#### Table of contents

### Introduction

- In general, the ratio of a number x to a number y is defined as the quotient of the numbers x and y.
- Ratios can be expressed as percentages. To express the value of a ratio as a percentage, we multiply the ratio by 100.

Thus, 4/5 = 0.8 = 80%

### Some Important Properties

- If we multiply the numerator and the denominator of a ratio by the same number, the ratio remains unchanged.

That is, a/b = ma/mb - If we divide the numerator and the denominator of a ratio by the same number, the ratio remains unchanged. Thus,

a/b= (a/m)/(b/m) - Denominator equation method: The magnitudes of two ratios can be compared by equating the denominators of the two ratios and then checking for the value of the numerator.
- eg. Which is larger: 8/3 or 11/4

- If either or both the terms of a ratio are a surd quantity, then the ratio will never evolve into integral numbers unless the surd quantities are equal. use this principle to spot options in questions having surds.
- If a/b = c/d = e/f = g/h = k then

K = (a+c+e+g) / (b + d + f + h)

### Mathematical use of ratio and Proportions

- As a bridge between 3 or more quantities:

If A: B and B:C is given A:C can be found out - Ratio as a Multiplier

If A: B is 3:1, then the value of B has to be multiplied by 3 to get the value of A.

### Calculations

Calculation methods for Ratio comparisons

### The Cross Multiplication Method

Two ratios can be compared using the cross-multiplication method as follows:

- Suppose you have to compare 12/17 with 15/19. Then, to test which ratio is higher, cross multiply and compare 12 * 19 and 15 * 17.

If 12 * 19 is bigger the Ratio 12/17 will be bigger. If 15 * 17 is higher, the ratio 15/19 will be higher.

In this case, 15 * 17 being higher, the Ratio 15/19 is higher.

### Proportions

When two ratios are equal, then the quantities composing them are said to be proportional. Thus, if a/b = c/d, then a, b, c, d is proportional. This is expressed as

a: b: c: d or a: b = c: d

### Variations

- Direct Proportion (A∝ B)

When it is said that A varies directly as B, you should understand the following implications: - Logical implication: When A increases B increases
- Calculation implication: If A increases by 10%, B will also increase by 10%
- Graphical implications: See following graph
- Equation implication: The ratio A/B is constant

- Inverse Proportion (A∝1/B):

When A varies inversely as B, the following implication arises:

a. Logical implication: When A increases B decreases

b. Calculation implication: If A decreases by 9.09%, B will increase by 10%.

c. Graphical implications: see graph above

d. Equation implication: The product A * B is constant.

### Practice Questions:

**1. Rs. 5783 is divided among Aman, Balan, and Chintan in such a way that if Rs. 28, Rs. 37 and Rs. 18 are deducted from their respective shares, they have money in the ratio 4:6:9. Find Aman’s share.**

a.1228

b. 1245.5

c. 1328

d. 1356

**2. If 10 persons can clean 10 floors by 10 mops in 10 days, in how many days can 8 persons clean 8 floors by 8 mops**

a. 12 days

b. 10 days

c. 11 days

d. 8 days

**3. Rs. 3650 is divided among 4 engineers, 3 MBAs and 5 CAs such that 3 CAs get as much as 2 MBAs and 3 Engineers as much as 2 CAs. Find the share of an MBA.**

a. 300

b. 450

c. 475

d. None of these

**4. Divide Rs. 1400 into three parts in such a way that half of the first part, one-fourth of the second part and one-eighth of the third part are equal.**

a. 300, 600, 500

b. 200, 400, 800

c.100, 400, 1000

d. None of these

**5. The ratio of water and milk in a 30-litre mixture is 7: 3. Find the quantity of water to be added to the mixture in order to make this ratio 6:1**

a. 30

b. 32

c. 33

d. 35

**6. A student appeared in 6 papers. The maximum marks are the same for each paper. His marks in these papers are in the proportion of 5: 6: 7: 8: 9: 10. overall he scored 60%. In how many numbers of papers did he score less than 60% of the maximum marks? (UPSC 2021)**

a. 2

b. 3

c. 4

d. 5

**7. Out of 120 applications for a post, 70 are males and 80 have a driver license. What is the ratio of between the minimum to maximum number of males having driver's license.
(UPSC 2013)**

a.1:2

b.2:3

c.3:7

d.5:7

**8. In a rare coin collection, there is one gold coin for every three non-gold coins. 10 more gold coins are added to the collection and the ratio of gold coins to non-gold coins would be 1 : 2. Based on the information; the total number of coins in the collection now becomes.
(UPSC 2013)**

a. 90

b. 80

c. 60

d. 50

**9. The ratio of a two-digit natural number to a number formed by reversing its digits is 4:7. The number of such pairs is (UPSC 2019)**

a. 5

b. 4

c. 3

d. 2

**10. The monthly incomes of X and Y are in the ratio of 4 : 3 and their monthly expenses are in the ratio of 3: 2. However, each saves Rs. 6,000 per month. What is their total monthly income?
(UPSC 2017)**

a. 28,000

b. 42,000

c. 56,000

d. 84,000

**11. The monthly incomes of Peter and Paul are in the ratio of 4:3. Their expenses are in the ratio of 3:2. If each saves Rs. 6,000 at the end of the month, their monthly incomes respectively are (in Rs.) (UPSC 2015)**

a. 24,000 and 18,000

b. 28000 and 21000

c. 32,000 and 24,000

d. 34,000 and 26,000

**12. A sum of Rs. 2,500 is distributed among X, Y and Z in two ratio 1/2 ∶ 3/4 ∶ 5/6. What is the difference between the maximum share and the minimum share? (UPSC 2020)**

a. Rs 300

b. Rs. 350

c. Rs. 400

d. Rs. 450

**13. A 6 student appeared in 6 papers. The maximum marks are the same for each paper. His marks in these are in the proportion of 5:6:7: 8:9:10 overall he scored 60%. In how many numbers of papers did he score less than 60% of the maximum marks?
(UPSC 2021)**

a. 2

b. 3

c. 4

d. 5

**14. Seats for Mathematics, Physics and Biology in a school are in the ratio 5 : 7 : 8. There is a proposal to increase these seats by 40%, 50% and 75% respectively. What will be the ratio of increased seats?**

a. 2 : 3 : 4

b. 6 : 7 : 8

c. 6 : 8 : 9

d. None of these

**15. The ratio of the number of boys and girls in a college is 7: 8. If the percentage increase in the number of boys and girls be 20% and 10% respectively, what will be the new ratio?**

a. 8: 9

b. 17: 18

c. 21: 22

d. Cannot be determined

**16. If x varies as y, and y = 4 when x =12, find x when y = 15.**

a. 45

b. 54

c. 70

d. 15

**17. The ratio of water and milk in a 30-litre mixture is 7:3. Find the quantity of water to be added to the mixture in order to make this ratio 6:1.**

a. 6

b. 4

c. 5

d. 7

**18. Rahul has coins of the denomination of ₹ 1, 50 p and 25 p in the ratio of 12:10:7. The total worth of the coins he has is ₹ 112.5. Find the number of 25 p coins that Rahul has**

a. 48

b. 72

c. 60

d. 42

**19. Four numbers in the ratio 1:2:4:8 add up to give a sum of 120. Find the value of the biggest number.**

a. 40

b. 30

c. 64

d. 60

**20. The ratio between two numbers is 7:11 and their LCM is 154. The first number is:**

a. 14

b. 7

c. 22

d. 32

**21. A fort has provisions for 60 days. If after 15 days 500 men strengthen them and the food lasts 40 days longer, how many men are there in the fort?**

a. 3500

b. 4000

c. 6000

d. None of these

**22. The ratio of the age of a man and his wife is 4 : 3. After 4 years, this ratio will be 9 : 7. If at the time of the marriage, the ratio was 5:3, then how many years ago they were married?**

a. 8 years

b. 12 years

c. 10 years

d. 15 years

**23. 60 litres of diesel is required to travel 600 km using a 800 cc engine. If the volume of diesel required to cover a distance varies directly as the capacity of the engine, then how many litres of diesel is required to travel 800 km using 1200 cc engine?**

a. 80 litres

b. 90 litres

c. 120 litres

d. 170 litres

**24. If 20 men or 24 women or 40 boys can do a task in 12 days working for 8 hours a day, how many men working with 6 women and 2 boys take to do a task four times as big working for 5 hours a day for 12 days?**

a. 8 men

b. 2 men

c. 122 men

d. 24 men

**25. If ₹ 232 is divided among 150 children such that each girl and each boy gets ₹1 and ₹2 respectively. Then hoe many girls are there?**

a. 52

b. 54

c. 68

d. 62

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