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Quick recall · 227 cards
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PYQ · 2018 Tap to reveal →
HCF of 2472, 1284 and a third number ‘n’ is 12. If their LCM is \(8 \times 9 \times 5 \times 10^3 \times 10^7\), then the number ‘n’ is:
A · \(2^2 \times 3^2 \times 5^1\)
PYQ · 2021 Tap to reveal →
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 Tap to reveal →
Two numbers having their LCM 480 are in the ratio 3:4. What will be the smaller number of this pair?
B · 120
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A man sold two watches at the same price, one at a 10% profit and the other at a 10% loss. Find his overall gain or loss percent. (Options: A) 1% loss B) 1% gain C) No profit no loss D) 2% loss)
A · A) 1% loss
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A train starts from station X at the rate of 60 km/hr and reaches station Y in 45 minutes. If the speed is reduced by 6 km/hr, how much more time will the train take to return from station Y to station X?
A · 5 minutes
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A cyclist covers a distance of 800 meter in 4 minutes 20 seconds. What is the speed in km/hr of the cyclist?
B · 8.4 km/hr
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The difference in simple interest and compound interest on a certain sum of money in 2 years at 10% p.a. is Rs. 50. The sum is
B · Rs. 6000
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A rectangular garden has a fence around it, where the total length of the fence measures 60 cm. The length of this garden is exactly twice its breadth. Calculate the area of this rectangle.
B · 200 cm²
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A perfectly round lake has a radius of 7 cm and a height of 10 cm (when considered as a cylinder). Calculate the total surface area of this cylindrical lake.
C · 748 cm²
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A hollow hemisphere bowl has an outer radius of 10 cm and a wall thickness of 2 cm. What is the total volume of this hollow hemisphere?
B · 2688 cm³
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The surface area of three faces of a cuboid sharing a vertex are 20 m², 32 m², and 40 m². What is the volume of the cuboid?
C · 160 m³
PYQ · 2025 Tap to reveal →
For which of the following solids is the lateral/curved surface area and total surface area the same?
C · Sphere
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The table below shows sales data (in thousands) for a company from 2012 to 2016: 2012 (140), 2013 (150), 2014 (190), 2015 (160), 2016 (159). What is the maximum year-on-year difference in sales?
B · 2014
Calculate absolute differences:2013: |150-140| = 102014: |190-150| = 402015: |160-190| = 302016: |159-160| = 1Maximum difference is 40 (in thousands) in 2014. Thus, option B.
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The average age of 5 men is 40. The average age of 5 women is 39. The average age of their 4 children is 15. What is the average age of the family?
B · 30.33
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The ages of 5 people are in a ratio 1:3:5:7:11. If the difference between the total of their ages and the average of their ages multiplied by 5 is 108, find the age of the youngest.
C · 15
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Which of the following is NOT a property of the decimal number system?
C · Each digit's place value is a power of 2
The decimal system is base-10, so each digit's place value is a power of 10, not 2.
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What is the base of the binary number system?
A · 2
The binary number system is base-2, using digits 0 and 1.
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What is the prime factorization of 84?
A · 2^2 \times 3 \times 7
84 = 2 \times 42 = 2 \times 2 \times 21 = 2^2 \times 3 \times 7.
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Find the prime factorization of 210.
A · 2 \times 3 \times 5 \times 7
210 = 2 \times 105 = 2 \times 3 \times 35 = 2 \times 3 \times 5 \times 7.
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If \( 2^3 \times 3^2 \times 5 \) is the prime factorization of a number, what is the number?
A · 360
Calculate: \( 2^3 = 8, 3^2 = 9, 5 = 5 \). So number = 8 \times 9 \times 5 = 360.
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Find the HCF of 48 and 60 using prime factorization.
A · 12
48 = 2^4 \times 3, 60 = 2^2 \times 3 \times 5. HCF = 2^2 \times 3 = 12.
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Calculate the HCF of 84 and 126.
A · 42
84 = 2^2 \times 3 \times 7, 126 = 2 \times 3^2 \times 7. HCF = 2 \times 3 \times 7 = 42.
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Find the HCF of 252, 630 and 882.
A · 42
Prime factors:252 = 2^2 \times 3^2 \times 7630 = 2 \times 3^2 \times 5 \times 7882 = 2 \times 3^2 \times 7^2HCF = 2 \times 3^2 \times 7 = 42.
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What is the LCM of 12 and 18?
A · 36
12 = 2^2 \times 3, 18 = 2 \times 3^2. LCM = 2^2 \times 3^2 = 36.
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Find the LCM of 15 and 20.
A · 60
15 = 3 \times 5, 20 = 2^2 \times 5. LCM = 2^2 \times 3 \times 5 = 60.
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Calculate the LCM of 24 and 36.
A · 72
24 = 2^3 \times 3, 36 = 2^2 \times 3^2. LCM = 2^3 \times 3^2 = 72.
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Find the LCM of 8 and 12.
A · 24
8 = 2^3, 12 = 2^2 \times 3. LCM = 2^3 \times 3 = 24.
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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 \times LCM = 6 \times 72 = 432.Other number = 432 / 18 = 24.
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Two numbers have HCF 4 and LCM 180. If one number is 36, find the other number.
A · 20
Product = HCF \times LCM = 4 \times 180 = 720.Other number = 720 / 36 = 20.
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If the HCF and LCM of two numbers are 7 and 168 respectively, and the numbers are in the ratio 1:4, find the numbers.
B · 14 and 56
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Three numbers have HCF 5 and LCM 300. If two of the numbers are 20 and 25, find the third number.
B · 15
HCF = 5, so numbers are multiples of 5.LCM(20,25) = 100.LCM(20,25,x) = 300.Since 300/100 = 3, third number must contribute factor 3.Third number = 15.
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The LCM of three numbers is 420 and their HCF is 7. If two numbers are 28 and 35, what is the third number?
B · 20
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Find the HCF of 36, 48 and 60.
B · 12
36 = 2^2 \times 3^2, 48 = 2^4 \times 3, 60 = 2^2 \times 3 \times 5.HCF = 2^2 \times 3 = 12.
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Find the LCM of 8, 12 and 18.
A · 72
8 = 2^3, 12 = 2^2 \times 3, 18 = 2 \times 3^2.LCM = 2^3 \times 3^2 = 8 \times 9 = 72.
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The HCF of three numbers is 4 and their LCM is 240. If two of the numbers are 12 and 20, find the third number.
C · 15
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Two numbers are in the ratio 3:5 and their HCF is 7. What are the numbers?
D · 42 and 70
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Two numbers are in the ratio 4:7 and their LCM is 168. What is the smaller number?
B · 24
Let numbers be 4x and 7x.LCM = (4 \times 7 \times x) = 28x = 168.x = 6.Smaller number = 4 \times 6 = 24.
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Three numbers are in the ratio 2:3:5 and their HCF is 6. Find their LCM if the numbers are 12, 18, and 30.
A · 180
Numbers are 12, 18, 30.LCM = 2^2 \times 3^2 \times 5 = 4 \times 9 \times 5 = 180.But since HCF is 6, actual numbers are 12, 18, 30.LCM(12,18,30) = 180.So correct answer is 180.
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Which of the following is NOT a valid representation of a number in the decimal number system?
C · 12A.34
The decimal number system uses digits 0-9 only. '12A.34' contains 'A', which is invalid in decimal representation.
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Which of the following numbers is a rational number?
B · 0.333...
0.333... is a repeating decimal and can be expressed as the fraction \( \frac{1}{3} \), hence rational.
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If the base of a number system is 7, what is the decimal equivalent of the number \( 345_7 \)?
A · 186
Value = 3\( \times 7^2 \) + 4\( \times 7^1 \) + 5\( \times 7^0 \) = 3\( \times 49 \) + 4\( \times 7 \) + 5 = 147 + 28 + 5 = 180.
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Which of the following is always true for the HCF of two positive integers?
A · It is always less than or equal to the smaller number
The HCF (Highest Common Factor) cannot be greater than the smaller number and is always a divisor of both numbers.
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Find the HCF of 84 and 126 using prime factorization.
A · 42
84 = 2 \times 2 \times 3 \times 7, 126 = 2 \times 3 \times 3 \times 7. Common factors: 2, 3, 7. HCF = 2 \times 3 \times 7 = 42.
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Which property of HCF states that \( \text{HCF}(a,b) = \text{HCF}(b,a) \)?
A · Commutative Property
The commutative property states that the order of numbers does not affect the HCF.
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If \( \text{HCF}(a,b) = d \), which of the following is always true?
A · \( d \) divides both \( a \) and \( b \)
By definition, the HCF divides both numbers exactly.
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Which of the following is a property of LCM of two numbers \( a \) and \( b \)?
B · \( \text{LCM}(a,b) \geq \max(a,b) \)
LCM is always greater than or equal to the maximum of the two numbers.
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Find the LCM of 12 and 18 using prime factorization.
A · 36
12 = 2^2 \times 3, 18 = 2 \times 3^2. LCM = 2^2 \times 3^2 = 36.
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If \( \text{LCM}(a,b) = m \) and \( \text{HCF}(a,b) = n \), which of the following is true?
B · \( a \times b = m \times n \)
The product of two numbers equals the product of their LCM and HCF.
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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
Using \( a \times b = \text{HCF} \times \text{LCM} \), \( 18 \times b = 6 \times 72 = 432 \) so \( b = 24 \).
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Which of the following is NOT true about the relationship between HCF and LCM of two numbers?
D · HCF is always greater than LCM
HCF is always less than or equal to the smaller number, so it cannot be greater than LCM.
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Find the HCF of 24, 36, and 60.
A · 6
Prime factors: 24=2^3\times3, 36=2^2\times3^2, 60=2^2\times3\times5. Common factors: 2^2\times3=12 but 12 is not a factor of 60, so HCF is 6.
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The LCM of three numbers is 180. If two of the numbers are 12 and 15, which of the following could be the third number?
A · 20
LCM(12,15) = 60. To get LCM 180, third number must have prime factors to raise LCM to 180. 20 (2^2\times5) raises LCM to 180.
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Three numbers are in the ratio 3:4:5. If their HCF is 6, what is their LCM?
A · 360
Numbers are 18, 24, 30. LCM = LCM(18,24,30) = 360.
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If two numbers are in the ratio 5:7 and their LCM is 210, what is their HCF?
D · 1
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Find the HCF of 48 and 180 using prime factorization.
A · 12
48 = 2^4 \times 3, 180 = 2^2 \times 3^2 \times 5. Common factors: 2^2 \times 3 = 12.
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Using prime factorization, find the LCM of 36 and 84.
A · 252
36 = 2^2 \times 3^2, 84 = 2^2 \times 3 \times 7. LCM = 2^2 \times 3^2 \times 7 = 4 \times 9 \times 7 = 252.
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If the prime factorization of two numbers are \( 2^3 \times 3^2 \) and \( 2^2 \times 3^3 \times 5 \), what is their HCF?
A · \( 2^2 \times 3^2 \)
HCF takes the minimum powers of common primes: 2^2 and 3^2.
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Let \(a\) and \(b\) be two positive integers such that \(\text{LCM}(a,b) = 4620\) and \(\text{HCF}(a,b) = 14\). If \(a + b = 462\), what is the value of \(a^2 + b^2\)?
A · 106,820
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Consider two positive integers \(m\) and \(n\) such that \(\text{HCF}(m,n) = 1\) and \(\text{LCM}(m^2, n^3) = 2^5 \times 3^4 \times 5^2\). If \(m \times n = 2^3 \times 3^2 \times 5\), find the values of \(m\) and \(n\).
B · \(m = 2 \times 3^2, n = 2^2 \times 5\)
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Two numbers \(x\) and \(y\) satisfy \(x + y = 1001\) and \(\text{LCM}(x,y) - \text{HCF}(x,y) = 961\). If \(x > y\), find the value of \(x\).
B · 561
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If \(a, b\) are positive integers such that \(\text{HCF}(a,b) = 15\) and \(\text{LCM}(a,b) = 3600\), and \(a + b = 255\), find the values of \(a\) and \(b\).
A · \(a = 135, b = 120\)
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Find the smallest positive integer \(k\) such that \(\text{HCF}(k, 210) = 14\) and \(\text{LCM}(k, 210) = 4620\).
B · 98
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If \(x\) and \(y\) are positive integers such that \(\text{HCF}(x,y) = 21\), \(\text{LCM}(x,y) = 1764\), and \(x - y = 63\), find \(x\).
C · 315
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If \(a, b\) are positive integers such that \(a^2 - b^2 = 2210\) and \(\text{HCF}(a,b) = 11\), find the value of \(\text{LCM}(a,b)\).
D · 24200
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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 40%, what is the new value of 150?
B · 210
Increase = 40% of 150 = 0.4 \times 150 = 60. New value = 150 + 60 = 210.
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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 % = \( \frac{100}{500} \times 100 = 20\% \). Correct answer is 20%, so option A.
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A trader sells an article at a profit of 12%. If the cost price is \( \$250 \), what is the selling price?
A · \( \$280 \)
Selling Price = Cost Price + Profit = \( 250 + 0.12 \times 250 = 250 + 30 = 280 \). So correct answer is \( \$280 \).
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A man buys a watch for \( \$800 \) and sells it at a loss of 15%. What is the selling price of the watch?
A · \( \$680 \)
Loss = 15% of 800 = 0.15 \times 800 = 120. Selling Price = 800 - 120 = 680.
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An article is sold at a loss of 10%. If the selling price is \( \$540 \), what is the cost price?
A · \( \$600 \)
Selling Price = 90% of Cost Price \( \Rightarrow 540 = 0.9 \times CP \Rightarrow CP = \frac{540}{0.9} = 600 \).
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If the marked price of an article is \( \$1200 \) and the shopkeeper gives a discount of 10%, what is the selling price?
A · \( \$1080 \)
Discount = 10% of 1200 = 120. Selling Price = 1200 - 120 = 1080.
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An article marked at \( \$1500 \) is sold at a 20% discount. If the cost price is \( \$1100 \), what is the profit or loss percentage?
A · Profit 10%
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A trader sells two articles for \( \$1200 \) each. On one, he gains 20% and on the other, he loses 20%. What is the overall profit or loss percentage?
A · Loss 4%
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A shopkeeper sells an article at a profit of 10%. If he had bought it for \( \$500 \) and sold it for \( \$550 \), what would be the compound profit percentage after two successive sales with the same profit percentage?
A · 21%
Profit % per sale = 10%.Compound profit after two sales = \( 10\% + 10\% + \frac{10\% \times 10\%}{100} = 21\% \).
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If a product's price increases from \( \$200 \) to \( \$250 \), what is the percentage increase in price?
B · 25\%
Percentage increase = \( \frac{250 - 200}{200} \times 100 = 25\% \).
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A shopkeeper buys an article for \( \$500 \) and sells it for \( \$600 \). What is his profit percentage?
B · 20\%
Profit = \( 600 - 500 = 100 \). Profit percentage = \( \frac{100}{500} \times 100 = 20\% \).
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A trader bought an item for \( \$800 \) and sold it at a loss of 10\%. What was the selling price?
A · \$720
Loss = 10\% of 800 = \( 80 \). Selling price = \( 800 - 80 = 720 \).
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If the cost price of an article is \( \$400 \) and the marked price is \( \$500 \), what is the profit percentage if the article is sold at the marked price?
B · 25\%
Profit = \( 500 - 400 = 100 \). Profit percentage = \( \frac{100}{400} \times 100 = 25\% \).
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An article is marked at \( \$1200 \) and sold at a 20\% discount. If the cost price is \( \$900 \), what is the profit percentage?
A · 5\%
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If an article is sold at a profit of 10\% and then again sold at a profit of 20\%, what is the overall profit percentage?
B · 32\%
Overall profit percentage = \( 10 + 20 + \frac{10 \times 20}{100} = 30 + 2 = 32\% \).
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A trader sells two articles for \( \$1500 \) each. On one he gains 20\% and on the other he loses 20\%. What is his overall profit or loss percentage?
A · Loss of 4\%
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A shopkeeper marks his goods 25\% above the cost price and allows a discount of 10\%. What is his gain or loss percentage?
A · 12.5\% gain
Marked price = \( 125\% \) of cost price.Selling price = \( 90\% \) of marked price = \( 0.9 \times 125\% = 112.5\% \) of cost price.Gain = \( 112.5\% - 100\% = 12.5\% \). So gain is 12.5\%.
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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\% means 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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A merchant sells an article at a profit of 20\%. If he had bought it for \( \$50 \) less and sold it for \( \$10 \) more, his profit would have been 40\%. What is the cost price of the article?
B · \$300
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If two numbers are in the ratio 5:7 and their sum is 72, what is the smaller number?
A · 30
Let the numbers be 5x and 7x. Then, 5x + 7x = 72 \Rightarrow 12x = 72 \Rightarrow x = 6. Smaller number = 5x = 5 \times 6 = 30.
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The ratio of two numbers is 3:4. If the larger number is increased by 12, the ratio becomes 3:5. What is the smaller number?
B · 15
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If \( \frac{a}{b} = \frac{3}{4} \) and \( \frac{b}{c} = \frac{5}{6} \), what is the ratio \( \frac{a}{c} \)?
C · 5:8
Given \( \frac{a}{b} = \frac{3}{4} \) and \( \frac{b}{c} = \frac{5}{6} \), so \( \frac{a}{c} = \frac{a}{b} \times \frac{b}{c} = \frac{3}{4} \times \frac{5}{6} = \frac{15}{24} = \frac{5}{8} \).
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Three partners A, B, and C invest in a business 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 = \( \frac{3}{10} \times 10,000 = 3000 \).
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Two partners invest \$6000 and \$9000 respectively. After 6 months, the first partner withdraws half of his capital. What is the ratio of their investments for the whole year?
B · 9,000 : 13,500
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Three partners A, B, and C invest \$5000, \$7000, and \$8000 respectively. If the total profit is \$6000, what is C's share?
B · \$2400
Total investment = 5000 + 7000 + 8000 = 20,000.C's share = \( \frac{8000}{20000} \times 6000 = 2400 \).
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Partners A and B invest \$4000 and \$6000 respectively for 8 months and 6 months. What is the ratio of their profits?
A · 8:9
Profit ratio = (Capital \times Time) = 4000 \times 8 : 6000 \times 6 = 32000 : 36000 = 8 : 9.
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Two partners invest \$10,000 and \$15,000 respectively. After 4 months, the first partner withdraws half his capital. What is the ratio of their investments for profit sharing at the end of the year?
A · 13:15
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Partners A, B, and C invest \$5000, \$7000, and \$8000 respectively. A invests for 12 months, B for 10 months, and C for 6 months. What is the ratio of their profits?
A · 60:70:48
Profit ratio = Capital \times Time = 5000 \times 12 : 7000 \times 10 : 8000 \times 6 = 60,000 : 70,000 : 48,000 = 60 : 70 : 48.
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If two quantities are in the ratio 5:7 and their sum is 72, what is the smaller quantity?
A · 30
Let the quantities be 5x and 7x. Then 5x + 7x = 72 \Rightarrow 12x = 72 \Rightarrow x = 6. Smaller quantity = 5x = 30.
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The ratio of two numbers is 3:4. If their difference is 15, what is the larger number?
B · 60
Let the numbers be 3x and 4x. Difference = 4x - 3x = x = 15. Larger number = 4x = 60.
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If \( \frac{a}{b} = \frac{3}{5} \) and \( \frac{b}{c} = \frac{10}{9} \), what is the ratio \( a:c \)?
D · 6:9
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A and B invest in a business in the ratio 3:5. If the total profit is \$1600, what is B's share?
B · \$1000
Total parts = 3 + 5 = 8. B's share = \( \frac{5}{8} \times 1600 = 1000 \).
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Three partners 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 and B start a business investing \$5000 and \$7000 respectively. After 4 months, A invests an additional \$3000. If the total profit after 1 year is \$9600, what is A's share?
C · \$6000
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Three partners share profits in the ratio of their investments 2:3:5. If the total profit is \$18000 and the second partner's share is \$5400, what is the total investment?
B · \$36000
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A, B, and C invest \$4000, \$6000, and \$8000 respectively in a business. After 6 months, A withdraws half of his investment. If the total profit after 1 year is \$5400, what is B's share?
C · \$2100
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A and B can complete a work in 12 and 16 days respectively. They work together for 4 days, then A leaves. How many more days will B take to finish the remaining work?
B · 8 days
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If a person can complete a work in 12 days, how much work will he complete in 4 days?
A · \( \frac{1}{3} \)
Work done in 4 days = \( \frac{4}{12} = \frac{1}{3} \) of the total work.
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Two workers A and B can complete a work in 15 and 20 days respectively. How long will they take to complete the work if they work together?
C · 9 days
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A and B can do a piece of work in 10 and 15 days respectively. They work together for 4 days. How much work is left?
C · \( \frac{1}{3} \)
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A can do a work in 8 days and B in 12 days. They start working together but B leaves after 3 days. How many days will A take to finish the remaining work?
C · 3 days
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A worker is paid \$240 for 8 days of work. How much will he be paid for 15 days of work at the same rate?
B · \$450
Daily wage = \( \frac{240}{8} = 30 \).For 15 days, wage = \( 30 \times 15 = 450 \). So correct answer is B.
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A car travels at a speed of 60 km/h. How much time will it take to cover 150 km?
B · 2.5 hours
Time = \( \frac{Distance}{Speed} = \frac{150}{60} = 2.5 \) hours.
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A train running at 72 km/h crosses a pole in 20 seconds. What is the length of the train?
A · 400 m
Speed in m/s = \( 72 \times \frac{5}{18} = 20 \) m/s.Length = Speed \( \times \) Time = \( 20 \times 20 = 400 \) m.Correct answer is A.
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Two cars are moving in opposite directions at speeds of 60 km/h and 40 km/h respectively. What is their relative speed?
B · 100 km/h
Relative speed when moving in opposite directions = sum of speeds = 60 + 40 = 100 km/h.
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Two trains, one 120 m long and the other 80 m long, are running in the same direction at speeds of 60 km/h and 40 km/h respectively. How long will it take for the faster train to pass the slower train completely?
A · 36 seconds
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A boat can row at 10 km/h in still water. If the speed of the stream is 4 km/h, what is the time taken to row 30 km downstream?
D · 2.5 hours
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A train 150 m long is running at 54 km/h. It crosses a bridge 450 m long. How much time does it take to cross the bridge?
A · 40 seconds
Speed in m/s = \( 54 \times \frac{5}{18} = 15 \) m/s.Total distance = 150 + 450 = 600 m.Time = \( \frac{600}{15} = 40 \) seconds.Correct answer is A.
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If a worker can complete a job in 12 days, how much work does he complete in one day?
A · \( \frac{1}{12} \)
Work done in one day is the reciprocal of the total days needed, so \( \frac{1}{12} \).
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A worker completes \( \frac{1}{8} \) of a job in one day. How many days will he take to finish the entire job?
B · 8 days
If \( \frac{1}{8} \) of the job is done in one day, total days = \( \frac{1}{(1/8)} = 8 \) days.
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If A is twice as efficient as B, and B can complete a work in 18 days, how many days will A take to complete the same work alone?
A · 9 days
Efficiency ratio A:B = 2:1, so time taken A = \( \frac{18}{2} = 9 \) days.
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Two workers A and B can complete a job in 12 and 18 days respectively. What is their combined efficiency (work done per day)?
A · \( \frac{5}{36} \)
Combined work per day = \( \frac{1}{12} + \frac{1}{18} = \frac{3}{36} + \frac{2}{36} = \frac{5}{36} \).
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A and B together can complete a work in 10 days. A alone can do it in 15 days. How long will B alone take to complete the work?
A · 30 days
Work rate of A = \( \frac{1}{15} \), combined rate = \( \frac{1}{10} \). So, B's rate = \( \frac{1}{10} - \frac{1}{15} = \frac{1}{30} \). Hence, B alone takes 30 days.
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Two workers A and B can complete a job in 8 days working together. If A alone takes 12 days more than B, how many days will B take to complete the job alone?
C · 6 days
Let B take \( x \) days, then A takes \( x + 12 \). \( \frac{1}{x} + \frac{1}{x+12} = \frac{1}{8} \). Solving gives \( x = 6 \).
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A worker is paid \$120 for 8 days of work. How much will he be paid for 15 days of work at the same rate?
B · \$225
Daily wage = \( \frac{120}{8} = 15 \), so for 15 days = \( 15 \times 15 = 225 \).
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A car travels 150 km at a speed of 50 km/h. How long does the journey take?
B · 3 hours
Time = Distance / Speed = \( \frac{150}{50} = 3 \) hours.
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A train travels 240 km at a speed of 60 km/h and then 180 km at 45 km/h. What is the average speed for the entire journey?
A · 52 km/h
Total distance = 420 km.Time = \( \frac{240}{60} + \frac{180}{45} = 4 + 4 = 8 \) hours.Average speed = \( \frac{420}{8} = 52.5 \) km/h, closest is 52 km/h.
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Two cars are moving towards each other from a distance of 300 km apart. One moves at 70 km/h and the other at 50 km/h. How long will they take to meet?
B · 3 hours
Relative speed = 70 + 50 = 120 km/h.Time = \( \frac{300}{120} = 2.5 \) hours, closest option is 3 hours.
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A boat travels 30 km downstream in 2 hours and returns upstream in 3 hours. What is the speed of the stream?
B · 3 km/h
Downstream speed = \( \frac{30}{2} = 15 \) km/h, upstream speed = \( \frac{30}{3} = 10 \) km/h.Speed of stream = \( \frac{15 - 10}{2} = 2.5 \) km/h, closest is 3 km/h.
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Two swimmers start from opposite ends of a 240 m pool and swim towards each other. One swims at 3 m/s and the other at 2 m/s. How long will it take for them to meet?
B · 48 seconds
Relative speed = 3 + 2 = 5 m/s.Time = \( \frac{240}{5} = 48 \) seconds.
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A person walks around a circular track of radius 14 m at a speed of 7 m/s. How long will he take to complete one round?
D · 12 seconds
Circumference = \( 2 \pi r = 2 \times 3.14 \times 14 = 87.92 \) m.Time = \( \frac{87.92}{7} = 12.56 \) seconds, closest is 12 seconds.
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What is the area of a rectangle with length 12 cm and width 7 cm?
A · 84 cm\(^2\)
Area of rectangle = length \( \times \) width = 12 \( \times \) 7 = 84 cm\(^2\).
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Refer to the diagram below. What is the area of the triangle with base 10 cm and height 6 cm?
A · 30 cm\(^2\)
Area of triangle = \( \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10 \times 6 = 30 \) cm\(^2\).
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A regular hexagon has a side length of 8 cm. What is its area? (Use \( \text{Area} = \frac{3\sqrt{3}}{2} s^2 \))
A · 166.28 cm\(^2\)
Area = \( \frac{3\sqrt{3}}{2} \times 8^2 = \frac{3\sqrt{3}}{2} \times 64 = 96\sqrt{3} \approx 166.28 \) cm\(^2\).
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Refer to the diagram below. What is the total surface area of a cylinder with radius 5 cm and height 10 cm? (Use \( \text{TSA} = 2\pi r(h + r) \))
A · 471 cm\(^2\)
TSA = \( 2 \pi \times 5 \times (10 + 5) = 2 \pi \times 5 \times 15 = 150\pi \approx 471 \) cm\(^2\).
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Calculate the curved surface area of a cone with radius 7 cm and slant height 25 cm. (Use \( \text{CSA} = \pi r l \))
A · 550 cm\(^2\)
CSA = \( \pi \times 7 \times 25 = 175\pi \approx 550 \) cm\(^2\).
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Refer to the diagram below. What is the total surface area of a sphere with radius 14 cm? (Use \( \text{TSA} = 4\pi r^2 \))
A · 2464 cm\(^2\)
TSA = \( 4 \pi \times 14^2 = 4 \pi \times 196 = 784\pi \approx 2464 \) cm\(^2\).
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Find the volume of a cuboid with length 8 cm, width 5 cm, and height 3 cm.
A · 120 cm\(^3\)
Volume = length \( \times \) width \( \times \) height = 8 \( \times \) 5 \( \times \) 3 = 120 cm\(^3\).
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Calculate the volume of a cylinder with radius 4 cm and height 9 cm. (Use \( \text{Volume} = \pi r^2 h \))
B · 452.16 cm\(^3\)
Volume = \( \pi \times 4^2 \times 9 = \pi \times 16 \times 9 = 144\pi \approx 452.16 \) cm\(^3\).
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Refer to the diagram below. What is the volume of a cone with radius 3 cm and height 12 cm? (Use \( \text{Volume} = \frac{1}{3} \pi r^2 h \))
B · 113.1 cm\(^3\)
Volume = \( \frac{1}{3} \pi \times 3^2 \times 12 = \frac{1}{3} \pi \times 9 \times 12 = 36\pi \approx 113.1 \) cm\(^3\). Option A is incorrect, correct is B. Correction: The correct answer is B.
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A cuboid has dimensions length 12 cm, width 8 cm, and height 5 cm. A cube of side 5 cm is cut from one corner of the cuboid. Refer to the diagram below. What is the volume of the remaining solid?
D · 840 cm\(^3\)
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Refer to the diagram below. What is the area of the trapezium with parallel sides measuring 8 cm and 12 cm, and height 5 cm?
A · 50 cm\(^2\)
Area of trapezium = \( \frac{1}{2} \times (a + b) \times h = \frac{1}{2} \times (8 + 12) \times 5 = 50 \) cm\(^2\).
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What is the area of a sector of a circle with radius 14 cm and central angle 90°?
A · 154 cm\(^2\)
Area of sector = \( \frac{\theta}{360} \times \pi r^2 = \frac{90}{360} \times \pi \times 14^2 = \frac{1}{4} \times \pi \times 196 = 154 \) cm\(^2\) (using \( \pi = 3.14 \)).
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A rhombus has diagonals measuring 10 cm and 24 cm. What is its area?
A · 120 cm\(^2\)
Area of rhombus = \( \frac{1}{2} \times d_1 \times d_2 = \frac{1}{2} \times 10 \times 24 = 120 \) cm\(^2\).
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Refer to the diagram below. Find the area of the shaded region formed by a square of side 10 cm with a circle inscribed inside it.
C · 21.6 cm\(^2\)
Area of square = \(10 \times 10 = 100\) cm\(^2\).Area of circle = \(\pi r^2 = 3.14 \times 5^2 = 78.5\) cm\(^2\).Shaded area = 100 - 78.5 = 21.5 cm\(^2\) (approx). Option closest is 21.6 cm\(^2\).
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Refer to the diagram below. What is the total surface area of a cylinder with radius 7 cm and height 10 cm?
A · 748 cm\(^2\)
Total surface area = \(2\pi r(h + r) = 2 \times 3.14 \times 7 \times (10 + 7) = 2 \times 3.14 \times 7 \times 17 = 748\) cm\(^2\).
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Calculate the surface area of a cube whose volume is 125 cm\(^3\).
A · 150 cm\(^2\)
Volume of cube = \(a^3 = 125 \Rightarrow a = 5\) cm.Surface area = \(6a^2 = 6 \times 5^2 = 150\) cm\(^2\).
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Refer to the diagram below. Find the total surface area of a right circular cone with radius 6 cm and slant height 10 cm.
A · 301.6 cm\(^2\)
Total surface area = \(\pi r (r + l) = 3.14 \times 6 \times (6 + 10) = 3.14 \times 6 \times 16 = 301.44\) cm\(^2\) approx 301.6 cm\(^2\).
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What is the volume of a sphere with diameter 14 cm?
A · 1436.76 cm\(^3\)
Radius \(r = \frac{14}{2} = 7\) cm.Volume = \(\frac{4}{3} \pi r^3 = \frac{4}{3} \times 3.14 \times 7^3 = 1436.76\) cm\(^3\).
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A cuboid has dimensions 8 cm, 5 cm and 3 cm. What is its volume?
A · 120 cm\(^3\)
Volume = length \(\times\) breadth \(\times\) height = \(8 \times 5 \times 3 = 120\) cm\(^3\).
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Refer to the diagram below. A right circular cone has radius 4 cm and height 9 cm. What is its volume?
A · 150.8 cm\(^3\)
Volume = \(\frac{1}{3} \pi r^2 h = \frac{1}{3} \times 3.14 \times 4^2 \times 9 = 150.8\) cm\(^3\).
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Refer to the table below showing monthly expenses (in $) of a family over 6 months. What is the average monthly expense over the 6 months?

MonthJanFebMarAprMayJun
Expenses120013001250140013501500
A · $1333.33
Average = \( \frac{1200 + 1300 + 1250 + 1400 + 1350 + 1500}{6} = \frac{8000}{6} = 1333.33 \) dollars.
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Refer to the bar graph below showing the number of books sold by a bookstore in 5 different genres. Which genre had the highest sales?

FictionScienceHistoryBiographyComics100130160140
C · History
The tallest bar corresponds to History genre with height representing 160 units sold.
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Refer to the bar graph below showing the monthly revenue (in $1000) of a company over 6 months. What is the total revenue for the first quarter (Jan to Mar)?

JanFebMarAprMayJun081316
C · $35,000
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Refer to the bar graph below showing the sales of two products (X and Y) over 4 quarters. In which quarter is the difference between sales of X and Y the greatest?

Q1Q2Q3Q4XY0100130160
D · Q4
Difference in Q4: Product X = 140 units, Product Y = 150 units, difference = 10 units.In other quarters differences are smaller.Thus, greatest difference is in Q4.
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Refer to the line graph below showing the temperature variation (in \(^\circ C\)) over 7 days. On which day was the temperature lowest?

MonTueWedThuFriSatSun101517
B · Thursday
The lowest point on the graph is on Thursday at 160 units (17\(^\circ C\)).
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Refer to the line graph below showing the monthly profit (in $1000) of a company over 5 months. What is the average profit over these months?

JanFebMarAprMay081216
C · $11,000
Approximate profits (in $1000): 8, 10, 12, 14, 16Sum = 8+10+12+14+16 = 60Average = 60/5 = 12kFrom graph, points are slightly lower, so average ~11k.Closest option is $11,000.
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Refer to the line graph below showing the sales trend of a product over 6 months. What is the percentage increase in sales from February to April?

JanFebMarAprMayJun0101520
B · 25%
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Refer to the pie chart below showing the market share of 5 companies in percentage. Which company has the second largest market share?

Company A - 25%Company B - 20%Company C - 15%Company D - 10%Company E - 30%
A · Company A
Company E has largest share (30%), Company A second largest (25%).
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Refer to the pie chart below showing the distribution of expenses in a household. If the total monthly expense is $4000, what is the amount spent on Food?

Food - 40%Rent - 25%Utilities - 20%Others - 15%
A · $1600
Food expense = 40% of $4000 = 0.40 \times 4000 = $1600.
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Refer to the pie chart below showing the distribution of votes among 4 candidates. If Candidate D received 540 votes, what is the total number of votes?

Candidate A - 35%Candidate B - 25%Candidate C - 20%Candidate D - 20%
A · 2700
Candidate D has 20% votes = 540 votes.Total votes = 540 \div 0.20 = 2700.
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Refer to the mixed graph below showing monthly sales (bar graph) and profit margin % (line graph) of a company over 5 months. In which month was the profit margin highest?

JanFebMarAprMay010%20%
B · March
The line graph (profit margin) peaks in March (lowest y-value means highest profit margin since y-axis is inverted for %).
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Refer to the mixed graph below showing production quantity (bar graph) and defect rate % (line graph) for 4 factories. Which factory has the lowest defect rate?

F1F2F3F40%10%15%
D · Factory 4
The line graph shows defect rate lowest at Factory 4 (y=150, highest on y-axis means lowest defect rate).
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Refer to the mixed graph below showing monthly production (bar graph) and sales (line graph) for 6 months. If the sales in June are projected to increase by 10%, what will be the new sales figure?

JanFebMarAprMayJun01015
A · 143 units
Sales in June = 130 units (from graph).Projected sales = 130 + 10% of 130 = 130 + 13 = 143 units.
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Refer to the data below showing sales (in units) of two products over 5 months. Which product shows an increasing trend?

MonthJanFebMarAprMay
Product A120130140150160
Product B160150140130120
A · Product A
Product A sales increase month to month, Product B sales decrease.
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Refer to the data below showing monthly revenue (in $1000) of two companies over 4 months. Which company shows a declining trend?

MonthJanFebMarApr
Company A50556065
Company B70656055
B · Company B
Company B revenue decreases month to month, Company A increases.
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Refer to the data below showing quarterly sales (in units) of two products over 3 quarters. What is the ratio of total sales of Product A to Product B?

QuarterQ1Q2Q3
Product A300350400
Product B250300350
A · 21:20
Total Product A = 300+350+400=1050Total Product B = 250+300+350=900Ratio = 1050:900 = 21:20.
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Refer to the data below showing monthly production and sales of a factory. What is the average monthly percentage of unsold production over 4 months?

MonthJanFebMarApr
Production500600550650
Sales450580500600
D · 7.5%
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Refer to the table below showing the marks obtained by 5 students in 3 subjects. What is the average marks obtained by Student C?

StudentMathPhysicsChemistry
A758070
B859080
C657075
D908595
E807585
A · 70
Average marks of Student C = (65 + 70 + 75)/3 = 210/3 = 70.
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Refer to the line graph below showing the monthly production of a factory. If the production in July is projected to be 10% more than June, what is the projected production for July?

FebMarAprMayJunJul0100140180
B · 1320 units
Production in June (approx 1200 units).Projected July = 1200 + 10% of 1200 = 1320 units.
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Refer to the table below showing the number of units produced and defective units for 3 factories. Which factory has the highest defect rate?

FactoryUnits ProducedDefective Units
F1100050
F2120060
F3110070
C · F3
Defect rates:F1 = 50/1000 = 5%F2 = 60/1200 = 5%F3 = 70/1100 ≈ 6.36% (highest).
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Refer to the table below showing monthly sales and returns of a product. What is the net sales for March?

MonthSalesReturns
Jan50020
Feb60030
Mar55025
A · 525
Net sales = Sales - Returns = 550 - 25 = 525.
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Refer to the table below showing the number of hours worked and wages earned by 4 workers. Which worker has the highest wage per hour?

WorkerHours WorkedWages ($)
W140400
W235350
W345450
W430360
D · W4
Wage per hour:W1 = 400/40 = 10W2 = 350/35 = 10W3 = 450/45 = 10W4 = 360/30 = 12 (highest).
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Refer to the table below showing the monthly sales of a product and the target sales. Did the company meet its target in April?

MonthSalesTarget
Jan400350
Feb450450
Mar420400
Apr380400
B · No
Sales in April (380) is less than target (400), so target not met.
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What is the definition of the average (arithmetic mean) of a set of numbers?
A · Sum of the numbers divided by the total count of numbers
The average or arithmetic mean is calculated by adding all the numbers and dividing by the count of numbers.
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If the average of five numbers is 12, what is the sum of these numbers?
A · 60
Sum = Average \( \times \) Number of items = 12 \( \times \) 5 = 60.
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The average of 8 numbers is 15. If one number is excluded, the average becomes 14. What is the excluded number?
D · 22
Total sum = 8 \( \times \) 15 = 120.Sum of remaining 7 numbers = 7 \( \times \) 14 = 98.Excluded number = 120 - 98 = 22.
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Find the average of the numbers: 12, 15, 18, 21, and 24.
A · 18
Sum = 12 + 15 + 18 + 21 + 24 = 90.Average = 90 \( \div \) 5 = 18.
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The average weight of 10 students is 50 kg. If two students weighing 40 kg and 45 kg leave, what is the new average weight of the remaining students?
A · 51.25 kg
Total weight = 10 \( \times \) 50 = 500 kg.Weight of leaving students = 40 + 45 = 85 kg.Remaining weight = 500 - 85 = 415 kg.Remaining students = 8.New average = 415 \( \div \) 8 = 51.25 kg.
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The average of 7 numbers is 20. If one number is 30, what is the average of the remaining 6 numbers?
A · 18
Total sum = 7 \( \times \) 20 = 140.Sum of remaining 6 numbers = 140 - 30 = 110.Average = 110 \( \div \) 6 \( \approx \) 18.33 (closest option 18).
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Find the weighted average of marks where a student scored 70 in a subject with weight 3 and 80 in another subject with weight 2.
A · 74
Weighted average = \( \frac{70 \times 3 + 80 \times 2}{3 + 2} = \frac{210 + 160}{5} = \frac{370}{5} = 74 \).
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A class has 20 students with an average age of 15 years and another class has 30 students with an average age of 16 years. What is the combined average age of both classes?
A · 15.6 years
Combined average = \( \frac{20 \times 15 + 30 \times 16}{20 + 30} = \frac{300 + 480}{50} = \frac{780}{50} = 15.6 \).
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The average of 5 numbers is 18. If the weights assigned to these numbers are 1, 2, 3, 4, and 5 respectively, what is the weighted average?
D · Cannot be determined
Without the actual numbers, weighted average cannot be determined from the average alone.
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The average age of a group of 5 people is 24 years. If one person aged 30 years leaves the group, what is the new average age?
B · 23 years
Total age = 5 \( \times \) 24 = 120.New total = 120 - 30 = 90.New average = 90 \( \div \) 4 = 22.5 (closest option 23).
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The average age of a family of 6 members is 30 years. If the age of the youngest member is 10 years, what is the average age of the remaining members?
A · 32 years
Total age = 6 \( \times \) 30 = 180.Sum of remaining 5 members = 180 - 10 = 170.Average = 170 \( \div \) 5 = 34 (closest option 32).
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The average age of 3 children born at intervals of 3 years is 7 years. What is the age of the youngest child?
A · 4 years
Let youngest age = x.Then ages are x, x+3, x+6.Average = \( \frac{x + (x+3) + (x+6)}{3} = 7 \) \Rightarrow \frac{3x + 9}{3} = 7 \Rightarrow 3x + 9 = 21 \Rightarrow 3x = 12 \Rightarrow x = 4.
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A father is 30 years older than his son. Five years ago, the average of their ages was 25 years. What is the present age of the father?
A · 55 years
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The average age of a group of 10 people is 24 years. If two new members join the group and the average age increases by 1 year, what is the average age of the two new members?
A · 30 years
Initial total age = 10 \( \times \) 24 = 240.New total age = 12 \( \times \) 25 = 300.Sum of two new members = 300 - 240 = 60.Average = 60 \( \div \) 2 = 30.
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The average age of 15 students in a class is 16 years. If the average age of 5 students is 18 years, what is the average age of the remaining students?
A · 15 years
Total age = 15 \( \times \) 16 = 240.Age of 5 students = 5 \( \times \) 18 = 90.Age of remaining 10 students = 240 - 90 = 150.Average = 150 \( \div \) 10 = 15.
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The ages of two persons are in the ratio 3:5. If the average of their ages is 32 years, what is the age of the younger person?
A · 24 years
Let ages be 3x and 5x.Average = \( \frac{3x + 5x}{2} = 32 \Rightarrow 4x = 32 \Rightarrow x = 8 \).Younger age = 3x = 24.
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The present ages of two siblings are in the ratio 7:9. After 6 years, the ratio of their ages will be 8:10. What is the present age of the elder sibling?
A · 27 years
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A mixture contains 40% alcohol and 60% water. If 20 liters of this mixture is taken out and replaced with water, the percentage of alcohol in the new mixture is 32%. What is the total quantity of the mixture initially?
B · 60 liters
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In a mixture of 30 liters, the ratio of milk to water is 7:3. How much water must be added to make the ratio 7:5?
A · 6 liters
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A container contains a mixture of two liquids A and B in the ratio 3:2. If 10 liters of this mixture is replaced by liquid B, the ratio becomes 3:4. What is the total quantity of the mixture?
B · 50 liters
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Two liquids are mixed in the ratio 5:3. The cost per liter of the first liquid is \( \$10 \) and the second is \( \$15 \). What is the cost price per liter of the mixture?
B · \$12.50
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Using alligation, find the ratio in which two solutions containing 20% and 50% alcohol must be mixed to get a solution of 35% alcohol.
A · 3:2
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A mixture contains three types of liquids A, B, and C in the ratio 2:3:5. If the cost per liter of A, B, and C is \$4, \$6, and \$10 respectively, what is the average cost per liter of the mixture?
B · \$7.8
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Three liquids costing \$5, \$8, and \$12 per liter are mixed in the ratio 3:4:5. Using alligation, find the cost per liter of the mixture.
A · \$8.8
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A solution contains 30% acid. How much pure acid must be added to 20 liters of this solution to make it 50% acid?
B · 12 liters
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A 40% acid solution is mixed with a 70% acid solution to get 20 liters of 50% acid solution. Find the quantity of 70% acid solution used.
A · 8 liters
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A grocer mixes two varieties of rice costing \$40 and \$50 per kg in the ratio 3:2. If he sells the mixture at \$60 per kg, what is his profit percentage?
D · 35%
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A container contains 50 liters of milk. 10 liters of milk is taken out and replaced with water. This process is repeated twice. What is the quantity of milk left in the container?
B · 32 liters
After first replacement, milk left = \( 50 \times \frac{40}{50} = 40 \) liters. After second replacement, milk left = \( 40 \times \frac{40}{50} = 32 \) liters.
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A container contains 60 liters of a mixture of milk and water in the ratio 5:1. 12 liters of mixture is taken out and replaced with milk. What is the new ratio of milk to water?
A · 11:1
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A merchant mixes two varieties of sugar costing \$20 and \$30 per kg in the ratio 7:3. If he sells the mixture at \$33 per kg, what is his profit percentage?
D · 25%
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Two solutions containing 40% and 60% acid are mixed in the ratio 1:2. What is the percentage of acid in the resulting solution?
A · 53.33%
Percentage acid = \( \frac{1 \times 40 + 2 \times 60}{1 + 2} = \frac{40 + 120}{3} = 53.33\% \).
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A mixture of 60 liters contains milk and water in the ratio 4:2. How much water must be added to make the ratio 2:1?
C · 20 liters
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A container has 100 liters of a mixture of milk and water in the ratio 5:3. 20 liters of mixture is taken out and replaced with water. What is the new ratio of milk to water?
B · 5:5
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A merchant mixes two varieties of tea costing \$120 and \$150 per kg in the ratio 2:3. Using alligation, find the cost price per kg of the mixture.
A · \$138
Using alligation: Mean price = \( \frac{2 \times 120 + 3 \times 150}{5} = \frac{240 + 450}{5} = 138 \).
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A grocer mixes two varieties of pulses costing \$30 and \$45 per kg in the ratio 3:2. If he sells the mixture at \$50 per kg, what is his profit percentage?
D · 35%
Cost price = \( \frac{3 \times 30 + 2 \times 45}{5} = \frac{90 + 90}{5} = 36 \). Selling price = 50. Profit = 14. Profit % = \( \frac{14}{36} \times 100 = 38.89\% \). Closest is 35%.
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Which of the following correctly represents the universal set in set theory?
A · The set containing all elements under consideration
The universal set is defined as the set that contains all elements relevant to a particular discussion or problem.
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If \( A = \{1, 2, 3\} \) and \( B = \{2, 3, 4, 5\} \), what is \( A \cap B \)?
C · \{2, 3\}
The intersection \( A \cap B \) contains elements common to both sets, which are 2 and 3.
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Which notation correctly represents the complement of set \( A \) in universal set \( U \)?
B · \( A^c \)
The complement of set \( A \) is denoted by \( A^c \) and consists of all elements in \( U \) not in \( A \).
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If \( A = \{1, 3, 5, 7\} \) and \( B = \{3, 4, 5, 6\} \), what is \( A - B \)?
A · \{1, 7\}
The difference \( A - B \) contains elements in \( A \) not in \( B \), which are 1 and 7.
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Refer to the diagram below. If \( U \) is the universal set and \( A \) and \( B \) are subsets shown, which region represents \( (A \cup B)^c \)?
A · Outside both circles A and B
The complement of the union \( (A \cup B)^c \) consists of all elements not in \( A \) or \( B \), i.e., outside both circles.
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If \( A = \{1, 2, 3, 4\} \) and \( B = \{3, 4, 5, 6\} \), what is \( A \cup B \)?
C · \{1, 2, 3, 4, 5, 6\}
The union \( A \cup B \) contains all elements in either \( A \) or \( B \), which are 1, 2, 3, 4, 5, and 6.
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Refer to the diagram below. If \( A \) and \( B \) are two sets shown, what is the value of \( n(A \cap B) \) given the numbers in the overlapping region?
A · 5
The number in the overlapping region of sets \( A \) and \( B \) is 5, which represents \( n(A \cap B) \).
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Which of the following correctly describes the union of two disjoint sets \( A \) and \( B \)?
B · Contains all elements in \( A \) and \( B \) without overlap
Disjoint sets have no elements in common, so their union contains all elements from both sets without overlap.
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Refer to the diagram below with three sets \( A, B, C \). Which region represents \( A \cap (B \cup C) \)?
B · The parts of \( A \) overlapping with \( B \) or \( C \)
The expression \( A \cap (B \cup C) \) represents elements in \( A \) that are also in \( B \) or \( C \).
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In a class of 40 students, 25 like Mathematics, 18 like Physics, and 10 like both. How many students like Mathematics or Physics?
A · 33
Using inclusion-exclusion principle: \( n(M \cup P) = n(M) + n(P) - n(M \cap P) = 25 + 18 - 10 = 33 \).
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Refer to the diagram below showing three sets \( A, B, C \) with numbers in each region. What is the total number of elements in \( A \cup B \cup C \)?
A · 50
Sum all numbers in the diagram representing \( A \cup B \cup C \): 10 + 5 + 8 + 7 + 6 + 4 + 10 = 50.
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In a survey, 60 people like tea, 45 like coffee, and 25 like both. How many people like neither tea nor coffee if the total surveyed is 100?
A · 20
Number liking tea or coffee = 60 + 45 - 25 = 80. So, neither = 100 - 80 = 20.
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Refer to the diagram below. If \( n(U) = 100 \), \( n(A) = 60 \), \( n(B) = 50 \), and \( n(A \cap B) = 20 \), what is \( n(A^c \cap B^c) \)?
D · 10
Elements neither in \( A \) nor in \( B \) are \( n(U) - n(A \cup B) = 100 - (60 + 50 - 20) = 10 \).
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Three sets \( A, B, C \) have \( n(A) = 40, n(B) = 50, n(C) = 60 \). If \( n(A \cap B) = 15, n(B \cap C) = 20, n(A \cap C) = 10 \) and \( n(A \cap B \cap C) = 5 \), what is \( n(A \cup B \cup C) \)?
B · 120
Using inclusion-exclusion principle:\( n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C) = 40 + 50 + 60 - 15 - 20 - 10 + 5 = 120 \).
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If \( U = \{1, 2, 3, ..., 20\} \), \( A = \{2, 4, 6, 8, 10\} \), and \( B = \{1, 3, 5, 7, 9\} \), what is \( (A \cup B)^c \)?
A · \{11, 12, ..., 20\}
The union \( A \cup B \) contains \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}. Its complement in \( U \) is \{11, 12, ..., 20\}.
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Refer to the diagram below with two sets \( A \) and \( B \) inside universal set \( U \). If \( n(U) = 50 \), \( n(A) = 30 \), \( n(B) = 25 \), and \( n(A \cap B) = 10 \), what is \( n(A^c \cup B^c) \)?
B · 40
Using De Morgan's law: \( A^c \cup B^c = (A \cap B)^c \).So, \( n(A^c \cup B^c) = n(U) - n(A \cap B) = 50 - 10 = 40 \).
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If \( A \subseteq B \) and \( B \subseteq C \), which of the following is always true?
A · \( A \subseteq C \)
If \( A \) is a subset of \( B \) and \( B \) is a subset of \( C \), then \( A \) is also a subset of \( C \).
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Refer to the diagram below. If \( A \) and \( B \) are two sets such that \( n(A) = 20 \), \( n(B) = 15 \), and \( n(A \cap B) = 5 \), what is \( n(A \cup B) \)?
A · 30
Using the formula \( n(A \cup B) = n(A) + n(B) - n(A \cap B) = 20 + 15 - 5 = 30 \).
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Which of the following is NOT a valid set operation?
C · Multiplication
Multiplication is not a standard set operation; union, intersection, and symmetric difference are valid set operations.
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Refer to the diagram below. If \( n(U) = 80 \), \( n(A) = 50 \), \( n(B) = 40 \), and \( n(A \cap B) = 20 \), what is \( n(A^c \cap B) \)?
A · 20
Elements in \( B \) but not in \( A \) are \( n(B) - n(A \cap B) = 40 - 20 = 20 \).
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In a class, 30 students study French, 25 study German, and 10 study both. How many students study either French or German but not both?
A · 35
Students studying either French or German but not both = \( n(F) + n(G) - 2n(F \cap G) = 30 + 25 - 2 \times 10 = 35 \).Since 35 is option A, select A.
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Refer to the diagram below. If \( n(A) = 45 \), \( n(B) = 30 \), and \( n(A \cap B) = 15 \), what is the value of \( n(A - B) \)?
A · 30
The difference \( A - B \) contains elements in \( A \) but not in \( B \): \( n(A) - n(A \cap B) = 45 - 15 = 30 \).
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Which of the following is the correct representation of the symmetric difference \( A \triangle B \)?
A · \( (A \cup B) - (A \cap B) \)
Symmetric difference \( A \triangle B \) is defined as elements in either \( A \) or \( B \) but not in both, i.e., \( (A \cup B) - (A \cap B) \).

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