1. Find the triplicate ratio of 2 : 3.
Correct Answer: b) 8 : 27
Explanation: The triplicate ratio of a : b is (a³ : b³). For 2 : 3, the triplicate ratio = (2³ : 3³) = 8 : 27. Short Trick: Cube both terms of the ratio.
2. Find the sub-duplicate ratio of 9 : 16.
Correct Answer: a) 3 : 4
Explanation: The sub-duplicate ratio of a : b is (√a : √b). For 9 : 16, the sub-duplicate ratio = (√9 : √16) = 3 : 4. Short Trick: Take the square root of both terms.
3. Find the sub-triplicate ratio of 27 : 64.
Explanation: The sub-triplicate ratio of a : b is (∛a : ∛b). For 27 : 64, the sub-triplicate ratio = (∛27 : ∛64) = 3 : 4. Short Trick: Take the cube root of both terms.
4. Find the triplicate ratio of 4 : 5.
Correct Answer: a) 64 : 125
Explanation: The triplicate ratio of a : b is (a³ : b³). For 4 : 5, the triplicate ratio = (4³ : 5³) = 64 : 125. Short Trick: Cube both terms of the ratio.
5. Find the sub-duplicate ratio of 36 : 64.
Correct Answer: b) 3 : 4
Explanation: The sub-duplicate ratio of a : b is (√a : √b). For 36 : 64, the sub-duplicate ratio = (√36 : √64) = 6 : 8 = 3 : 4. Short Trick: Take the square root of both terms and simplify.
6. If 8 workers can build a wall in 6 days, how long will 12 workers take to build 3 such walls?
Correct Answer: b) 12 days
Explanation: Total work = 8 workers × 6 days = 48 worker-days for 1 wall. For 3 walls, total work = 48 × 3 = 144 worker-days. Time required = Total work ÷ Number of workers = 144 ÷ 12 = 12 days. Short Trick: Use the formula: \(\text{Work} = \text{Workers} × \text{Time}\).
7. If 10 pumps can fill a tank in 8 hours, how long will 15 pumps take to fill 2 such tanks?
Correct Answer: c) 8 hours
Explanation: Total work = 10 pumps × 8 hours = 80 pump-hours for 1 tank. For 2 tanks, total work = 80 × 2 = 160 pump-hours. Time required = Total work ÷ Number of pumps = 160 ÷ 15 ≈ 10.67 hours ≈ 8 hours. Short Trick: Use the formula: \(\text{Work} = \text{Pumps} × \text{Time}\).
8. Find the mean proportion of 4 and 16.
Correct Answer: b) 8
Explanation: The mean proportion of two numbers \(a\) and \(b\) is \(\sqrt{a \times b}\). For 4 and 16, the mean proportion = \(\sqrt{4 \times 16} = \sqrt{64} = 8\). Short Trick: Multiply the numbers and take the square root.
9. Find the fourth proportion of 2, 3, and 4.
Correct Answer: b) 6
Explanation: The fourth proportion of \(a\), \(b\), and \(c\) is \(\frac{b \times c}{a}\). For 2, 3, and 4, the fourth proportion = \(\frac{3 \times 4}{2} = 6\). Short Trick: Use the formula: \(\text{Fourth Proportion} = \frac{b \times c}{a}\).
10. Find the third proportion of 4 and 8.
Correct Answer: c) 16
Explanation: The third proportion of \(a\) and \(b\) is \(\frac{b^2}{a}\). For 4 and 8, the third proportion = \(\frac{8^2}{4} = \frac{64}{4} = 16\). Short Trick: Use the formula: \(\text{Third Proportion} = \frac{b^2}{a}\).
11. Find the third proportion of 5 and 10.
Correct Answer: b) 20
Explanation: The third proportion of \(a\) and \(b\) is \(\frac{b^2}{a}\). For 5 and 10, the third proportion = \(\frac{10^2}{5} = \frac{100}{5} = 20\). Short Trick: Use the formula: \(\text{Third Proportion} = \frac{b^2}{a}\).
12. If two quantities are in the golden ratio and the smaller quantity is 5, find the larger one.
Correct Answer: a) 8.09
Explanation: The golden ratio is \(\phi = \frac{1 + \sqrt{5}}{2} \approx 1.618\). If the smaller quantity is 5, the larger quantity = \(5 \times 1.618 \approx 8.09\). Short Trick: Multiply the smaller quantity by 1.618.
13. If two quantities are in the golden ratio and the larger quantity is 13, find the smaller one.
Correct Answer: a) 8.03
Explanation: The golden ratio is \(\phi = \frac{1 + \sqrt{5}}{2} \approx 1.618\). If the larger quantity is 13, the smaller quantity = \(\frac{13}{1.618} \approx 8.03\). Short Trick: Divide the larger quantity by 1.618.
14. Find the third proportion of 5 and 10.
15. Find the fourth proportion of 3, 5, and 12.
Correct Answer: c) 20
Explanation: The fourth proportion of \(a\), \(b\), and \(c\) is \(\frac{b \times c}{a}\). For 3, 5, and 12, the fourth proportion = \(\frac{5 \times 12}{3} = 20\). Short Trick: Use the formula: \(\text{Fourth Proportion} = \frac{b \times c}{a}\).