Starting your preparation for competitive exams!! Check the most important concept of aptitude here i.e., Ratios and Proportions. Ratio and Proportion are the topics that we use in our day-to-day life but never concentrate on them theoretically.

When it comes to a matter of tests regarding aptitude or something, we need to formulate them to get the required solution for a respective problem. You will even be surprised to know that it is one of the important topics for you to score in some competitive exams. Follow the below-provided sections to know more details like Formulae, Definitions, Tricks etc.

## Ratio and Proportion – Introduction

In the real world, Ratios and Proportions are used in daily life. Though we use it in our day-to-day life, we don’t notice that we use it. If you are preparing for the competitive exams, you must be perfect in ratios and proportions. This is an important and easy concept to score top marks in the exam. You must be good at calculations to solve these type of questions. Solving proportions is the fundamental building block for most problems.

We are providing in-detail material of ratios and proportions, models, estimation in problem-solving situations etc. Go through the complete article to know various Ratio and Proportion concepts, shortcuts and tricks. Moreover, we are also providing preparation tips, so that you can solve the problems in a fraction of seconds.

### Ratio and Proportion Important Formulas

#### 1. Ratio:

The ratio is nothing but the simplified form or comparison of two quantities of a similar kind. A ratio is a number, which indicates one quantity as the fraction of other quantity. **Example:** The Ratio of two numbers 5 to 6 is 5:6. It also expresses the no of times one quantity is equal to other quantity.

“Terms” are the term that indicates numbers forming the ratios. The upper part of the ratio(numerator) is called antecedent and the lower part of the fraction(denominator) is called consequent or descendent. **Example:** If the ratio is 4:6, then 4 is called the antecedent and 6 is called the consequent.

#### 2. Proportion:

Proportion is indicated as ‘::’ or ‘=’. If the ratio x:y is equal to the ratio of a:b, then x,y, a,b are in proportion.

We mention it using the symbols is as x:y=a:b or x:y:: a:b

When four terms are in proportion, then the product of two middle values(i.e., 2nd and 3rd values) must be equal to the product of two extremes(i.e., 1st and 4th values)

#### 3. Fourth Proportional:

If x:y = a:b, then b is called the fourth proportion to x,y and a

**Third Proportional:**

If x:y = a:b, then a is called the fourth proportion to x,y and b

**Mean Proportional:**

The mean proportional between x and y is the root(xy).

#### 4. Comparision of Ratios

We define that x:y>a:b if and only if x/y>a/b

**Compounded Ratios:**

Compounded Ratio of two ratios: (x:y), (a:b), (c:d) is (xac:ybd)

#### 5. Duplicate Ratios

The duplicated ratio of (x:y) is ((square(x): square(y)).

The sub-duplicated ratio of (x:y) is ((root(x): root(y)).

Triplicate ratio of (x:y) is ((cube(x):cube(y)).

The subtriplicate ratio of (x:y) is ((cuberoot(x):cuberoot(y)).

**Componendo and dividend rule**

If x/y=a/b, then x=y/x-y = a+b/a-b.

#### 6. Variations

We define that a is directly proportional to b, if a=kb for some constant k and we write it as – a is proportional to b.

We say that a is indirectly proportional to b, if ay=b for some constant k and we write it as – a is proportional to 1/b.

### Best Books for Ratios and Proportions

- Ratios and Proportions Workbook by Maria Miller
- Axel Tracy’s Ratio Analysis Fundamentals: How 17 financial ratios can allow you to analyse any business on the planet
- Rajesh Verma/Arihant Publications: Fourth Edition
- RS Aggarwal Publications
- S. Chand Publications
- McGraw Hill Publications
- Disha Publications
- Kiran Prakashan
- Sarvesh K Varma Publications

### Important Properties of Ratio and Proportion

Check the important properties of proportions and know how to apply them.

**Componendo and dividendo:**

If x:y=a:b then x+y:x–y=a+b:a-b

**Invertendo:**

If x:y=a:b, then y:x=b:a

**Alternendo:**

If x:y=a:b then x:a=y:b

**Componendo:**

If x:y=a:b then x+y:x=a+b:a

**Dividendo:**

If x:y=a:b then x-y:a=a-b:a

**Subtrahendo:**

If x:y=a:b then x-a:y-b

**Addendo:**

If x:y=a:b, then x+a:y+b

### Key Points to Remember

- The ratio between two quantities should exist with the same kind.
- While comparing two ratios, their units must be similar.
- Significant order of terms must be there.
- If the ratios are equal like a fraction, then only a comparison of 2 ratios can be performed.

### Tips and Tricks for Ratios and Proportions

- If a/b=x/y, then ay=bx
- If a/b=x/y, then a/x=b/y
- Suppose a/b=x/y, then b/a=y/x
- If a/b=x/y, then (a+b)/b=(x+y)/y
- If a/b=x/y, then (a-b)/b=(x-y)/y
- Suppose a/b=x/y, then (a+b)/(a-b)=(x+y)/(x-y)
- If x/(y+z)=y/(z+a)=z/(x+y) and x+y+z is not equal to 0 then x=y=z

### Important Solved Questions

**Question 1:** Find if the ratios are in proportion. The ratios are 4:5 and 8:10.

A. Yes

B. No

C. Data Insufficient

D. None of the above

**Solution:**

**A(Yes)**

Given 4:5 and 8:10 are the ratios

4:5=4/5 and 8:10=8/10

4/5=0.8 and 8/10=0.8

Therefore, both proportions are equal and they are said to be in proportion.

**Question 2:** Given Ratios are

x:y=2:3

y:z=5:2

z:a=1:4

Find x:y:z:a

A. 15:20:14:22

B. 15:25:14:26

C. 10:15:6:24

D. 10:20:6:24

**Solution:**

**C(10:15:6:24)**

Given the ratios are

x:y=2:3

y:z=5:2

z:a=1:4

Multiplying the 1st ratio by 5, 2nd by 3 and 3rd by 6, we have

x:y=10:15

y:z=15:6

z:a=6:24

In the above equations, all the mean terms are similar, so

x:y:z:a=10:15:6:24

### Word Problems on Ratio and Proportion

**Question 3:** From the total strength of the class, if the no of boys in the class is 5 and no of girls in the class is 3, then find the ratio between girls and boys.

A. 3/5

B. 4/6

C. 8/10

D. 2/5

**Solution:**

A(3/5)

The ratio of girls and boys can be written as 3:5(Girls: Boys). The ratio can be written in the fraction form like 3/5.

**Question 4:** Suppose 2 numbers are in the ratio 2:3. If the sum of two numbers is 60. Find the numbers?

A. 40, 36

B. 24, 36

C. 25, 40

D. 44, 36

**Solution:**

B(24,36)

Given, the ratio of two numbers is 2:3

As, per the given question, the sum of 2 numbers = 60

Therefore, 2x+3x=60

5x=60

x=12

Hence, 2 numbers are;

2x=2*12=24

3x=3*12=36

24 and 36 are the required numbers.

**Question 5:** 20% and 50% are 2 numbers respectively more than a 3rd number. Find the ratios of 2 numbers.

A. 4:5

B. 2:5

C. 6:7

D: 3:5

**Solution:**

A(4:5)

Suppose the third number is x

Let the 1st number = 120% of x=120x/100 = 6x/5

Let the 2nd number = 150% of x=150x/100 = 3x/2

Therefore, ratios of 2 numbers = (6x/5:3x/2) = 12x:15x = 4:5

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