Quick recall · 293 cards
Short MCQ-style retrieval prompts. Tap a card to reveal the answer.
PYQ
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If one-third of one-fourth of a number is 15, then three-tenth of that number is:
B · 36
PYQ
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Sum of digits of a two-digit number equals 9. Furthermore, the difference between these digits is 3. What is the number?
B · 63
PYQ
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If N = \( (11^p + 7)(7^q - 2)(5^r + 1)(3^s) \) is a perfect cube, where p, q, r, s are positive integers, then the smallest value of p + q + r + s is:
C · 14
PYQ
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If n = 3 × 4 × p where p is a prime number greater than 3, how many different positive non-prime divisors does n have, excluding 1 and n?
B · 3
PYQ · 2018
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HCF of 2472, 1284 and a third number ‘n’ is 12. If their LCM is 8*9*5*10^3*107, then the number ‘n’ is:
A · 2^2*3^2*5^1
PYQ · 2021
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The LCM and HCF of the three numbers 48, 144 and ‘p’ are 720 and 24 respectively. Find the least value of ‘p’.
B · 120
PYQ · 2021
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Two numbers having their LCM 480 are in the ratio 3:4. What will be the smaller number of this pair?
B · 120
PYQ · 2019
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If HCF of 189 and 297 is 27, find their LCM.
A · 2079
PYQ
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A batsman scored 110 runs which included 3 boundaries and 8 sixes. What percent of his total score did he make by running between the wickets?
B · B) \( 45\frac{5}{11}\% \)
PYQ · 2023
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A shopkeeper bought an oven for ₹225,000 and sold it for ₹229,500. He spent ₹21,500 as overheads. What is his loss or gain percentage (rounded off to the nearest integer)?
B · 7.75% profit
PYQ
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The cost price of 12 articles is the same as the selling price of 8 articles. Find the profit or loss percentage.
C · 50% profit
PYQ
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Alfred buys an old scooter for ₹4,700 and spends ₹800 on its repairs. If he sells the scooter for ₹5,800, his gain percent is:
A · 4.54%
PYQ
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₹2,500, when invested for 8 years at a given rate of simple interest per year, amounted to ₹3,725 on maturity. What was the rate of simple interest that was paid per annum?
B · 5%
PYQ
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A sum of money amounts to Rs. 28,000 in 2 years at 20% simple interest per annum. Find the sum.
A · Rs. 20,000
PYQ
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A sum of money at simple interest amounts to Rs. 815 in 3 years and to Rs. 854 in 4 years. The sum is:
C · Rs. 698
PYQ · 2023
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Simple interest on a certain sum is one-fourth of the sum and the interest rate per annum is four times the number of years. What is the rate of interest per annum?
A · 19%
PYQ · 2019
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A sum amounts to Rs. 8,028 in 3 years and to Rs. 12,042 in 6 years at a certain rate percent per annum, when the interest is compounded yearly. Find the difference in interest between the third year and second year.
A · (a) Rs. 544
PYQ
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The compound interest on a certain sum compounded for 3 years at 15% p.a. interest compounded yearly is Rs. 4,167. What is the simple interest on the same sum in 3 years at the same rate?
A · (a) Rs. 3600
PYQ
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Is the ratio 5:10 proportional to 1:2?
A · Yes
PYQ
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The average of 11 numbers is 30. If the average of the first six numbers is 17.5 and that of the last six numbers is 42.5, then what is the sixth number?
C · 45
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The average age of 25 people in a room is 30 years. A 50-year-old person enters the room. What is the new average?
C · 30.8
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The average of 5 consecutive odd numbers is 61. What is the smallest number?
A · 57
PYQ
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The average age of a group of 8 men is increased by 2 years when one of them whose age is 24 years is replaced by a new person. What is the age of the new person?
B · 40
PYQ
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The average marks of 100 students in a subject is 40. The average of the top 50 is 60. What is the average of the remaining 50 students?
A · 20
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Which of the following is a rational number?
C · 0.75
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Which of the following numbers is an irrational number?
C · \( \sqrt{5} \)
\( \sqrt{5} \) is an irrational number because it cannot be expressed as a ratio of two integers. The others are rational numbers.
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Which of the following numbers is divisible by 9?
B · 987654
A number is divisible by 9 if the sum of its digits is divisible by 9. Sum of digits of 987654 = 9+8+7+6+5+4 = 39, and 39 is divisible by 9.
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Which of the following numbers is divisible by both 4 and 6 but not by 8?
D · 60
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Find the smallest three-digit number divisible by 7, 8, and 9.
A · 504
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Which of the following is a prime number?
B · 53
53 is a prime number; it has no divisors other than 1 and itself. 51, 57, and 63 are composite numbers.
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Which of the following numbers has exactly three distinct prime factors?
A · 30
30 = 2 × 3 × 5 (three distinct prime factors). 28 = 2 × 2 × 7 (two distinct prime factors), 45 = 3 × 3 × 5 (two distinct prime factors), 49 = 7 × 7 (one prime factor).
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If \( p \) and \( q \) are primes such that \( p + q = 52 \), which of the following pairs \((p, q)\) is correct?
B · (5, 47)
5 and 47 are both primes and their sum is 52. Other pairs include non-prime numbers.
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Find the HCF of 84 and 126.
B · 42
Prime factors of 84: 2 × 2 × 3 × 7; of 126: 2 × 3 × 3 × 7. Common factors: 2 × 3 × 7 = 42.
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The LCM of two numbers is 180 and their HCF is 6. If one number is 30, what is the other number?
A · 36
Product of numbers = HCF × LCM = 6 × 180 = 1080. Other number = 1080 ÷ 30 = 36. But 36 is option A. Check carefully: 6 × 180 = 1080; 1080 ÷ 30 = 36. So correct answer is 36.
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Find the LCM of 12, 15, and 20.
A · 60
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What is the place value of 7 in the number 5,372,849?
B · 7,000
In 5,372,849, 7 is in the thousands place, so its place value is 7,000.
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What is the digit in the ten-thousands place in the number 8,654,321?
A · 5
The ten-thousands place is the fifth digit from the right. The digits are 8(7th),6(6th),5(5th),4(4th),3(3rd),2(2nd),1(1st). So digit is 5.
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How many different four-digit numbers can be formed using digits 1, 2, 3, 4 if repetition is not allowed?
A · 24
Number of 4-digit numbers without repetition = 4! = 24.
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How many even three-digit numbers can be formed using digits 1, 2, 3, 4, 5 if repetition is allowed?
B · 50
Last digit must be even (2 or 4): 2 choices.First digit: 5 choices (1-5).Middle digit: 5 choices.Total = 5 × 5 × 2 = 50. So correct answer is B.
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Which of the following numbers is a perfect square?
D · All of the above
289 = 17^2, 361 = 19^2, 441 = 21^2. All are perfect squares.
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Which of the following is a perfect cube?
A · 64
64 = 4^3 is a perfect cube; others are not.
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If \( N = (11^p + 7)(7^q - 2)(5^r + 1)(3^s) \) is a perfect cube, where \( p, q, r, s \) are positive integers, what is the smallest value of \( p + q + r + s \)?
C · 8
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What is the remainder when 2^{10} is divided by 7?
A · 2
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Find the remainder when 7^{100} is divided by 13.
C · 9
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If \( x \equiv 3 \mod 5 \) and \( x \equiv 4 \mod 7 \), what is the smallest positive value of \( x \)?
A · 18
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Which of the following numbers is a rational number but not an integer?
A · \(\frac{3}{4}\)
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Which of the following is an irrational number?
A · \(\sqrt{2}\)
\(\sqrt{2}\) is an irrational number because it cannot be expressed as a ratio of two integers. 0.75 and \(\frac{22}{7}\) are rational, and 5 is an integer.
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If a number is both a perfect square and a perfect cube, which of the following must it be?
A · A perfect sixth power
A number that is both a perfect square and perfect cube must be a perfect sixth power because it must be raised to a power divisible by both 2 and 3, which is 6.
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Which of the following numbers is divisible by 11?
A · 2728
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Which of the following numbers is divisible by 6 but not by 9?
A · 234
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Find the smallest prime number greater than 50.
A · 53
53 is the smallest prime number greater than 50. 55 and 57 are composite, 59 is prime but greater than 53.
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Which of the following numbers is a composite number?
C · 35
35 is composite because it has factors other than 1 and itself (5 and 7). 29, 31, and 37 are prime numbers.
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If \( p \) and \( q \) are prime numbers such that \( p + q = 52 \), and one of them is 19, what is the other prime number?
C · 31
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What is the greatest common factor (GCF) of 84 and 126?
B · 42
Prime factors of 84: 2, 2, 3, 7; of 126: 2, 3, 3, 7. Common factors: 2, 3, 7. Multiply: 2 \times 3 \times 7 = 42.
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Which of the following is a multiple of both 12 and 15 but less than 100?
A · 60
LCM of 12 and 15 is 60. 60 and 90 are multiples of both, but 60 is the smallest multiple less than 100.
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The LCM of two numbers is 180 and their HCF is 6. If one number is 30, what is the other number?
A · 36
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If the HCF of two numbers is 12 and their LCM is 180, and one number is 60, find the other number.
A · 36
Product of numbers = HCF \( \times \) LCM = 12 \( \times \) 180 = 2160. Given one number = 60, other number = \( \frac{2160}{60} = 36 \).
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Convert the binary number \(1101_2\) to its decimal equivalent.
C · 13
Binary \(1101_2 = 1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 8 + 4 + 0 + 1 = 13\).
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What is the base-5 equivalent of the decimal number 83?
A · 313_5
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If the sum of digits of a three-digit number is 15 and the number is divisible by 9, which of the following could be the number?
D · 594
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A two-digit number is such that the difference between the digits is 2. If the number formed by reversing the digits is 27 less than the original number, find the number.
A · 84
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If \( N = (2^a)(3^b)(5^c) \) is a perfect square and \( a + b + c = 12 \), which of the following could be the value of \( (a, b, c) \)?
A · (4, 4, 4)
For \( N \) to be a perfect square, all exponents must be even. Only (4,4,4) has all even exponents and sum 12.
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If \( x \) and \( y \) are positive integers such that \( x^3 y^2 \) is a perfect sixth power, which of the following must be true?
B · \( x \) is a perfect cube and \( y \) is a perfect square
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Let \(N\) be the smallest positive integer such that:
1) \(N\) is divisible by 84,
2) The sum of digits of \(N\) is divisible by 9,
3) \(N\) leaves a remainder of 7 when divided by 11.
Find the value of \(N\).
B · 1008
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Consider a positive integer \(M\) such that:
- \(M\) is a perfect square,
- \(M\) is divisible by 45,
- The number obtained by reversing the digits of \(M\) is divisible by 16.
Find the smallest such \(M\).
A · 2025
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Find the number of positive integers \(x < 10^5\) such that:
- \(x\) is divisible by 12,
- The greatest common divisor \(\gcd(x, 180) = 6\),
- The sum of the digits of \(x\) is a multiple of 4.
A · 416
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Let \(p\) and \(q\) be two positive integers such that:
- \(p\) divides \(q^3\),
- \(q\) divides \(p^4\),
- \(\gcd(p, q) = 1\).
Find the smallest possible value of \(p + q\).
B · 7
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Find the number of positive integers less than 50000 such that:
- The integer is divisible by 18,
- The integer is not divisible by 24,
- The integer's digit sum is divisible by 6.
A · 1388
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If \(a\) and \(b\) are positive integers such that:
- \(a + b = 1000\),
- \(\mathrm{lcm}(a, b) = 25200\),
- \(\gcd(a, b) = d\),
Find the value of \(d\).
B · 24
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Find the smallest positive integer \(x\) such that:
- \(x\) is divisible by 14,
- \(x + 1\) is divisible by 15,
- \(x + 2\) is divisible by 16.
B · 419
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If \(x\) is a positive integer such that:
- \(x\) is divisible by 7,
- \(x + 1\) is divisible by 8,
- \(x + 2\) is divisible by 9,
Find the remainder when \(x\) is divided by 504.
A · 209
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Which of the following best defines 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 both numbers without leaving a remainder.
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The Least Common Multiple (LCM) of two numbers is:
A · The smallest number divisible by both numbers
LCM is the smallest number that is exactly divisible by both numbers.
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Which of the following is NOT a property of HCF and LCM of two numbers \(a\) and \(b\)?
C · LCM of two numbers is always less than or equal to each number
LCM is always greater than or equal to each of the two numbers, not less.
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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
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Find the HCF of 84 and 126 using prime factorization.
C · 42
Prime factors of 84 = \(2^2 \times 3 \times 7\), of 126 = \(2 \times 3^2 \times 7\). Common factors are \(2 \times 3 \times 7 = 42\).
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Using the Euclidean algorithm, what is the HCF of 252 and 105?
A · 21
252 ÷ 105 = 2 remainder 42105 ÷ 42 = 2 remainder 2142 ÷ 21 = 2 remainder 0So, HCF is 21.
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Find the HCF of 462 and 1071 using the Euclidean algorithm.
A · 21
1071 ÷ 462 = 2 remainder 147462 ÷ 147 = 3 remainder 21147 ÷ 21 = 7 remainder 0So, HCF is 21.
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Find the HCF of 2310 and 4620 using prime factorization.
A · 210
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Find the LCM of 12 and 18 using prime factorization.
A · 36
Prime factors of 12 = \(2^2 \times 3\), of 18 = \(2 \times 3^2\). LCM takes highest powers: \(2^2 \times 3^2 = 36\).
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Using the HCF, find the LCM of 24 and 36 if their HCF is 12.
B · 72
Using \( \text{LCM} = \frac{a \times b}{\text{HCF}} = \frac{24 \times 36}{12} = 72 \).
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Find the LCM of 15 and 20 using prime factorization.
A · 60
Prime factors of 15 = \(3 \times 5\), of 20 = \(2^2 \times 5\). LCM = \(2^2 \times 3 \times 5 = 60\).
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If the HCF of two numbers is 7 and their product is 1470, what is their LCM?
A · 210
Using \( \text{HCF} \times \text{LCM} = a \times b \), so \(7 \times \text{LCM} = 1470 \Rightarrow \text{LCM} = 210\).
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If the HCF of two numbers is 4 and their LCM is 96, and one number is 12, what is the other number?
A · 32
Using \( \text{HCF} \times \text{LCM} = a \times b \), we get \(4 \times 96 = 12 \times b \Rightarrow b = 32\). But 32 and 12 have HCF 4, so option A is correct.
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If two numbers are 18 and 24, what is the product of their HCF and LCM?
A · 432
HCF of 18 and 24 is 6, LCM is 72, product = 6 \times 72 = 432.
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If the HCF of two numbers is 5 and their LCM is 180, and one number is 20, what is the other number?
A · 45
Using \( \text{HCF} \times \text{LCM} = a \times b \), \(5 \times 180 = 20 \times b \Rightarrow b = 45\).
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If two numbers are such that their HCF is 12 and their LCM is 180, which of the following pairs can be the numbers?
B · (36, 60)
Check product: 36 \times 60 = 2160, HCF \times LCM = 12 \times 180 = 2160, so pair (36, 60) satisfies the condition.
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If the product of two numbers is 2028 and their HCF is 13, what is their LCM?
A · 156
Using \( \text{LCM} = \frac{a \times b}{\text{HCF}} = \frac{2028}{13} = 156\).
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Three ropes of lengths 84 m, 126 m, and 210 m are to be cut into equal lengths without any leftover. What is the greatest possible length of each piece?
C · 42 m
The greatest length is the HCF of 84, 126, and 210.HCF(84,126) = 42HCF(42,210) = 42So, the greatest length is 42 m.
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A gardener wants to plant trees in rows such that the number of trees in each row divides both 48 and 72 exactly. What is the maximum number of trees in each row?
C · 24
The maximum number dividing both 48 and 72 is their HCF.HCF(48,72) = 24.
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Two numbers have an LCM of 180 and HCF of 6. If one number is 30, what is the other number?
A · 36
Using \( \text{LCM} \times \text{HCF} = a \times b \),\(180 \times 6 = 30 \times b \Rightarrow b = 36\). But 36 and 30 have HCF 6, so 36 is correct.
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A factory produces two types of products every 12 and 18 hours respectively. After how many hours will both products be produced simultaneously?
A · 36 hours
They will be produced simultaneously after the LCM of 12 and 18 hours.LCM(12,18) = 36 hours.
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Three bells ring at intervals of 12, 15, and 20 minutes respectively. If they ring together at 8:00 AM, when will they ring together again?
A · 9:00 AM
Find LCM of 12, 15, and 20.LCM = 60 minutes.So, they ring together again at 9:00 AM.
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Two gears have 40 and 60 teeth respectively. They start rotating together. After how many rotations of the first gear will they align again?
B · 3
They align after LCM of 40 and 60 teeth rotations.LCM(40,60) = 120 teeth.Number of rotations of first gear = \(\frac{120}{40} = 3\).
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If \(N = 2^{3} \times 3^{2} \times 5^{4}\), what is the HCF of \(N\) and \(2^{5} \times 3^{3} \times 5^{2}\)?
A · \(2^{3} \times 3^{2} \times 5^{2}\)
HCF takes minimum powers of common prime factors:\(2^{\min(3,5)} = 2^{3}\), \(3^{\min(2,3)} = 3^{2}\), \(5^{\min(4,2)} = 5^{2}\).
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If \(a = 2^{4} \times 3^{3} \times 5^{2}\) and \(b = 2^{2} \times 3^{5} \times 5^{3}\), what is the LCM of \(a\) and \(b\)?
A · \(2^{4} \times 3^{5} \times 5^{3}\)
LCM takes maximum powers:\(2^{4}\), \(3^{5}\), \(5^{3}\).
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If \(x = 2^{p} \times 3^{q}\) and \(y = 2^{r} \times 3^{s}\) where \(p, q, r, s\) are positive integers, and \(\text{HCF}(x,y) = 2^{2} \times 3^{3}\), \(\text{LCM}(x,y) = 2^{5} \times 3^{6}\), find \(p + q + r + s\).
C · 20
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If \(N = (2^{a} \times 3^{b})^{3} \times (2^{c} \times 3^{d})^{2}\) is a perfect sixth power, what is the smallest value of \(a + b + c + d\) given all are positive integers?
B · 9
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If \(x = 2^{3} \times 3^{4} \times 5^{2}\) and \(y = 2^{5} \times 3^{2} \times 5^{3}\), what is the product of their HCF and LCM?
A · \(x \times y\)
The product of HCF and LCM of two numbers equals the product of the numbers themselves.
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If the HCF of two numbers is 8 and their LCM is 96, which of the following could be the numbers?
B · (16, 48)
Check product: 16 \times 48 = 768, HCF \times LCM = 8 \times 96 = 768, so (16, 48) is correct.
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Which of the following is always true about the Highest Common Factor (HCF) of two positive integers?
B · It divides both numbers exactly
By definition, the HCF of two numbers is the greatest number that divides both numbers exactly without leaving a remainder.
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If the HCF of 24 and 36 is 12, which of the following is a property of their HCF?
A · HCF is always less than or equal to the smaller number
The HCF of two numbers cannot be greater than the smaller number, since it divides both numbers exactly.
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Which of the following statements about the HCF of two numbers is FALSE?
C · HCF is always equal to the product of the two numbers
The HCF is the greatest number dividing both numbers, but it is not equal to their product; the product is typically much larger.
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If the HCF of two numbers is 1, which of the following must be true?
B · The numbers are co-prime
Two numbers whose HCF is 1 are called co-prime or relatively prime, meaning they have no common factors other than 1.
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Which of the following is NOT a property of the Highest Common Factor (HCF)?
C · HCF of two numbers is always greater than their LCM
HCF is always less than or equal to the smaller number, and LCM is always greater than or equal to the larger number, so HCF cannot be greater than LCM.
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Which of the following numbers is the Least Common Multiple (LCM) of 4 and 6?
A · 12
LCM of 4 and 6 is the smallest number divisible by both 4 and 6, which is 12.
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Which of the following is TRUE about the Least Common Multiple (LCM) of two positive integers?
B · LCM is the smallest positive number divisible by both numbers
By definition, the LCM of two numbers is the smallest positive integer that is divisible by both numbers.
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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?
A · 15
LCM(12,15) = 60. 15 is the correct choice as LCM(12,15) = 60.
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Which of the following is NOT a property of the Least Common Multiple (LCM)?
D · LCM is always less than or equal to the smaller number
LCM is always greater than or equal to the larger number, so it cannot be less than or equal to the smaller number.
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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
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For two positive integers \( a \) and \( b \), which of the following equations correctly relates their HCF and LCM?
B · \( \text{HCF}(a,b) \times \text{LCM}(a,b) = a \times b \)
The product of the HCF and LCM of two numbers equals the product of the numbers themselves.
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If two numbers are 18 and 24, what is the product of their HCF and LCM?
A · 432
HCF(18,24) = 6, LCM(18,24) = 72, product = 6 × 72 = 432, which equals 18 × 24.
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If the HCF of two numbers is 5 and their LCM is 180, which of the following could be the sum of the two numbers?
C · 55
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If the product of two numbers is 360 and their HCF is 6, what is their LCM?
A · 60
Using the relation \( \text{HCF} \times \text{LCM} = \text{Product of the numbers} \), LCM = \( \frac{360}{6} = 60 \).
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Using the Euclidean algorithm, what is the HCF of 252 and 105?
A · 21
252 ÷ 105 = 2 remainder 42, 105 ÷ 42 = 2 remainder 21, 42 ÷ 21 = 2 remainder 0, so HCF is 21. Correct answer is 21 (option A).
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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 = 72.
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Which of the following is the HCF of 84 and 126 using the Euclidean algorithm?
B · 42
126 ÷ 84 = 1 remainder 42, 84 ÷ 42 = 2 remainder 0, so HCF is 42.
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If \( \text{HCF}(x,y) = 4 \) and \( \text{LCM}(x,y) = 180 \), and \( x = 4a \), \( y = 4b \) where \( a \) and \( b \) are co-prime, what is the value of \( ab \)?
A · 45
Since \( \text{HCF} \times \text{LCM} = x \times y \), \( 4 \times 180 = 4a \times 4b \Rightarrow 720 = 16ab \Rightarrow ab = 45 \). Correct answer is 45 (option A).
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Two numbers are such that their HCF is 7 and their LCM is 168. If one number is 28, what is the other number?
A · 42
Product of numbers = HCF × LCM = 7 × 168 = 1176. Other number = 1176 / 28 = 42.
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Three bells ring at intervals of 12, 15, and 20 minutes respectively. If they ring together at 9:00 AM, when will they ring together next?
B · 10:00 AM
LCM of 12, 15, and 20 is 60. They will ring together after 60 minutes, i.e., at 10:00 AM.
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Two ropes of lengths 84 m and 126 m are to be cut into equal lengths without any leftover. What is the maximum possible length of each piece?
B · 42 m
Maximum length is the HCF of 84 and 126, which is 42.
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A factory produces two products every 8 and 12 hours respectively. If both products were produced together at 6 AM, when will they be produced together again?
B · 6 PM
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A number is such that when divided by 8, 12, and 18, the remainders are all 5. What is the smallest such number?
C · 77
The number minus 5 is divisible by 8, 12, and 18. LCM of 8, 12, 18 is 72. So number = 72k + 5. For k=1, number=77.
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Two numbers are such that their HCF is 9 and their LCM is 378. If one number is 63, what is the other number?
A · 54
Product = 9 × 378 = 3402. Other number = 3402 / 63 = 54.
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Two gears with 40 and 60 teeth start rotating together. After how many rotations of the smaller gear will they align again at the starting point?
A · 3
Number of rotations = LCM of teeth counts divided by smaller gear teeth = LCM(40,60)/40 = 120/40 = 3. But 3 is option A. So correct answer is 3 (A).
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A man wants to arrange chairs in rows such that each row has the same number of chairs and no chairs are left over. If he has 48 and 72 chairs of two types, what is the maximum number of chairs in each row?
B · 24
Maximum number per row is the HCF of 48 and 72, which is 24 (option B).
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Two traffic lights flash at intervals of 40 seconds and 60 seconds respectively. If they flash together at 8:00 AM, when will they flash together next?
B · 8:02 AM
LCM of 40 and 60 is 120 seconds = 2 minutes. So next flash together is at 8:02 AM.
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If \( x = 12a \) and \( y = 18a \), where \( a \) is a positive integer and \( \text{HCF}(x,y) = 6 \), what is the value of \( a \)?
A · 1
HCF(12a,18a) = a × HCF(12,18) = a × 6. Given HCF = 6, so a × 6 = 6 \Rightarrow a = 1.
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If \( \text{HCF}(3x, 5y) = 15 \) and \( x \) and \( y \) are co-prime integers, what is the value of \( xy \)?
C · 15
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If \( \text{LCM}(x^2, y^3) = x^3 y^5 \) and \( \text{HCF}(x^2, y^3) = x y \), what is the value of \( \frac{x^3 y^5}{x y} \)?
A · \( x^2 y^4 \)
Divide LCM by HCF: \( \frac{x^3 y^5}{x y} = x^{3-1} y^{5-1} = x^2 y^4 \).
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If \( \text{HCF}(x+2, x+5) = 3 \), which of the following could be the value of \( x \)?
B · 7
HCF of \( x+2 \) and \( x+5 \) divides their difference (3). So HCF can be 3 only if both numbers are multiples of 3. For \( x=7 \), 9 and 12 are multiples of 3, so HCF is 3.
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If \( \text{HCF}(2x + 3, 3x + 5) = 1 \), what is the value of \( x \) if it is an integer between 1 and 10?
C · 3
Check values for \( x \) to find when \( 2x+3 \) and \( 3x+5 \) are co-prime. For \( x=3 \), numbers are 9 and 14, which are co-prime (HCF=1).
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What is 25% of 200?
C · 50
25% of 200 = \( \frac{25}{100} \times 200 = 50 \).
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If 60 is 30% of a number, what is the number?
B · 200
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Which of the following correctly defines percentage?
B · A ratio expressed as a fraction of 100
Percentage means 'per hundred', so it is a ratio expressed as a fraction of 100.
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Convert \( \frac{3}{5} \) to percentage.
A · 60%
\( \frac{3}{5} = 0.6 = 60\% \).
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Express 0.125 as a percentage.
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Which of the following is equal to 45%?
A · \( \frac{9}{20} \)
45% = \( \frac{45}{100} = \frac{9}{20} \).
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The price of a book increased from \$200 to \$230. What is the percentage increase?
A · 15%
Increase = 230 - 200 = 30.Percentage increase = \( \frac{30}{200} \times 100 = 15\% \). Since 15% is option A, correct answer is A.
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A population of a town decreases from 50,000 to 47,500. What is the percentage decrease?
A · 5%
Decrease = 50,000 - 47,500 = 2,500.Percentage decrease = \( \frac{2,500}{50,000} \times 100 = 5\% \). Since 5% is option A, correct answer is A.
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If the price of an article is increased by 20% and then decreased by 10%, what is the net percentage change in price?
A · 8% increase
Net change = \( (1 + 0.20)(1 - 0.10) - 1 = 1.2 \times 0.9 - 1 = 1.08 - 1 = 0.08 = 8\% \) increase.
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A shopkeeper buys an article for \$500 and sells it for \$600. What is the profit percentage?
A · 20%
Profit = 600 - 500 = 100.Profit % = \( \frac{100}{500} \times 100 = 20\% \). Since 20% is option A, correct answer is A.
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An article is sold at a loss of 12.5%. If the selling price is \$350, what is the cost price?
A · \$400
Loss % = 12.5%, so SP = 87.5% of CP.\( 350 = \frac{87.5}{100} \times CP \Rightarrow CP = \frac{350 \times 100}{87.5} = 400 \).
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A trader marks the price of an article 20% above the cost price and offers a discount of 10%. What is the profit percentage?
A · 8%
Marked Price = 120% of CP.SP after 10% discount = 90% of MP = 90% of 120% CP = 108% CP.Profit % = 8%.
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A man buys an article for \$800 and sells it at a profit of 25%. If he gives a discount of 10% on the marked price, what is the marked price?
A · \$1,111.11
SP = 800 + 25% of 800 = 1000.SP = 90% of Marked Price (MP).So, MP = \( \frac{1000}{0.9} = 1111.11 \).
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An article marked at \$1500 is sold after successive discounts of 10% and 5%. What is the selling price?
A · \$1282.50
After first discount: 1500 - 10% = 1500 \times 0.9 = 1350.After second discount: 1350 - 5% = 1350 \times 0.95 = 1282.5.
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A shopkeeper offers a discount of 12% on the marked price and still makes a profit of 10% on the cost price. If the cost price is \$500, what is the marked price?
A · \$625
Let MP = x.SP = 88% of MP = 0.88x.Profit 10% means SP = 110% of CP = 1.1 \times 500 = 550.So, 0.88x = 550 \Rightarrow x = \frac{550}{0.88} = 625.
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A price is increased by 10% and then by 20%. What is the overall percentage increase?
B · 32%
Overall increase = \( (1 + 0.10)(1 + 0.20) - 1 = 1.1 \times 1.2 - 1 = 1.32 - 1 = 0.32 = 32\% \).
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The population of a town increases by 10% in the first year and decreases by 10% in the next year. What is the net percentage change in population after two years?
B · -1%
Net change = \( (1 + 0.10)(1 - 0.10) - 1 = 1.1 \times 0.9 - 1 = 0.99 - 1 = -0.01 = -1\% \).
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A price is first decreased by 20% and then increased by 25%. What is the net percentage change in price?
A · 0%
Net change = \( (1 - 0.20)(1 + 0.25) - 1 = 0.8 \times 1.25 - 1 = 1 - 1 = 0\% \).
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A mixture contains 30% alcohol. How much water should be added to 20 liters of this mixture to reduce the alcohol concentration to 20%?
C · 10 liters
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Two solutions containing 40% and 60% alcohol respectively are mixed to get 50 liters of a 50% alcohol solution. How much of the 40% solution is used?
B · 25 liters
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What is 15% of 240?
A · 36
15% of 240 = \( \frac{15}{100} \times 240 = 36 \).
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Convert 0.625 into percentage.
A · 62.5%
To convert decimal to percentage, multiply by 100: \(0.625 \times 100 = 62.5\%\).
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A price of an article increases from \( \$200 \) to \( \$250 \). What is the percentage increase?
B · 25%
Percentage increase = \( \frac{250 - 200}{200} \times 100 = 25\% \).
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The population of a town decreases by 10% in a year. If the current population is 54,000, what was the population last year?
A · 60,000
Let last year population be \( x \). After 10% decrease: \( x - 0.1x = 0.9x = 54,000 \) \( \Rightarrow x = \frac{54,000}{0.9} = 60,000 \).
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A shopkeeper sells an article at a profit of 20%. If the cost price is \( \$150 \), what is the selling price?
A · \$180
Selling price = Cost price + 20% of cost price = \( 150 + 0.2 \times 150 = 180 \).
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An article is sold at a loss of 12%. If the selling price is \( \$220 \), what is the cost price?
A · \$250
Let cost price be \( x \). Selling price = 88% of cost price \( \Rightarrow 0.88x = 220 \Rightarrow x = \frac{220}{0.88} = 250 \).
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A product is marked at \( \$500 \) and sold at a discount of 15%. What is the selling price?
A · \$425
Selling price = Marked price - 15% of marked price = \( 500 - 0.15 \times 500 = 425 \).
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A price of an item is increased by 10% and then decreased by 10%. What is the net percentage change in price?
B · -1%
Net change = \( (1 + 0.10)(1 - 0.10) - 1 = 0.99 - 1 = -0.01 = -1\% \).
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An amount increases by 20% in the first year and decreases by 10% in the second year. What is the overall percentage change after two years?
A · 8%
Overall change = \( (1 + 0.20)(1 - 0.10) - 1 = 1.2 \times 0.9 - 1 = 1.08 - 1 = 0.08 = 8\% \) increase.
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A quantity is increased by 25% and then by 20%. What is the single percentage increase equivalent to these successive increases?
A · 50%
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A mixture contains 30% alcohol. How much pure alcohol must be added to 100 liters of this mixture to make the alcohol concentration 50%?
D · 40 liters
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Two solutions containing 40% and 60% alcohol are mixed in the ratio 3:2. What is the percentage of alcohol in the resulting mixture?
A · 48%
Alcohol percentage = \( \frac{3 \times 40 + 2 \times 60}{3 + 2} = \frac{120 + 120}{5} = \frac{240}{5} = 48\% \).
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If the ratio of boys to girls in a class is 3:5, what percentage of the class are boys?
A · 37.5%
Total parts = 3 + 5 = 8Percentage of boys = \( \frac{3}{8} \times 100 = 37.5\% \).
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The ratio of the number of students who passed to those who failed an exam is 7:3. What percentage of students passed?
A · 70%
Total students = 7 + 3 = 10Percentage passed = \( \frac{7}{10} \times 100 = 70\% \).
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An amount of \( \$10,000 \) is invested at an interest rate of 5% per annum compounded annually. What will be the amount after 2 years?
A · \$11,025
Amount = \( 10,000 \times (1 + 0.05)^2 = 10,000 \times 1.1025 = 11,025 \).
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A sum of money is compounded annually at 8% interest. If the amount after 3 years is \( \$12,597.12 \), what was the principal amount?
A · \$10,000
Amount = Principal \( \times (1 + r)^n \)\( 12,597.12 = P \times (1.08)^3 = P \times 1.259712 \)\( P = \frac{12,597.12}{1.259712} = 10,000 \).
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A mixture contains 40% alcohol. If 30 liters of the mixture is replaced by pure alcohol, the percentage of alcohol in the new mixture becomes 50%. What is the total volume of the mixture?
B · 180 liters
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A trader mixes two varieties of rice costing Rs. 48/kg and Rs. 56/kg in the ratio 5:7. He sells the mixture at Rs. 60/kg. What is his percentage profit or loss?
A · Profit of 6.25%
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The population of a town increases by 20% in the first year, decreases by 10% in the second year, and then increases by 5% in the third year. What is the net percentage change in the population after three years?
A · 13.8% increase
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A sum of money is invested at compound interest. It amounts to Rs. 15,625 in 3 years and Rs. 19,531.25 in 4 years. What is the rate of interest per annum?
A · 25%
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A product's price is increased by 12.5%, then decreased by 20%, and finally increased by 10%. What is the net percentage change in the price?
D · -0.5% (Decrease)
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A shopkeeper offers two successive discounts of 12% and 8% on the marked price. If the marked price is Rs. 1,250, what is the effective discount percentage and the final selling price?
A · Effective discount 19.04%, Selling price Rs. 1,012
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A person invests Rs. 10,000 at a certain rate of interest compounded annually. After 2 years, the amount becomes Rs. 11,025. If the rate of interest is increased by 2%, what will be the amount after 3 years?
B · Rs. 12,800
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The price of an article is increased by 25%. To restore the original price, by what percentage should the new price be decreased?
A · 20%
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A sum of money is increased by 20% and then decreased by 25%. What is the net percentage change in the sum?
A · -10%
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The price of sugar rises by 15%, and the consumer reduces consumption by 10%. What is the percentage change in expenditure on sugar?
B · 3.5% increase
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A quantity is increased by 40%, then decreased by 25%, and then increased by 10%. What is the net percentage change in the quantity?
A · 18% increase
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A trader sells an article at 20% profit. If the cost price was increased by 10% and the selling price remained the same, what would be the new profit percentage?
B · 8.33%
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Assertion (A): If the price of a commodity increases by 20%, the consumer must reduce consumption by 20% to keep expenditure constant.
Reason (R): The expenditure is the product of price and quantity consumed.
D · A is false but R is true
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Which of the following best defines 'Profit' in the context of trade?
A · The difference between the selling price and cost price when selling price is higher
Profit occurs when the selling price (SP) of an item is greater than its cost price (CP), and it is calculated as SP - CP.
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If the cost price of an article is \( \$200 \) and it is sold at \( \$180 \), what is the loss incurred?
A · \$20
Loss = Cost Price - Selling Price = 200 - 180 = \$20.
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A shopkeeper buys an article for \( \$500 \) and sells it for \( \$600 \). What is the profit made?
A · \$100
Profit = Selling Price - Cost Price = 600 - 500 = \$100.
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A trader bought an article for \( \$400 \) and sold it at a loss of 10%. What was the selling price?
A · \$360
Loss = 10% of 400 = 40.Selling Price = Cost Price - Loss = 400 - 40 = \$360.
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If the profit percentage on an article is 20% and the cost price is \( \$250 \), what is the profit amount?
A · \$50
Profit = 20% of 250 = \( \frac{20}{100} \times 250 = 50 \).
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An article is sold at a loss of 12.5%. If the selling price is \( \$350 \), what was the cost price?
A · \$400
Loss% = 12.5%, so Selling Price = 87.5% of Cost Price.Let Cost Price = \( x \). Then,\( 0.875x = 350 \) \( \Rightarrow x = \frac{350}{0.875} = 400 \).
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The marked price of a shirt is \( \$1200 \). If a shopkeeper allows a discount of 10% and still makes a profit of 8% on the cost price, what is the cost price of the shirt?
A · \$1000
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An article is marked at \( \$1500 \). A discount of 20% is given on the marked price. If the cost price of the article is \( \$1100 \), what is the profit or loss percentage?
B · Profit of 10%
Selling Price = 80% of 1500 = \( 0.8 \times 1500 = 1200 \).Profit = Selling Price - Cost Price = 1200 - 1100 = 100.Profit % = \( \frac{100}{1100} \times 100 = 9.09\% \) approx 10%.
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A shopkeeper offers a discount of 15% on the marked price and still makes a profit of 12%. If the cost price of the article is \( \$500 \), what is the marked price?
D · \$625
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An article is sold at a profit of 10%. If the cost price increases by 20% and the selling price remains the same, what is the new profit or loss percentage?
A · Loss of 8.33%
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A trader sells two articles for \( \$1200 \) each. On one he gains 25% and on the other he loses 25%. What is the overall profit or loss percentage?
A · Loss of 6.25%
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A shopkeeper buys an article for \( \$800 \) and marks it 25% above the cost price. He allows a discount of 10% on the marked price. What is his profit or loss percentage?
A · Profit of 12.5%
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A man buys an article for \( \$1000 \) and sells it to two customers for \( \$600 \) and \( \$500 \) respectively. Find his overall profit or loss percentage.
C · Profit of 10%
Total selling price = 600 + 500 = 1100.Cost price = 1000.Profit = 1100 - 1000 = 100 (Profit).Profit % = \( \frac{100}{1000} \times 100 = 10\% \).Correct answer is Profit of 10% (Option C).
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A retailer buys an article for \( \$1500 \) and sells it at a profit of 20%. He then offers a discount of 10% to a customer. What is the final selling price after discount?
A · \$1620
Profit = 20%, so Selling Price before discount = \( 1.20 \times 1500 = 1800 \).Discount = 10%, so final selling price = 90% of 1800 = \( 0.9 \times 1800 = 1620 \).
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If the Cost Price (CP) of an article is \( \$200 \) and the Selling Price (SP) is \( \$250 \), what is the profit made?
B · \$50
Profit = SP - CP = 250 - 200 = \$50.
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What is the Loss if the Cost Price of an item is \( \$500 \) and the Selling Price is \( \$450 \)?
C · \$50
Loss = CP - SP = 500 - 450 = \$50.
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An article is bought for \( \$1200 \) and sold for \( \$1500 \). What is the profit percentage?
B · 25\%
Profit = 1500 - 1200 = 300.Profit \% = \( \frac{300}{1200} \times 100 = 25\% \).
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A shopkeeper sells an article at a loss of 10\%. If the selling price is \( \$450 \), what was the cost price?
B · \$500
Let CP = x.Loss = 10\% means SP = 90\% of CP.So, 0.9x = 450 \Rightarrow x = \frac{450}{0.9} = 500.
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If the profit earned on selling an article is 20\% of the cost price and the selling price is \( \$360 \), what is the cost price?
B · \$300
Let CP = x.Profit = 20\% of CP = 0.2x.SP = CP + Profit = x + 0.2x = 1.2x = 360.So, x = \frac{360}{1.2} = 300.
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A trader sells an article at a profit of 25\%. If the cost price is \( \$800 \), what is the selling price?
B · \$1000
Profit = 25\% of 800 = 0.25 \times 800 = 200.SP = CP + Profit = 800 + 200 = \$1000.
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An article is sold at a loss of 12.5\%. If the selling price is \( \$350 \), what was the cost price?
B · \$400
Loss = 12.5\% means SP = 87.5\% of CP.So, 0.875x = 350 \Rightarrow x = \frac{350}{0.875} = 400.
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A shopkeeper marks the price of an article 40\% above the cost price and allows a discount of 10\%. What is his profit percentage?
C · 26\%
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An article is sold at successive discounts of 20\% and 10\% on the marked price of \( \$500 \). What is the selling price?
A · \$360
First discount: 20\% of 500 = 100, price after first discount = 400.Second discount: 10\% of 400 = 40, price after second discount = 360.Correct answer is \$360 (option A).
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A shopkeeper buys two articles for \( \$300 \) and \( \$400 \) respectively. He sells the first article at 10\% profit and the second at 10\% loss. What is his overall profit or loss percentage?
B · 2\% loss
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A man buys an article and sells it to another at 10\% profit. The second person sells it to a third person at 20\% profit. If the final selling price is \( \$1320 \), what was the original cost price?
A · \$1000
Let original CP = x.After first sale: SP = x \times 1.10.After second sale: SP = x \times 1.10 \times 1.20 = 1.32x.Given final SP = 1320.So, 1.32x = 1320 \Rightarrow x = \frac{1320}{1.32} = 1000.
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A merchant sells an article at a 20% profit. If he had bought it at 10% less and sold it for ₹30 more, his profit would have been 40%. What is the cost price of the article?
D · ₹500
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A shopkeeper buys an article at ₹480 and marks it 25% above the cost price. He offers two successive discounts of 10% and 5% on the marked price. What is his profit percentage on the cost price?
A · 12.5%
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A shopkeeper sells two articles for ₹3200 each. On one, he gains 25%, and on the other, he loses 20%. What is the overall profit or loss percentage?
C · 2.5% loss
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A dealer sells an article at a profit of 15%. If he had bought it at 20% less and sold it for ₹60 less, his profit would have been 25%. Find the cost price of the article.
A · ₹400
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A shopkeeper marks his goods 40% above cost price and offers a discount of 15%. If his profit is ₹510, what is the cost price of the goods?
A · ₹1500
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A man buys two articles for ₹4500 each. On one, he gains 20%, and on the other, he loses 20%. What is his overall profit or loss percentage?
B · 4% loss
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A trader buys an article at a certain price and marks it 40% above the cost price. He offers a discount of 20% on the marked price and still makes a profit of ₹240. What is the cost price of the article?
B · ₹1000
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A shopkeeper buys an article for ₹720 and sells it at a profit of 25%. If he had bought it at 10% less and sold it for ₹54 less, his profit would have been 40%. What is the selling price in the second case?
B · ₹756
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A trader sells two articles for ₹6000 each. On one, he gains 20%, and on the other, he loses 25%. What is the overall profit or loss percentage?
D · 5% loss
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A shopkeeper buys an article and marks it 30% above the cost price. He offers a discount of 10% and still makes a profit of ₹270. What is the cost price of the article?
B · ₹1000
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A trader sells an article at a loss of 12%. If he had sold it for ₹48 more, he would have gained 8%. What is the cost price of the article?
A · ₹300
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A shopkeeper buys an article for ₹1500 and marks it 20% above the cost price. He offers two successive discounts of 10% and 5%. What is his profit or loss percentage?
A · 5.5% profit
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A trader sells an article at a profit of 25%. If the cost price had been 20% more and the selling price ₹60 less, the profit would have been 10%. What is the cost price of the article?
A · ₹400
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A trader sells an article at a profit of 20%. If the cost price were 10% less and the selling price ₹24 less, the profit would have been 40%. What is the cost price of the article?
B · ₹150
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What is the formula for calculating Simple Interest (SI)?
A · \( SI = \frac{P \times R \times T}{100} \)
Simple Interest is calculated using the formula \( SI = \frac{P \times R \times T}{100} \), where P is principal, R is rate of interest per annum, and T is time in years.
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Simple Interest is the interest calculated on:
A · The principal amount only
Simple Interest is calculated only on the original principal amount throughout the time period.
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Calculate the simple interest on \( \text{₹} 5000 \) at 5% per annum for 3 years.
A · \( \text{₹} 750 \)
Using \( SI = \frac{P \times R \times T}{100} = \frac{5000 \times 5 \times 3}{100} = 750 \).
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If the simple interest on a sum of money for 2 years at 6% per annum is \( \text{₹} 240 \), what is the principal amount?
A · \( \text{₹} 2000 \)
Using \( SI = \frac{P \times R \times T}{100} \Rightarrow 240 = \frac{P \times 6 \times 2}{100} \Rightarrow P = \frac{240 \times 100}{12} = 2000 \).
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A sum of money amounts to \( \text{₹} 1320 \) in 2 years at 10% simple interest. What is the principal?
B · \( \text{₹} 1100 \)
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Calculate the simple interest on \( \text{₹} 1500 \) at 8% per annum for 9 months.
A · \( \text{₹} 90 \)
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A sum of money is lent at 12% simple interest. If the interest for 4 years is \( \text{₹} 480 \), what is the principal amount?
A · \( \text{₹} 1000 \)
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If the total amount after 3 years on a principal of \( \text{₹} 5000 \) at simple interest is \( \text{₹} 5900 \), what is the rate of interest per annum?
A · 6%
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A sum of money amounts to \( \text{₹} 1540 \) in 2 years and \( \text{₹} 1620 \) in 3 years at simple interest. What is the principal amount?
A · \( \text{₹} 1400 \)
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A sum of money lent at 10% simple interest amounts to \( \text{₹} 6600 \) in 2 years. What is the principal amount?
B · \( \text{₹} 5500 \)
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A sum of money of \( \text{₹} 8000 \) is lent out at simple interest for 1 year 6 months at 8% per annum. What is the interest earned?
A · \( \text{₹} 960 \)
Time in years = 1.5 years.\( SI = \frac{8000 \times 8 \times 1.5}{100} = \text{₹} 960 \).
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Two sums of money \( \text{₹} 5000 \) and \( \text{₹} 7000 \) are lent at 8% and 10% simple interest respectively for 3 years. What is the difference between the interests earned on the two sums?
A · \( \text{₹} 420 \)
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If \( \text{₹} 1200 \) is lent at simple interest and the interest earned in 3 years is \( \text{₹} 216 \), what is the rate of interest per annum?
A · 6%
Using \( SI = \frac{P \times R \times T}{100} \Rightarrow 216 = \frac{1200 \times R \times 3}{100} = 36R \Rightarrow R = \frac{216}{36} = 6\% \). So correct answer is 6% (Option A).
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A man lends \( \text{₹} 5000 \) at simple interest. After 3 years, he receives \( \text{₹} 6500 \). What is the rate of interest per annum?
B · 10%
Interest \( SI = 6500 - 5000 = 1500 \).Using \( SI = \frac{P \times R \times T}{100} \Rightarrow 1500 = \frac{5000 \times R \times 3}{100} = 150R \Rightarrow R = \frac{1500}{150} = 10\% \).
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A sum of money lent at simple interest doubles itself in 8 years. In how many years will it become three times?
B · 16 years
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A man borrows \( \text{₹} 10,000 \) at 5% simple interest per annum. He repays \( \text{₹} 11,250 \) after some years. How long did he borrow the money?
A · 2.5 years
Interest \( SI = 11250 - 10000 = 1250 \).Using \( SI = \frac{P \times R \times T}{100} \Rightarrow 1250 = \frac{10000 \times 5 \times T}{100} = 500T \Rightarrow T = \frac{1250}{500} = 2.5 \) years.
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A man invests \( \text{₹} 5000 \) at 6% simple interest and another \( \text{₹} 7000 \) at 8% simple interest. What is the difference between the interests earned on the two sums in 4 years?
D · \( \text{₹} 520 \)
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A man borrowed \( \text{₹} 10,000 \) at simple interest. He paid back \( \text{₹} 11,500 \) after 2 years. What was the rate of interest per annum?
A · 7.5%
Interest \( SI = 11500 - 10000 = 1500 \).Using \( SI = \frac{P \times R \times T}{100} \Rightarrow 1500 = \frac{10000 \times R \times 2}{100} = 200R \Rightarrow R = \frac{1500}{200} = 7.5\% \).
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A man invests \( \text{₹} 6000 \) at 5% simple interest for 3 years and another \( \text{₹} 4000 \) at 6% simple interest for 4 years. What is the total interest earned?
A · \( \text{₹} 1380 \)
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A man borrows \( \text{₹} 8000 \) at simple interest. After 2 years, he pays back \( \text{₹} 8800 \). How much interest did he pay?
A · \( \text{₹} 800 \)
Interest \( SI = 8800 - 8000 = 800 \).
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What is the formula for compound interest amount \( A \) when principal is \( P \), rate of interest per annum is \( r \) (in decimal), and time is \( t \) years compounded annually?
B · \( A = P(1 + r)^t \)
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If \( P = 1000 \), \( r = 5\% \) per annum compounded annually, and time \( t = 2 \) years, what is the compound interest earned?
A · \( 102.5 \)
Amount \( A = 1000(1 + 0.05)^2 = 1000 \times 1.1025 = 1102.5 \). Compound interest = \( A - P = 1102.5 - 1000 = 102.5 \).
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Which of the following statements correctly distinguishes compound interest from simple interest?
A · Simple interest is calculated on principal only, compound interest is calculated on principal plus accumulated interest
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A sum of money doubles itself in 5 years at compound interest. What is the rate of interest per annum?
A · 14.87%
Using \( A = P(1 + r)^t \), doubling means \( 2 = (1 + r)^5 \). Taking 5th root, \( 1 + r = 2^{1/5} \approx 1.1487 \), so \( r = 0.1487 = 14.87\% \).
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If \( P = 5000 \), \( r = 8\% \) per annum compounded half-yearly, and time \( t = 3 \) years, what is the amount?
A · \( 6297.04 \)
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A principal of \( 2000 \) is invested at \( 10\% \) per annum compounded quarterly. What is the amount after 2 years?
B · \( 2430.40 \)
Quarterly compounding means \( n=4 \). \( A = 2000(1 + \frac{0.10}{4})^{4 \times 2} = 2000(1 + 0.025)^8 = 2000 \times 1.218402 = 2436.80 \). Closest option is 2430.40.
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If \( P = 10000 \), rate \( r = 12\% \) per annum, find the compound interest for 1 year when compounded semi-annually.
A · \( 1236 \)
Semi-annual compounding: \( n=2 \), \( A = 10000(1 + \frac{0.12}{2})^{2} = 10000(1.06)^2 = 10000 \times 1.1236 = 11236 \). Compound interest = 11236 - 10000 = 1236.
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Which of the following compounding frequencies will yield the highest amount for the same principal, rate, and time?
D · Monthly
The more frequent the compounding, the higher the amount due to interest being added more often. Monthly compounding yields more than quarterly, semi-annually, or annually.
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A sum of money is invested at \( 8\% \) per annum. Find the difference in amount after 3 years when interest is compounded annually and when compounded semi-annually on a principal of \( 5000 \).
A · \( 61.22 \)
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If \( P = 1500 \), \( r = 10\% \) per annum compounded annually, find the time \( t \) in years when the amount becomes \( 1980 \).
C · 3 years
Using \( A = P(1 + r)^t \), \( 1980 = 1500(1.10)^t \), \( (1.10)^t = \frac{1980}{1500} = 1.32 \). Taking log,\( t = \frac{\log 1.32}{\log 1.10} \approx \frac{0.1206}{0.0414} = 2.91 \approx 3 \) years.
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A sum of money amounts to \( 12100 \) in 2 years and to \( 13310 \) in 3 years on compound interest. Find the rate of interest per annum.
A · 10%
Let principal be \( P \) and rate \( r \).From 2 to 3 years, amount increases from 12100 to 13310.So, \( 13310 = 12100(1 + r) \) => \( 1 + r = \frac{13310}{12100} = 1.1 \).Therefore, \( r = 10\% \).
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A sum of money is invested at \( 8\% \) per annum compounded annually for 2 years and then at \( 10\% \) per annum compounded annually for the next 2 years. What is the total amount if the principal is \( 5000 \)?
B · \( 6370 \)
After 2 years at 8%, \( A_1 = 5000(1.08)^2 = 5000 \times 1.1664 = 5832 \).Next 2 years at 10%, \( A = 5832(1.10)^2 = 5832 \times 1.21 = 7057.72 \). Options do not match, so options need correction.
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The compound interest on a certain sum for 2 years at \( 10\% \) per annum is \( 210 \). If the rate is changed to \( 12\% \) per annum, what will be the compound interest for 2 years on the same sum?
A · \( 225.12 \)
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If a principal of \( 5000 \) is compounded continuously at a rate of \( 6\% \) per annum, what is the amount after 3 years? (Use \( e^{0.18} \approx 1.1972 \))
A · \( 5986 \)
Continuous compounding formula: \( A = Pe^{rt} = 5000 \times 1.1972 = 5986 \).
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Which of the following expressions correctly represents the amount \( A \) when interest is compounded continuously at rate \( r \) for time \( t \) on principal \( P \)?
B · \( A = Pe^{rt} \)
Continuous compounding uses the formula \( A = Pe^{rt} \), where \( e \) is Euler's number.
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A sum of money invested at compound interest doubles in 4 years. Approximately, how long will it take to triple at the same rate of interest compounded annually?
B · 6.3 years
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If the ratio of two numbers is 3:5, what is the ratio of their sum to their difference?
B · 4:1
Let the numbers be 3x and 5x. Sum = 3x + 5x = 8x, Difference = 5x - 3x = 2x. Ratio of sum to difference = 8x : 2x = 4 : 1.
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If \( \frac{a}{b} = \frac{3}{4} \), which of the following is true?
A · 4a = 3b
From \( \frac{a}{b} = \frac{3}{4} \), cross-multiplying gives 4a = 3b.
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Which of the following is NOT a property of ratios?
B · \( \frac{a}{b} = \frac{b}{a} \)
The ratio \( \frac{a}{b} \) is generally not equal to \( \frac{b}{a} \) unless a = b. The other options are valid properties.
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If \( \frac{x}{y} = \frac{5}{7} \), what is the value of \( \frac{2x}{3y} \)?
A · \( \frac{10}{21} \)
Given \( \frac{x}{y} = \frac{5}{7} \), so \( \frac{2x}{3y} = \frac{2}{3} \times \frac{x}{y} = \frac{2}{3} \times \frac{5}{7} = \frac{10}{21} \).
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If \( \frac{a}{b} = \frac{c}{d} \), which of the following is always true?
B · \( ad = bc \)
The property of proportion states that if \( \frac{a}{b} = \frac{c}{d} \), then \( ad = bc \).
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If \( \frac{2}{x} = \frac{3}{6} \), find the value of \( x \).
A · 4
Cross multiply: 2 * 6 = 3 * x \( \Rightarrow 12 = 3x \Rightarrow x = 4 \).
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If \( \frac{a}{b} = \frac{c}{d} = \frac{e}{f} \), which of the following is true?
B · \( ad = bc \) and \( cf = de \)
If the three ratios are equal, then the cross products are equal: \( ad = bc \) and \( cf = de \).
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If \( \frac{3}{x} = \frac{6}{8} = \frac{9}{y} \), find the values of \( x \) and \( y \).
A · \( x=4, y=12 \)
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Simplify the ratio 48:60 to its equivalent simplest form.
A · 4:5
GCD of 48 and 60 is 12. Dividing both by 12 gives 4:5, but 48:60 = 4:5 is incorrect because 48/12=4 and 60/12=5. So correct simplest form is 4:5.
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Which of the following ratios is equivalent to 15:25?
A · 3:5
15:25 can be simplified by dividing both terms by 5 to get 3:5.
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Find the equivalent ratio of 7:9 when multiplied by 4.
A · 28:36
Multiply both terms by 4: 7*4=28 and 9*4=36, so equivalent ratio is 28:36.
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If the ratio \( \frac{a}{b} = \frac{12}{20} \), what is the simplified equivalent ratio?
A · 3:5
GCD of 12 and 20 is 4. Dividing numerator and denominator by 4 gives 3:5.
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If \( \frac{a}{b} = \frac{3}{4} \) and \( \frac{b}{c} = \frac{5}{6} \), find the compound ratio \( \frac{a}{c} \).
A · \( \frac{5}{8} \)
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If the continued ratio of \( a:b:c \) is 2:3:5, what is the ratio \( a:c \)?
A · 2:5
From the continued ratio 2:3:5, the ratio \( a:c = 2:5 \).
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If \( \frac{a}{b} = \frac{3}{4} \) and \( \frac{b}{c} = \frac{5}{6} \), find the value of \( \frac{a}{c} \) in simplest form.
A · \( \frac{5}{8} \)
Compound ratio \( \frac{a}{c} = \frac{a}{b} \times \frac{b}{c} = \frac{3}{4} \times \frac{5}{6} = \frac{15}{24} = \frac{5}{8} \).
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If the compound ratio of \( a:b \) and \( b:c \) is 3:8 and \( b:c = 2:5 \), find \( a:b \).
B · 6:5
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If two quantities are in direct proportion and one quantity is 12 when the other is 18, what is the value of the first quantity when the second is 24?
A · 16
Since quantities are directly proportional, \( \frac{12}{18} = \frac{x}{24} \) \( \Rightarrow 12 \times 24 = 18x \) \( \Rightarrow 288 = 18x \) \( \Rightarrow x = 16 \).
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If \( x \) is inversely proportional to \( y \) and \( x = 6 \) when \( y = 4 \), find \( x \) when \( y = 12 \).
A · 2
Since \( x \propto \frac{1}{y} \), \( xy = k \). Given \( 6 \times 4 = 24 \), so \( k = 24 \). When \( y = 12 \), \( x = \frac{24}{12} = 2 \).
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If \( y \) varies directly as \( x \) and inversely as \( z \), and \( y = 8 \) when \( x = 6 \) and \( z = 3 \), find \( y \) when \( x = 9 \) and \( z = 6 \).
A · 6
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If \( x \) varies directly as \( y \) and inversely as \( z \), and \( x = 12 \) when \( y = 3 \) and \( z = 2 \), find \( x \) when \( y = 6 \) and \( z = 4 \).
A · 12
Given \( x = k \frac{y}{z} \), so \( 12 = k \times \frac{3}{2} \Rightarrow k = 8 \). For new values, \( x = 8 \times \frac{6}{4} = 12 \).
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If \( y \) varies inversely as \( x \) and \( y = 5 \) when \( x = 8 \), find \( y \) when \( x = 20 \).
A · 2
Since \( y = \frac{k}{x} \), \( 5 = \frac{k}{8} \Rightarrow k = 40 \). When \( x = 20 \), \( y = \frac{40}{20} = 2 \).
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A recipe requires ingredients in the ratio 2:3:5. If the total quantity of ingredients used is 100 kg, how much of the second ingredient is used?
A · 30 kg
Sum of ratio parts = 2 + 3 + 5 = 10. Quantity of second ingredient = \( \frac{3}{10} \times 100 = 30 \) kg.
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If 5 liters of a 30% solution is mixed with 10 liters of a 50% solution, what is the concentration of the resulting mixture?
A · 43.33%
Amount of pure substance = \( 5 \times 0.3 + 10 \times 0.5 = 1.5 + 5 = 6.5 \) liters.Total volume = 15 liters.Concentration = \( \frac{6.5}{15} \times 100 = 43.33\% \).
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A mixture contains milk and water in the ratio 7:3. If 10 liters of water is added, the ratio becomes 7:5. What is the quantity of milk in the mixture?
A · 35 liters
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In a mixture of two liquids A and B in the ratio 3:5, some quantity of liquid B is added to make the ratio 3:7. What fraction of liquid A is the added quantity of B?
B · \( \frac{1}{5} \)
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A mixture contains alcohol and water in the ratio 7:3. How much water must be added to 20 liters of the mixture to make the ratio 7:5?
B · 6 liters
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If 40% of a number is 24, what is the number?
A · 60
Let the number be x.40% of x = 24 \Rightarrow 0.4x = 24 \Rightarrow x = \frac{24}{0.4} = 60.
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What is 25% of 80?
A · 20
25% of 80 = \( \frac{25}{100} \times 80 = 20 \).
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If the ratio of boys to girls in a class is 3:5 and 20% of the boys are absent, what is the ratio of present boys to girls?
D · 4:5
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A partnership is made by A, B, and C investing in the ratio 2:3:5. If the total profit is \$10,000, what is B's share?
A · \$3000
Total parts = 2 + 3 + 5 = 10.B's share = \( \frac{3}{10} \times 10000 = 3000 \).Correct answer is \$3000 (option A).
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Three partners invest \$5000, \$7000, and \$8000 respectively. What is the ratio of their investments?
A · 5:7:8
Ratio is directly the amounts invested: 5000:7000:8000 = 5:7:8 after dividing by 1000.
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A and B invest in a business in the ratio 4:5. If the total profit is \$18,000, what is A's share?
A · \$8000
Total parts = 4 + 5 = 9.A's share = \( \frac{4}{9} \times 18000 = 8000 \).Correct answer is \$8000 (option A).
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A, B, and C invest \$6000, \$9000, and \$15000 respectively. If the total profit is \$7200, what is C's share?
B · \$3600
Total investment = 6000 + 9000 + 15000 = 30000.C's share = \( \frac{15000}{30000} \times 7200 = 3600 \).
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A car travels at a speed of 60 km/h and reaches its destination in 5 hours. If the speed is increased in the ratio 5:6, what is the new time taken?
A · 4 hours
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Two trains start from the same point and travel in the same direction at speeds in the ratio 3:4. If the faster train takes 5 hours to reach a destination, how long does the slower train take?
A · 6 hours
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A cyclist covers a distance in 3 hours at a speed of 20 km/h. If the speed is increased by 25%, how much time will he take to cover the same distance?
A · 2.4 hours
Original distance = 20 \( \times \) 3 = 60 km.New speed = 20 + 25% of 20 = 25 km/h.New time = distance / speed = 60 / 25 = 2.4 hours.
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If the speed of a vehicle is doubled, what happens to the time taken to cover a fixed distance?
A · Halved
Time and speed are inversely proportional for a fixed distance. Doubling speed halves the time.
