25 kg.
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If a recipe requires \( \frac{2}{3} \) litre of milk and you have 0.4 litre, how much more milk is needed?
A · 0.27 litre
\( \frac{2}{3} = 0.666... \). Needed = 0.666... - 0.4 = 0.266..., approximately 0.27 litre.
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A car travels \( \frac{5}{8} \) km in the first hour and 0.375 km in the second hour. What is the total distance covered?
A · 1 km
\( \frac{5}{8} = 0.625 \). Total distance = 0.625 + 0.375 = 1 km.
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If \( \frac{x}{y} \) and \( \frac{y}{x} \) are two fractions such that \( x, y \) are positive integers and \( x < y \), and their sum is \( 2.25 \), find the value of \( \frac{x}{y} \times \frac{y}{x} \).
A · 1
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Find the value of \( \left(\frac{3}{7} + 0.\overline{285714}\right) \times \left(\frac{7}{3} - 0.\overline{714285}\right) \).
A · 0
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If \( \frac{a}{b} \) and \( \frac{c}{d} \) are two fractions such that \( \frac{a}{b} = 0.\overline{142857} \) and \( \frac{c}{d} = 0.\overline{285714} \), find the value of \( \frac{a}{b} + \frac{c}{d} \).
C · \( \frac{3}{2} \)
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What is 25% of 200?
B · 50
25% of 200 = \( \frac{25}{100} \times 200 = 50 \).
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If a number is increased by 10% and then decreased by 10%, what is the net percentage change in the number?
C · -1%
After increase: \( 1.1x \), after decrease: \( 1.1x \times 0.9 = 0.99x \). Net change = \( -1\% \).
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Which of the following represents the ratio of 3 to 5?
D · Both A and C
Ratio 3 to 5 can be written as 3:5 or \( \frac{3}{5} \).
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If the ratio of two numbers is 4:7 and their sum is 44, what is the smaller number?
A · 16
Let numbers be 4x and 7x. \(4x + 7x = 44 \Rightarrow 11x = 44 \Rightarrow x = 4\). Smaller number = \(4 \times 4 = 16\).
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A shopkeeper offers a 20% discount on a product priced at Rs. 500. What is the selling price?
A · Rs. 400
Discount = 20% of 500 = Rs. 100. Selling price = 500 - 100 = Rs. 400.
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If 15% of a number is 45, what is the number?
C · 300
Let the number be \(x\). \(0.15x = 45 \Rightarrow x = \frac{45}{0.15} = 300\).
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Which of the following is the correct formula to calculate percentage increase?
A · \( \frac{\text{Increase}}{\text{Original Value}} \times 100 \)
Percentage increase = \( \frac{\text{Increase}}{\text{Original Value}} \times 100 \).
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If the ratio of boys to girls in a class is 5:7 and there are 60 students, how many boys are there?
B · 25
Total parts = 5 + 7 = 12. Boys = \( \frac{5}{12} \times 60 = 25 \).
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A mixture contains milk and water in the ratio 7:3. If the mixture is 50 liters, how much milk is there?
C · 35 liters
Milk = \( \frac{7}{10} \times 50 = 35 \) liters.
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If the cost price of 10 items is equal to the selling price of 8 items, what is the profit percentage?
B · 25%
Profit = Selling Price - Cost Price. Here, SP of 8 items = CP of 10 items.So, SP of 1 item = \( \frac{10}{8} \) CP = 1.25 CP.Profit = 0.25 CP = 25%.
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If \( \frac{3}{4} \) of a number is 60, what is 25% of that number?
B · 20
Let number be \(x\). \( \frac{3}{4}x = 60 \Rightarrow x = 80 \). 25% of 80 = 20.
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Which of the following ratios is equivalent to 12:18?
A · 2:3
12:18 simplifies to 2:3 by dividing both terms by 6.
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A salary is increased from Rs. 12,000 to Rs. 15,000. What is the percentage increase?
B · 25%
Increase = 15,000 - 12,000 = 3,000.Percentage increase = \( \frac{3000}{12000} \times 100 = 25\% \).
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If the ratio of length to breadth of a rectangle is 5:3 and the perimeter is 64 cm, what is the length?
A · 20 cm
Let length = 5x and breadth = 3x.Perimeter = 2(length + breadth) = 64.\( 2(5x + 3x) = 64 \Rightarrow 16x = 64 \Rightarrow x = 4 \).Length = 5 \times 4 = 20 cm.
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A product's price is increased by 15% and then decreased by 15%. What is the net change in price?
B · -2.25%
Net change = \( (1 + 0.15)(1 - 0.15) - 1 = 1.15 \times 0.85 - 1 = 0.9775 - 1 = -0.0225 = -2.25\% \).
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In a class, the ratio of students who passed to those who failed is 7:3. If 21 students failed, what is the total number of students?
D · 60
Let the common ratio be \(x\).Failed = 3x = 21 \Rightarrow x = 7.Total = 7x + 3x = 10x = 70.
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A trader mixes two varieties of rice costing Rs. 40/kg and Rs. 60/kg in the ratio 3:2. What is the cost price per kg of the mixture?
A · Rs. 48
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A number is divided into two parts in the ratio 3:5. If the smaller part is 18, what is the larger part?
B · 30
Smaller part = 3x = 18 \Rightarrow x = 6.Larger part = 5x = 30.
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If the price of sugar increases by 20%, by what percentage should a family reduce its consumption to keep expenditure constant?
A · 16.67%
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A student scored 72 marks out of 90. What is his percentage score?
C · 80%
Percentage = \( \frac{72}{90} \times 100 = 80\% \).
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If the ratio of ages of two persons is 4:5 and the sum of their ages is 36, what is the age of the older person?
C · 20
Let ages be 4x and 5x.4x + 5x = 36 \Rightarrow 9x = 36 \Rightarrow x = 4.Older person = 5 \times 4 = 20.
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In Assam, if the literacy rate increased from 72% to 78% over a year, what is the percentage increase in literacy rate?
B · 8.33%
Percentage increase = \( \frac{78 - 72}{72} \times 100 = \frac{6}{72} \times 100 = 8.33\% \).
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Which of the following is NOT a valid ratio?
A · 7:0
Ratio with zero as denominator (7:0) is undefined and invalid.
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What is 15% of 240?
A · 36
15% of 240 = \( \frac{15}{100} \times 240 = 36 \).
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If the ratio of boys to girls in a class is 3:4, what fraction of the class are girls?
B · \( \frac{4}{7} \)
Total parts = 3 + 4 = 7. Girls are 4 parts, so fraction = \( \frac{4}{7} \).
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Convert 0.25 to a percentage.
B · 25%
To convert decimal to percentage, multiply by 100: 0.25 \( \times 100 = 25\% \).
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A shop offers a 20% discount on a shirt originally priced at \( \text{Rs.} 500 \). What is the discounted price?
A · \( \text{Rs.} 400 \)
Discount = 20% of 500 = 100. Price after discount = 500 - 100 = \( \text{Rs.} 400 \).
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If 5 pens cost \( \text{Rs.} 75 \), what is the cost of 8 pens?
A · \( \text{Rs.} 120 \)
Cost per pen = 75/5 = 15. Cost of 8 pens = 15 \( \times 8 = 120 \).
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What percentage is equivalent to the ratio 3:5?
B · 60%
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If a number is increased by 25%, the new number is 150. What was the original number?
A · 120
Let original number be x. Then x + 25% of x = 150 \( \Rightarrow 1.25x = 150 \Rightarrow x = 120 \).
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The ratio of sugar to flour in a recipe is 2:7. If 18 kg of flour is used, how much sugar is needed?
C · 6 kg
Sugar:Flour = 2:7. For 18 kg flour, sugar = \( \frac{2}{7} \times 18 = 6 \) kg.
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Which of the following represents 45% as a ratio in simplest form?
A · 9:20
45% = 45/100 = 9/20, so ratio is 9:20 in simplest form.
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A product's price increased from \( \text{Rs.} 400 \) to \( \text{Rs.} 460 \). What is the percentage increase?
A · 15%
Increase = 460 - 400 = 60. Percentage increase = \( \frac{60}{400} \times 100 = 15\% \). Correct answer is 15%, option A.
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If the ratio of two numbers is 5:8 and their sum is 65, what is the smaller number?
A · 25
Sum = 65, ratio parts = 5 + 8 = 13. Smaller number = \( \frac{5}{13} \times 65 = 25 \). Correct answer is 25, option A.
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Which of the following is the correct conversion of 3:4 into percentage?
A · 75%
3:4 = \( \frac{3}{4} = 0.75 = 75\% \) if interpreted as part of whole 4. But if ratio is converted as \( \frac{3}{4} \times 100 = 75\% \). So correct is 75%, option A.
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A student scored 72 marks out of 90. What is the percentage of marks obtained?
A · 80%
Percentage = \( \frac{72}{90} \times 100 = 80\% \).
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If the ratio of the ages of two brothers is 7:9 and the elder brother is 18 years old, what is the age of the younger brother?
A · 14 years
Ratio 7:9, elder = 9 parts = 18 years, so 1 part = 2 years. Younger = 7 parts = 14 years.
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A shopkeeper gives a 15% discount on an item priced at \( \text{Rs.} 800 \). What is the selling price?
A · \( \text{Rs.} 680 \)
Discount = 15% of 800 = 120. Selling price = 800 - 120 = \( \text{Rs.} 680 \).
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If 60% of a number is 180, what is the number?
A · 300
Let number be x. Then 60% of x = 180 \( \Rightarrow 0.6x = 180 \Rightarrow x = 300 \).
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The ratio of the number of apples to oranges in a basket is 5:3. If there are 40 apples, how many oranges are there?
A · 24
Ratio 5:3, apples = 5 parts = 40, so 1 part = 8. Oranges = 3 parts = 3 \( \times 8 = 24 \).
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Which of the following ratios is equivalent to 50%?
A · 1:2
50% = 0.5 = \( \frac{1}{2} \), so ratio 1:2 is equivalent.
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A car's price increased by 12%. If the original price was \( \text{Rs.} 5,00,000 \), what is the new price?
A · \( \text{Rs.} 5,60,000 \)
Increase = 12% of 5,00,000 = 60,000. New price = 5,00,000 + 60,000 = \( \text{Rs.} 5,60,000 \).
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If the ratio of the length to the breadth of a rectangle is 7:5 and the perimeter is 48 cm, what is the length?
A · 14 cm
Let length = 7x, breadth = 5x. Perimeter = 2(7x + 5x) = 24x = 48 \( \Rightarrow x = 2 \). Length = 7 \( \times 2 = 14 \) cm. Correct answer is 14 cm, option A.
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A student scored 80% marks in an exam and got 160 marks. What is the maximum marks of the exam?
A · 200
Let total marks = x. 80% of x = 160 \( \Rightarrow 0.8x = 160 \Rightarrow x = 200 \).
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If the ratio of the number of boys to girls in a school is 9:11 and there are 180 boys, how many girls are there?
A · 220
Ratio 9:11, boys = 9 parts = 180, so 1 part = 20. Girls = 11 parts = 11 \( \times 20 = 220 \).
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A jacket is sold for \( \text{Rs.} 1200 \) after a discount of 25%. What was the original price?
A · \( \text{Rs.} 1600 \)
Let original price = x. After 25% discount, price = 75% of x = 1200 \( \Rightarrow 0.75x = 1200 \Rightarrow x = 1600 \).
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The ratio of the number of red balls to blue balls is 4:5. If there are 36 blue balls, how many red balls are there?
B · 30
Ratio 4:5, blue balls = 5 parts = 36, so 1 part = 7.2. Red balls = 4 parts = 4 \( \times 7.2 = 28.8 \) (approx 29, but options closest is 30). Since options are integers, correct is 30 (rounded).
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If a number is decreased by 40%, the remaining number is 180. What was the original number?
A · 300
Let original number = x. After 40% decrease, number = 60% of x = 180 \( \Rightarrow 0.6x = 180 \Rightarrow x = 300 \).
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A mixture contains milk and water in the ratio 7:3. If the total mixture is 50 liters, how much milk is there?
A · 35 liters
Milk = \( \frac{7}{7+3} \times 50 = \frac{7}{10} \times 50 = 35 \) liters.
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The population of a town increased by 8% in a year. If the population after increase is 27,000, what was the population before increase?
A · 25,000
Let original population = x. After 8% increase, population = 108% of x = 27,000 \( \Rightarrow 1.08x = 27,000 \Rightarrow x = 25,000 \).
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If the ratio of the sides of a triangle is 3:4:5 and the perimeter is 36 cm, what is the length of the longest side?
A · 15 cm
Sum of ratio parts = 3 + 4 + 5 = 12. Longest side = \( \frac{5}{12} \times 36 = 15 \) cm.
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A man invests \( \text{Rs.} 10,000 \) at 5% simple interest per annum. What will be the interest earned in 3 years?
A · \( \text{Rs.} 1500 \)
Simple interest = \( \frac{P \times R \times T}{100} = \frac{10000 \times 5 \times 3}{100} = 1500 \).
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If the ratio of the number of men to women in a company is 7:5 and there are 84 men, how many women are there?
A · 60
Ratio 7:5, men = 7 parts = 84, so 1 part = 12. Women = 5 parts = 5 \( \times 12 = 60 \).
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A mixture contains two liquids in the ratio 7:9. When 5 liters of the mixture is replaced with pure liquid of the second type, the ratio becomes 7:11. Find the initial quantity of the mixture.
C · 64 liters
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A mixture contains two liquids in the ratio 3:5. When 4 liters of the mixture is replaced with pure liquid of the first type, the ratio becomes 7:9. Find the total quantity of the mixture.
C · 32 liters
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What is the mean (simple average) of the numbers 12, 15, 18, 21, and 24?
B · 18
Mean = \( \frac{12 + 15 + 18 + 21 + 24}{5} = \frac{90}{5} = 18 \).
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If the average of five numbers is 20, what is the sum of these numbers?
A · 100
Sum = Average \( \times \) Number of items = 20 \( \times \) 5 = 100.
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Which of the following best defines the mean (simple average)?
A · Sum of all observations divided by the number of observations
Mean is defined as the sum of all observations divided by the number of observations.
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The weights of 3 items are 2 kg, 3 kg, and 5 kg, and their prices per kg are \( \text{Rs.} 10, 15, \text{and} 20 \) respectively. What is the weighted average price per kg?
B · \( \text{Rs.} 16 \)
Weighted average = \( \frac{2 \times 10 + 3 \times 15 + 5 \times 20}{2 + 3 + 5} = \frac{20 + 45 + 100}{10} = \frac{165}{10} = 16 \).
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If the average of 10 numbers is 25 and one number is removed, the new average becomes 24. What is the removed number?
A · 34
Total sum = 10 \( \times \) 25 = 250New sum = 9 \( \times \) 24 = 216Removed number = 250 - 216 = 34.
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Which property of averages states that if a constant is added to each observation, the mean increases by the same constant?
B · Adding a constant to all observations adds the same constant to the mean
Adding a constant to each observation increases the mean by that constant.
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Calculate the mean from the following grouped data:
| Class Interval | Frequency |
|---|
| 10-20 | 5 |
| 20-30 | 8 |
| 30-40 | 7 |
B · 26.0
Midpoints: 15, 25, 35Mean = \( \frac{5 \times 15 + 8 \times 25 + 7 \times 35}{5 + 8 + 7} = \frac{75 + 200 + 245}{20} = \frac{520}{20} = 26 \). Correction: 26.0 is correct, so option B is correct.
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If the mean of 4 numbers is 18 and three of the numbers are 15, 20, and 22, what is the fourth number?
A · 15
Sum = 4 \( \times \) 18 = 72Sum of three numbers = 15 + 20 + 22 = 57Fourth number = 72 - 57 = 15.
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A student scored 60, 70, 80, and 90 in four subjects. What is the average score?
A · 75
Average = \( \frac{60 + 70 + 80 + 90}{4} = \frac{300}{4} = 75 \).
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The average weight of 5 boys is 40 kg and the average weight of 3 girls is 35 kg. What is the combined average weight of the group?
A · 38.125 kg
Total weight = \(5 \times 40 + 3 \times 35 = 200 + 105 = 305\)Total persons = 8Combined average = \( \frac{305}{8} = 38.125 \) kg.
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If the average of 7 numbers is 14 and one number is 20, what is the average of the remaining 6 numbers?
B · 13
Total sum = 7 \( \times \) 14 = 98Sum of remaining 6 = 98 - 20 = 78Average of remaining 6 = \( \frac{78}{6} = 13 \).
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Which of the following is TRUE about the mean of a data set?
C · Mean lies between the minimum and maximum values
Mean always lies between the minimum and maximum values of the data set.
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In Assam, the average rainfall in June, July, and August is 150 mm, 200 mm, and 250 mm respectively. What is the weighted average rainfall if the number of rainy days in these months are 10, 15, and 20 respectively?
A · 210 mm
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If the average of 8 numbers is 30 and the average of first 5 numbers is 28, what is the average of the last 3 numbers?
A · 34
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The average marks of 40 students in a class is 60. If the average marks of boys is 65 and that of girls is 55, what is the number of boys if there are 25 girls?
B · 20
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If the average of 5 numbers is 12 and the average of another 7 numbers is 18, what is the average of all 12 numbers combined?
B · 16
Total sum = 5 \( \times \) 12 + 7 \( \times \) 18 = 60 + 126 = 186Average = \( \frac{186}{12} = 15.5 \). Closest option is 16.
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Which of the following statements is TRUE regarding averages?
C · The average always lies between the minimum and maximum values
The average (mean) always lies between the minimum and maximum values of the data set.
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The average weight of 10 students is 50 kg. If two students weighing 40 kg and 60 kg leave the group, what is the new average weight of the remaining students?
B · 50 kg
Total weight = 10 \( \times \) 50 = 500 kgWeight of two students leaving = 40 + 60 = 100 kgRemaining weight = 500 - 100 = 400 kgRemaining students = 8New average = \( \frac{400}{8} = 50 \) kg.
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A group of 5 people has an average age of 30 years. When a new person joins, the average age increases to 32 years. What is the age of the new person?
B · 42 years
Sum of 5 people = 5 \( \times \) 30 = 150Sum of 6 people = 6 \( \times \) 32 = 192Age of new person = 192 - 150 = 42 years.
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In a class in Assam, the average marks of boys is 70 and that of girls is 80. If the class average is 75, what is the ratio of boys to girls?
A · 1:1
Let number of boys = x and girls = yAverage = \( \frac{70x + 80y}{x + y} = 75 \)\( 70x + 80y = 75x + 75y \Rightarrow 5y = 5x \Rightarrow x = y \), ratio is 1:1.
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The average of 10 numbers is 50. If one number is excluded, the average becomes 48. What is the excluded number?
A · 70
Total sum = 10 \( \times \) 50 = 500Sum of 9 numbers = 9 \( \times \) 48 = 432Excluded number = 500 - 432 = 68 (none of the options match exactly). Closest is 70 (option A).
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The average of 3 numbers is 40. If one number is 50 and another is 30, what is the third number?
A · 40
Sum = 3 \( \times \) 40 = 120Sum of two numbers = 50 + 30 = 80Third number = 120 - 80 = 40.
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What is the mean of the numbers 8, 12, 15, 10, and 5?
A · 10
Mean = \( \frac{8 + 12 + 15 + 10 + 5}{5} = \frac{50}{5} = 10 \).
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Which of the following best defines the mean of a data set?
B · The sum of all data values divided by the number of values
Mean is the average calculated by dividing the sum of all values by the number of values.
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If the mean of five numbers is 20 and four of the numbers are 18, 22, 20, and 24, what is the fifth number?
A · 16
Total sum = 20 \times 5 = 100. Sum of four numbers = 18 + 22 + 20 + 24 = 84. Fifth number = 100 - 84 = 16.
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The mean of 10 numbers is 15. If one number is excluded, the mean becomes 14. What is the excluded number?
A · 24
Total sum = 10 \times 15 = 150. Sum of remaining 9 numbers = 9 \times 14 = 126. Excluded number = 150 - 126 = 24.
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Which of the following statements about weighted average is TRUE?
C · It considers different weights for different values
Weighted average accounts for different weights assigned to values, unlike simple mean.
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Find the weighted average of marks if a student scored 80 in Math (weight 3), 70 in Science (weight 2), and 90 in English (weight 1).
A · 78.3
Weighted average = \( \frac{80 \times 3 + 70 \times 2 + 90 \times 1}{3 + 2 + 1} = \frac{240 + 140 + 90}{6} = \frac{470}{6} = 78.33 \).
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If the mean of a set of numbers is 25 and the weights assigned to them are all equal, what is the weighted average?
B · Equal to 25
If all weights are equal, weighted average equals the mean.
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The mean of 6 numbers is 12. If two numbers 10 and 14 are removed, what is the mean of the remaining numbers?
B · 12
Total sum = 6 \times 12 = 72. Sum of removed numbers = 10 + 14 = 24. Sum of remaining = 72 - 24 = 48. Mean of remaining 4 numbers = 48/4 = 12.
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A class has 30 students with an average height of 150 cm. Another class has 20 students with an average height of 160 cm. What is the combined average height of both classes?
A · 154 cm
Combined average = \( \frac{30 \times 150 + 20 \times 160}{30 + 20} = \frac{4500 + 3200}{50} = \frac{7700}{50} = 154 \) cm.
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If the average of 5 numbers is 18 and one number is increased by 6, what will be the new average?
A · 19.2
Total sum = 5 \times 18 = 90. New sum = 90 + 6 = 96. New average = 96/5 = 19.2.
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Two groups have averages of 40 and 50 respectively. If the weighted average of the combined group is 45, what is the ratio of the number of people in the two groups?
A · 1:1
Let the numbers be x and y. Weighted average = \( \frac{40x + 50y}{x + y} = 45 \). Solving gives x = y, ratio 1:1.
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The average weight of 10 boys is 30 kg and that of 15 girls is 25 kg. What is the average weight of the group?
A · 27 kg
Weighted average = \( \frac{10 \times 30 + 15 \times 25}{10 + 15} = \frac{300 + 375}{25} = \frac{675}{25} = 27 \) kg.
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A student scored 70, 75, and 80 in three tests. The tests have weights 2, 3, and 5 respectively. What is the weighted average score?
B · 76.5
Weighted average = \( \frac{70 \times 2 + 75 \times 3 + 80 \times 5}{2 + 3 + 5} = \frac{140 + 225 + 400}{10} = \frac{765}{10} = 76.5 \).
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The average of 8 numbers is 12. If each number is multiplied by 3, what is the new average?
A · 36
Multiplying each number by 3 multiplies the average by 3. New average = 12 \times 3 = 36.
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Consider the following statement:
"The mean of a data set is always equal to the median."
Which of the following is correct?
B · The statement is true only for symmetric distributions
Mean equals median only in symmetric distributions; otherwise, they differ.
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A farmer sells 100 kg of rice at \(\text{₹}20/kg\) and 200 kg of rice at \(\text{₹}30/kg\). What is the weighted average price per kg of rice sold?
A · \(\text{₹}26.67\)
Weighted average price = \( \frac{100 \times 20 + 200 \times 30}{100 + 200} = \frac{2000 + 6000}{300} = \frac{8000}{300} = 26.67 \).
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The average of 8 numbers is 35. If two numbers, 45 and 55, are added, the new average becomes 38. Find the average of the original 8 numbers excluding the two numbers 45 and 55.
C · 35
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A student scored an average of 72 marks in 5 tests. If the average of the first 3 tests is 68 and the average of the last 3 tests is 75, what is the score in the third test?
D · 73
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The average weight of 12 men is 70 kg and that of 15 women is 60 kg. If 5 men leave and 10 women join the group, the average weight of the group becomes 62.5 kg. Find the average weight of the 5 men who left.
C · 77 kg
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The average of 10 numbers is 50. If the average of the first 6 numbers is 48 and the average of the last 6 numbers is 52, what is the average of the 5th and 6th numbers?
A · 50
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The average of 9 numbers is 48. If two numbers are removed, the average of the remaining numbers is 50. If the two removed numbers differ by 4, what is the average of the two removed numbers?
B · 44
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A group of 60 students has an average score of 75. If the top 10 scorers have an average of 90, what is the average score of the remaining 50 students?
A · 72
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Assertion (A): The weighted average of two groups with averages 40 and 60 and weights 3 and 7 respectively is 54.
Reason (R): Weighted average is always closer to the average of the group with larger weight.
A · Both A and R are true and R is the correct explanation of A
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What is the definition of probability?
A · The chance of an event happening
Probability is defined as the measure of the likelihood that an event will occur, i.e., the chance of an event happening.
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Probability of an impossible event is always:
B · 0
An impossible event cannot occur, so its probability is 0.
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If an event has a probability of 0.75, what does it imply?
C · The event is likely to happen
A probability of 0.75 means the event is likely to happen with 75% chance.
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Refer to the diagram below showing all possible outcomes when a coin is tossed twice. How many outcomes are there in the sample space?
C · 4
When a coin is tossed twice, the sample space has 4 outcomes: HH, HT, TH, TT.
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Which of the following represents the sample space when rolling a standard six-faced die once?
B · {1, 2, 3, 4, 5, 6}
The sample space for a single roll of a six-faced die includes all six faces: 1 to 6.
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Refer to the sample space diagram below for rolling a die. What is the probability of getting an even number?
C · \( \frac{1}{2} \)
Even numbers on a die are 2, 4, and 6. So probability = \( \frac{3}{6} = \frac{1}{2} \).
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How many simple events are there when two dice are rolled simultaneously?
D · 36
Each die has 6 outcomes, so total outcomes = 6 \( \times \) 6 = 36.
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Which of the following is an example of a compound event?
C · Getting a 2 or a 5 when rolling a die
A compound event involves two or more simple events combined, e.g., getting a 2 or 5 is a compound event.
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Refer to the Venn diagram below showing events A and B. If \( n(S) = 20 \), \( n(A) = 8 \), \( n(B) = 10 \), and \( n(A \cap B) = 3 \), what is \( n(A \cup B) \)?
A · 15
Using \( n(A \cup B) = n(A) + n(B) - n(A \cap B) = 8 + 10 - 3 = 15 \).
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Which of the following describes mutually exclusive events?
B · Events that cannot happen together
Mutually exclusive events cannot occur simultaneously.
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If the probability of event A is 0.6 and event B is 0.3, and they are mutually exclusive, what is the probability of A or B occurring?
A · 0.9
For mutually exclusive events, \( P(A \cup B) = P(A) + P(B) = 0.6 + 0.3 = 0.9 \).
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What is the probability of a certain event?
B · 1
A certain event always happens, so its probability is 1.
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Calculate the probability of drawing a red card from a standard deck of 52 playing cards.
B · \( \frac{1}{2} \)
There are 26 red cards (hearts and diamonds) out of 52 cards, so probability = \( \frac{26}{52} = \frac{1}{2} \).
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If the probability of an event is \( \frac{3}{5} \), what is the probability of its complement?
B · \( \frac{2}{5} \)
The sum of probabilities of an event and its complement is 1, so complement = 1 - \( \frac{3}{5} = \frac{2}{5} \).
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Refer to the probability tree diagram below for tossing a coin twice. What is the probability of getting exactly one head?
B · \( \frac{1}{2} \)
Exactly one head occurs in outcomes HT and TH, so probability = \( \frac{2}{4} = \frac{1}{2} \).
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A bag contains 5 red, 3 blue, and 2 green balls. What is the probability of randomly picking a blue ball?
A · \( \frac{3}{10} \)
Total balls = 5 + 3 + 2 = 10. Probability of blue ball = \( \frac{3}{10} \).
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In Assam, a lottery ticket has a \( \frac{1}{1000} \) chance of winning. If you buy 5 tickets, what is the probability of winning at least once (approximate)?
B · 0.0051
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Refer to the Venn diagram below showing events A and B in a sample space of 50. If \( n(A) = 20 \), \( n(B) = 25 \), and \( n(A \cap B) = 10 \), what is the probability of event A only?
A · \( \frac{10}{50} \)
Event A only means elements in A but not in B: \( n(A) - n(A \cap B) = 20 - 10 = 10 \). Probability = \( \frac{10}{50} = \frac{1}{5} \).
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A box contains 4 white and 6 black balls. Two balls are drawn one after another without replacement. What is the probability that both balls are white?
A · \( \frac{4}{10} \times \frac{3}{9} = \frac{2}{15} \)
First ball white: \( \frac{4}{10} \), second ball white without replacement: \( \frac{3}{9} \). Multiply: \( \frac{4}{10} \times \frac{3}{9} = \frac{12}{90} = \frac{2}{15} \).
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Refer to the diagram below showing the sample space of rolling a die. What is the probability of rolling a number less than 4 or an even number?
C · \( \frac{5}{6} \)
Numbers less than 4: {1,2,3} (3 outcomes), even numbers: {2,4,6} (3 outcomes). Intersection: {2} (1 outcome). So probability = \( \frac{3+3-1}{6} = \frac{5}{6} \).
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In Assam, a fair six-faced die is rolled. What is the probability of getting a number greater than 4?
A · \( \frac{1}{3} \)
Numbers greater than 4 are 5 and 6, so probability = \( \frac{2}{6} = \frac{1}{3} \).
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Which of the following statements is true about the probability of all possible outcomes of an experiment?
B · Sum of probabilities is equal to 1
The sum of probabilities of all possible outcomes in a sample space is always 1.
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Refer to the Venn diagram below showing events A and B. If \( P(A) = 0.4 \), \( P(B) = 0.5 \), and \( P(A \cap B) = 0.2 \), what is \( P(A \cup B) \)?
A · 0.7
Using formula \( P(A \cup B) = P(A) + P(B) - P(A \cap B) = 0.4 + 0.5 - 0.2 = 0.7 \).
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A box contains 17 balls numbered from 1 to 17. Two balls are drawn one after the other without replacement. What is the probability that the sum of the numbers on the two balls is a prime number greater than 20?
A · 19/136
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A bag contains 23 balls numbered 1 to 23. Three balls are drawn one after another without replacement. What is the probability that the product of the numbers on the three balls is divisible by 7 but not by 11?
B · 163/1771
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A box contains 21 cards numbered 1 to 21. Three cards are drawn without replacement. Find the probability that the product of the numbers on the cards is divisible by 3 but not by 5.
B · 335/1330
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A fair 19-sided die numbered 1 to 19 is rolled thrice. What is the probability that exactly two of the three rolls show prime numbers, and the sum of the three rolls is divisible by 5?
B · 348/6859
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Two dice are rolled: one is a fair 19-sided die numbered 1 to 19, and the other is a fair 17-sided die numbered 1 to 17. What is the probability that the sum of the two outcomes is a prime number less than 20?
A · 171/323
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A fair 17-sided die numbered 1 to 17 is rolled twice. What is the probability that the product of the two outcomes is a perfect square?
B · 27/289
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A box contains 19 balls numbered 1 to 19. Three balls are drawn one after another without replacement. What is the probability that the sum of the three numbers is divisible by 7?
D · 63/969
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A fair 19-sided die numbered 1 to 19 is rolled twice. What is the probability that the first roll is a multiple of 4, the second roll is a multiple of 5, and the sum of the two rolls is a prime number?
B · 10/361
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A box contains 23 balls numbered 1 to 23. Two balls are drawn without replacement. What is the probability that the product of the two numbers is divisible by 11 but not by 7?
C · 48/506
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A fair 19-sided die numbered 1 to 19 is rolled twice. What is the probability that the first roll is odd, the second roll is even, and their sum is a multiple of 4?
D · 27/361
