Ratio and Proportion Aptitude

Ratio and Proportion problems are key components of quantitative aptitude tests. They involve comparing two quantities by division and determining how one quantity is related to another. A ratio is an expression that compares quantities relative to each other, while a proportion states that two ratios are equal. These concepts are widely used in solving problems related to real-life scenarios, business, finance, and science.

1. The ratio of the number of boys to girls in a class is 3:4. If there are 18 boys, how many girls are there?

Number of girls = (4/3) × 18 = 24.

2. Two numbers are in the ratio 5:6. If their sum is 55, find the numbers.

Let the numbers be 5x and 6x.
Then, 5x + 6x = 55, 11x = 55, x = 5.
The numbers are 25 and 30.

3. In a mixture of 45 litres, the ratio of milk to water is 4:1. How much more water should be added to make the ratio 2:3?

Quantity of milk in the mixture = (4/5) × 45 = 36 litres. 
The initial quantity of water = 45 - 36 = 9 litres.
Let the amount of water to be added be x litres.
New ratio of milk to water = 36:(9 + x).
To make this ratio 2:3, 36/(9 + x) = 2/3.
Solving for x gives x = 30 litres. So, 30 litres of water should be added.

4. Divide $1200 in the ratio 3:2:5.

Total ratio = 3 + 2 + 5 = 10.
Shares are $1200 × (3/10) = $360, $1200 × (2/10) = $240, $1200 × (5/10) = $600.

5. The incomes of A and B are in the ratio 3:4 and their expenditures are in the ratio 4:5. If each saves $2000, find A’s income.

Let A's income be 3x and B's income be 4x. A's expenditure = 4y, B's expenditure = 5y.
3x - 4y = 2000 and 4x - 5y = 2000. Solving gives A's income = 3x = $8000.

6. A bag contains coins of denominations $1, $2 and $5 in the ratio 3:2:1. If the total amount is $96, find the number of $2 coins.

Let the number of $1, $2, and $5 coins be 3x, 2x, and x respectively.
Total amount = 3x(1) + 2x(2) + x(5) = $96.
Solving for x gives x = 8. Number of $2 coins = 2x = 16.

7. The ratio of two numbers is 3:5 and their difference is 40. Find the numbers.

Let the numbers be 3x and 5x.
Then, 5x - 3x = 40, 2x = 40, x = 20.
The numbers are 60 and 100.

8. A container has a mixture of two liquids A and B in the ratio 7:5. When 9 litres of the mixture is drawn off and replaced with liquid B, the ratio becomes 7:9. Find the initial quantity of the mixture.

Let the initial quantity be x litres.
Quantity of A in the mixture left = 7x/12 - 7(9/x) litres.
New ratio = (7x/12 - 7(9/x)):(5x/12 + 9) = 7:9.
Solving gives x = 63 litres.

9. In what ratio must a grocer mix two types of rice costing $15/kg and $20/kg so that by selling the mixture at $18/kg he gains 20%?

Cost price of 1 kg of mixture = $18/120 × 100 = $15.
Using the rule of alligation, the required ratio = (20 - 15):(15 - 15) = 5:0 = 2:1.

10. The ages of A and B are in the ratio 4:3. After 6 years, the ratio of their ages will be 5:4. Find the present age of A.

Let the present ages of A and B be 4x and 3x years respectively.
(4x + 6):(3x + 6) = 5:4. Solving gives x = 6. Present age of A = 4x = 24 years.

11. The ratio of milk and water in a mixture is 2:3. If 15 litres of water is added, the ratio becomes 2:5. Find the initial quantity of milk in the mixture.

Let the initial quantities of milk and water be 2x and 3x litres respectively.
2x:(3x + 15) = 2:5. Solving gives x = 5. Initial quantity of milk = 2x = 10 litres.

12. If A:B = 2:3 and B:C = 4:5, then A:C is:

A:B = 2:3 and B:C = 4:5. To find A:C, make B same in both ratios.
A:B = 2:3 = 8:12 and B:C = 4:5 = 12:15. Therefore, A:C = 8:15.

13. A sum of money is to be divided among A, B, and C in the ratio 2:3:5. If C gets $200 more than A, find B's share.

Difference between A's and C's share = 5 - 2 = 3 parts. 3 parts = $200, 1 part = $200/3.
B's share = 3 × $200/3 = $200.

14. The ratio of the number of boys to girls in a school is 5:4. If there are 200 students in the school, how many are girls?

Total ratio = 5 + 4 = 9. Number of girls = (4/9) × 200 = 80.

15. The speed of a boat in still water is to the speed of the current in the ratio 4:1. If the boat covers 120 km downstream in 6 hours, find the speed of the boat in still water.

Downstream speed = 120 km / 6 hr = 20 km/hr.
Let the speed of the boat be 4x and the speed of the current be x.
Then, 4x + x = 20 km/hr, 5x = 20 km/hr, x = 4 km/hr. Speed

15. The speed of a boat in still water is to the speed of the current in the ratio 4:1. If the boat covers 120 km downstream in 6 hours, find the speed of the boat in still water.

Downstream speed = 120 km / 6 hr = 20 km/hr.
Let the speed of the boat be 4x and the speed of the current be x. 
Then, 4x + x = 20 km/hr, 5x = 20 km/hr, x = 4 km/hr. Speed of the boat = 4x = 16 km/hr.

16. If two numbers are in the ratio 3:5 and their difference is 16, find the numbers.

Let the numbers be 3x and 5x. Then, 5x - 3x = 16, 2x = 16, x = 8. The numbers are 24 and 40.

17. In a mixture, the ratio of acid and water is 1:2. If 3 liters of water is added to the mixture, the ratio becomes 1:3. Find the initial quantity of acid in the mixture.

Let the initial quantities of acid and water be x and 2x liters respectively.
x:(2x + 3) = 1:3. Solving gives x = 3 liters.

18. A bag contains coins of $1, $2, and $5 in the ratio 4:3:2. If the total amount is $144, how many $5 coins are there?

Let the number of $1, $2, and $5 coins be 4x, 3x, and 2x respectively.
Total amount = 4x(1) + 3x(2) + 2x(5) = $144.
Solving for x gives x = 6. Number of $5 coins = 2x = 12.

19. The ages of A, B, and C are in the ratio 3:2:4. If the sum of their ages is 54 years, find A's age.

Total ratio = 3 + 2 + 4 = 9. A's age = (3/9) × 54 years = 18 years.

20. In a 60 litre mixture of milk and water, the ratio of milk to water is 2:1. How much water must be added to make the ratio 1:2?

Quantity of milk = (2/3) × 60 = 40 litres. To make the ratio of milk to water 1:2, the quantity of water should be 2 × 40 = 80 litres. Water to be added = 80 - 20 = 60 litres.