Ratio and Proportion
Q1. Two numbers are in the ratio 1½ : 5(2/3). When each one of these is increased by 15, their ratio become 1(2/3) : 2½ . The smaller of the number is
(a) 21
(b) 24
(c) 25
(d) 27
Answer: (d) 27
| 1 | 1 | : 2 | 2 | = | 3 | : | 8 |
| 2 | 3 | 2 | 3 |
Let the numebrs be 3x/2 and 8x/3
| A/Q | 3x | + 15 : | 8x | + 15 = 1 | 2 | : 2 | 1 |
| 2 | 3 | 3 | 2 |
| ⇒ | 3x + 30 | : | 8x + 45 | = | 5 | : | 5 |
| 2 | 3 | 3 | 2 |
| ⇒ | (3x + 30)/2 | = | 5/3 |
| (8x + 45)/3 | 5/2 |
| ⇒ | 3(3x + 30) | = | 5 X 2 |
| 2(8x + 45) | 3 X 5 |
| ⇒ | 3x + 30 | = | 4 |
| 8x + 45 | 9 |
⇒ 27x + 270 = 32x + 180
⇒ 5x = 90
⇒ x = 8
Numbers be (3/2) X 18, (8/3) X 18 = 27, 48
∴ Smaller number is 27
Q2. The expenses on rice, fish and oil of a family are in the ratio of 12 : 17 : 3. The prices of these articles are increased by 20%, 30% and 50% respectively. The total expenses of the family are increased by
(a) 25½ %
(b) 28(1/8) %
(c) 29¾ %
(d) 31(1/5) %
Answer: (b) 28(1/8) %
Let, the expenses on rice, fish and oil be 12x, 17x and 3x respectively
Total expenses = Rs. (12x + 17x + 3x) = Rs. 32x
New expenses = Rs. (120% of 12x + 130% of 17x + 150% of 3x)
| = Rs. ( | 120 | X 12x + | 130 | X 17x + | 150 | X 3x ) |
| 100 | 100 | 100 |
| = Rs. ( | 72x | + | 221x | + | 9x | ) |
| 5 | 10 | 2 |
| = Rs. | 144x + 221x + 45x | = Rs.41x |
| 10 |
| Increase % = | 9x | X 100 |
| 32x |
| = 28 | 1 | % |
| 8 |
Q3. If y varies directly as (x + 4) and y = 6 when x = 1. What is the value of y when x = 2 ?
(a) 36/5
(b) 37/9
(c) 41/7
(d) 43/5
Answer: (a) 36/5
Given y α (x + 4)
⇒ y = k (x+4)
⇒ 6 = k (1+4)
⇒ k = 6/5
Now y = k (x + 4) = 6/5 (2 + 4) = (6/5) X 6 = 36/5
Q4. The monthly income of A and B are in the ratio 2 : 3 and their monthly expenses are in the ratio 5 : 9. If each of them saves Rs. 600 per month, then their monthly incomes are
(a) Rs. 1600, Rs. 2400
(b) Rs. 1650, Rs. 2200
(c) Rs. 1800, Rs. 2600
(d) Rs. 1950, Rs. 2750
Answer: (a) Rs. 1600, Rs. 2400
Let, the monthly income of A and B are 2x and 3x respectively
Let, the monthly expenditure of A and B are 5y and 9y respectively
A/Q 2x - 5y = 600 ------- (1)
Ans 3x - 9y = 600
⇒ x - 3y = 200
⇒ 2x - 6y = 400 ------ (2)
(1) - (2) ⇒ 6y - 5y = 600 - 400
⇒ y = 200
(1) ⇒ 2x - 5 X 200 = 600
⇒ 2x - 1000 = 600
⇒ x = 1600/2 = 800
∴ A's monthly income = Rs. 2 X 800 = Rs. 1600
B's monthly income = Rs. 3 X 800 = Rs. 2400
∴ Answer is Rs. 1600 and Rs. 2400
Q5. A bag contains Rs. 216 in the form of one rupee, 50 paise and 25 paise coins in the ratio of 2 : 3 : 4. The number of 50 paise coins is
(a) 133
(b) 136
(c) 144
(d) 149
Answer: (c) 144
Let, the no. of one-rupee, 50 paise and 25 paise coins be 2x, 3x, 4x respctively.
| A/Q 2x + | 50 X 3x | + | 25 X 4x | = 216 |
| 100 | 100 |
| ⇒ 2x + | 3x | + x = 216 |
| 2 |
⇒ 4x + 3x + 2x = 2 X 216
⇒ 9x = 432
⇒ x = 48
∴ No. of 50 paise coins = 3 X 48 = 144
Q6. Two vessels A and B contain milk and water mixed in the ratio 5 : 3 and 2 :3. When these mixtures are mixed to form a new mixture containing half milk and half water, they must taken in the ratio
(a) 1 : 3
(b) 3 : 5
(c) 4 : 5
(d) 5 : 7
Answer: (c) 4 : 5
Let the required ratio be x : 1
Milk in x liters of A = (x X (5/8)) liters = 5x/8 liters
Water in A = 3x/8 liters
Milk in 1 liter of B = (1 X (2/5)) liters = 2/5 liters
Water in B = 3/5 liters
| ∴ | 5x/8 + 2/5 | = | 1 |
| 3x/8 + 3/5 | 1 |
| ⇒ | 25x + 16 | = | 15x + 24 |
| 40 | 40 |
⇒ 10x = 24 - 16 = 8
⇒ x = 8/10 = 4/5
∴ Required ratio = 4/5 : 1 = 4 : 5
Q7. In an alloy, the ratio of copper and zinc is 5 : 2. If 1.250 Kg of zinc is mixed in 17 Kg 500 gm of alloy, then the ratio of copper and zinc in the alloy will be
(a) 2 : 1
(b) 2 : 3
(c) 3 : 4
(d) 3 : 5
Answer: (a) 2 : 1
Copper in 17.5 Kg of alloy = (17.5 X (5/7)) Kg = 12.5 Kg
Zinc in 17.5 Kg of alloy = (17.5 X (2/7)) Kg = 5 Kg
After mixing amount of zinc = 5 Kg + 1.250 Kg = 6.250 Kg
| Required ratio of copper and zinc = 12.5 : 6.250 = | 12.5 | = | 1250 | = | 2 | = 2 : 1 |
| 6.250 | 625 | 1 |
Q8. In a mixture of 60 liters, the ratio of milk and water is 2 : 1. If the ratio is to be 1 : 2, the quantity of water to be further added is
(a) 30 liters
(b) 40 liters
(c) 50 liters
(d) 60 liters
Answer: (d) 60 liters
| Quantity of milk = ( 60 X | 2 | ) lts = 60 X | 2 | = 40 lts |
| 2 + 1 | 3 |
Quantity of water = 60 - 40 = 20 lts
Let, the required quantity of water x lts
| A/Q | 40 | = | 1 |
| 20 + x | 2 |
⇒ 20 + x = 40 X 2 = 80
⇒ x = 80 - 20 = 60 lts
Q9. A mixture contains milk and water in the ratio 5 : 1. On adding 5 liters of water, the ratio of milk to water becomes 5 : 2. The quantity of milk in the original mixture is
(a) 25 liters
(b) 29 liters
(c) 33 liters
(d) 37 liters
Answer: (a) 25 liters
Let, the quantity of milk be 5x liters and water be x liters
| A/Q | 5x | = | 5 |
| x + 5 | 2 |
⇒ 10x = 5x + 25
⇒ 5x = 25
⇒ x = 5
∴ Required quantity of milk = 5 X 5 = 25 lts
Q10. Three numbers are in the ratio 3:4:5. The sum of the largest and the smallest equals the sum of the third and 52. The smallest number is
(a) 36
(b) 39
(c) 42
(d) 47
Answer: (b) 39
Let the numbers be 3x, 4x, 5x respectively
A/Q 3x + 5x = 4x + 52
⇒ 8x = 4x + 52
⇒ 4x = 52
⇒ x = 13
∴ Smallest No. = 3 X 13 = 39
