Ratio and Proportion - Solved Questions and Answers
A ratio compares two quantities by division, showing their relative sizes.
The proportion states that the two ratios are equal.
Ration and proportion questions and answers are provided below for you to learn and practice.
Question 1: Is the ratio 5:10 proportional to 1:2?
Solution:
5:10 divided by 5 gives 1:2. Thus, they are same to each other. So we can say that 5:10 is proportional to 1:2.
Question 2: Given a constant k, such that k:5 is proportional to 10:25. Find the value of k.
Solution:
Since k:5 is proportional to 10:25, we can write,
k / 5 = 10 / 25
k = 10/25 × 5 = 2So, the value of k is 2.
Question 3: Divide 100 into two parts such that they are proportional to 3:5.
Solution:
Let's the value of two parts are 3k and 5k, where k is a constant.
Since the total sum of two parts is 100, we can write,3k + 5k = 100
8k = 100k = 12.5
So, the parts are 3k = 3 × 12.5 = 37.5 and 5k = 5 × 12.5 = 62.5
Question 4: If x2 + 6y2 = 5xy, then find the value of x/y.
Solution:
Given, x2 + 6y2 = 5xy.
Dividing the equation by y2, we get
(x/y)2 + 6 = 5 (x/y)Let's x/y = t
So, we can write,
t2 + 6 = 5t
t2 - 5t + 6 = 0
(t - 2)(t - 3) = 0t = 2 or t = 3
Since, t = x/y, we get
x/y = 2 or x/y = 3
Question 5: If a: b = 5: 9 and b: c = 7: 4, then find a: b: c.
Solution:
Here, we make the common term ‘b’ equal in both ratios.
Therefore, we multiply the first ratio by 7 and the second ratio by 9.
So, we have a : b = 35 : 63 and b : c = 63 : 36
Thus, a : b : c = 35 : 63 : 36
Question 6: Find the mean proportional between 0.23 and 0.24.
Solution:
We know that the mean proportional between ‘a’ and ‘b’ is the square root of (a x b).
Required mean proportional =\sqrt(0.23 \times 0.24) = 0.234946802
Question 7: Divide Rs. 981 in the ratio 5: 4.
Solution:
The given ratio is 5: 4
Sum of numbers in the ratio = 5 + 4 = 9
We divide Rs. 981 into 9 parts.
981 / 9 = 109
Therefore, Rs. 981 in the ratio 5: 4 = Rs. 981 in the ratio (5 / 9) : (4 / 9)
Rs. 981 in the ratio 5 : 4 = (5 x 109) : (4 x 109) = 545 : 436
Question 8: A bag contains 50 p, 25 p, and 10 p coins in the ratio 2 : 5 : 3, amounting to Rs. 510. Find the number of coins of each type.
Solution:
Let the common ratio be 100k.
Number of 50 p coins = 200 k
Number of 25 p coins = 500 k
Number of 10 p coins = 300 k
Value of 50 p coins = 0.5 x 200 k = 100 k
Value of 25 p coins = 0.25 x 500 k = 125 k
Value of 10 p coins = 0.1 x 300 k = 30 k
Total value of all coins = 100 k + 125 k + 30 k = 255 k = 510 (given)
k = 2
Therefore, Number of 50 p coins = 200 k = 400
Number of 25 p coins = 500 k = 1000
Number of 10 p coins = 300 k = 600
Question 9: A mixture contains sugar solution and colored water in the ratio of 4 : 3. If 10 liters of colored water is added to the mixture, the ratio becomes 4: 5. Find the initial quantity of sugar solution in the given mixture.
Solution:
The initial ratio is 4 : 3.
Let ‘k’ be the common ratio.
=> Initial quantity of sugar solution = 4 k
=> Initial quantity of colored water = 3 k
=> Final quantity of sugar solution = 4 k
=> Final quantity of colored water = 3 k + 10
Final ratio = 4 k : 3 k + 10 = 4 : 5
=> k = 5
Therefore, the initial quantity of sugar solution in the given mixture = 4 k = 20 liters
Question 10: Two friends A and B started a business with an initial capital contribution of Rs. 1 lac and Rs. 2 lacs. At the end of the year, the business made a profit of Rs. 30,000. Find the share of each in the profit.
Solution:
We know that if the time period of investment is the same, profit/loss is divided by the ratio of the value of the investment.
=> Ratio of value of investment of A and B = 1,00,000 : 2,00,000 = 1 : 2
=> Ratio of share in profit = 1 : 2
=> Share of A in profit = (1/3) x 30,000 = Rs. 10,000
=> Share of B in profit = (2/3) x 30,000 = Rs. 20,000
Question 11: Three friends A, B, and C started a business, each investing Rs. 10,000. After 5 months A withdrew Rs. 3000, B withdrew Rs. 2000 and C invested Rs. 3000 more. At the end of the year, a total profit of Rs. 34,600 was recorded. Find the share of each.
Solution:
We know that if the period of investment is not uniform, the gains/losses from the business are divided in the ratio of their inputs, where input is calculated as the product of an amount of investment and the time period of investment.
So, input = value of investment x period of investment, and here, the period of investment would be broken into parts as the investment is not uniform throughout the time period.
A’s input = (10,000 x 5) + (7,000 x 7) = 99,000
B’s input = (10,000 x 5) + (8,000 x 7) = 1,06,000
C’s input = (10,000 x 5) + (13,000 x 7) = 1,41,000
=> A : B : C = 99000 : 106000 : 141000
=> A : B : C = 99 : 106 : 141
=> A : B : C = (99 / 346) : (106 / 346) : (141 / 346)
Thus, A’s share = (99 / 346) x 34600 = Rs. 9900
B’s share = (106 / 346) x 34600 = Rs. 10600
C’s share = (141 / 346) x 34600 = Rs. 14100
Question 12: A invested Rs. 70,000 in a business. After a few months, B joined him with Rs. 60,000. At the end of the year, the total profit was divided between them in the ratio of 2: 1. After how many months did B join?
Solution:
Let A work alone for ‘n’ months.
=> A’s input = 70,000 x 12
=> B’s input = 60,000 x (12 – n)
So, (70,000 x 12) / [60,000 x (12 – n)] = 2 / 1
=> (7 x 12) / [6 x (12 – n)] = 2 / 1
=> 12 – n = 7
=> n = 5
Therefore, B joined after 5 months.
