# Tricks to Solve Ratio & Proportion

**Numerical ability section** is considered to be one of the toughest subjects of** SSC Exams** but it can be scored off well if prepared well. **Ratio & Proportion** is one of the toughest chapters which leaves candidates a bit confused and most of the aspirants leave these questions untouched.

To make the chapter easy for you all, we are providing you all some **Important Short Tricks to solve Ratio & Proportion Questions** which will surely make the chapter easy for you all.

**Ratio and Proportion**

*The number of times one quantity contains another quantity of the same kind is called the ratio of the two quantities.*

Observe carefully that the two quantities must be of the same kind. There can be a ratio between Rs.20 and Rs 30, but there can be no ratio between Rs 20 and 30 mangoes.

The ratio 2 to 3 is written **as 2 : 3 or 2/3.** 2 and 3 are called the **terms of the ratio**. 2 is the first term and 3 is the second term.

**Consequent **

In the **ratio 2 :3** , 2 is the antecedent and 3 is the consequent.

**Note: **

(1) The word ‘**consequent** ’ literally means ‘that which goes after’.

(2) since the quotient obtained on dividing one concrete quantity by another of the same kind is an abstract number, the ratio between two concrete quantities of the same kind is an abstract number. Thus, the ratio between Rs 5 and 7 is 5:7.

**Compound Ratio**

Ratios are **compound** by multiplying together the antecedents for a new antecedent, and the consequents for a new consequent.

**Ex: find the ratio compound of the ratio:**

4:3, 9: 13, 26 : 5 and 2:15

**Solution;**

The required ratio =

**Inverse Ratio**

- If 2:3 be the given ratio, then 1/2: 1/3 or 3 :2 is called its
**inverse**or**reciprocal ratio.** - If the antecedent = the consequent, the ratio is called the
**ratio of equality**, such as 3:3. - If the antecedent >the consequent, the ratio is called the
**ratio of greater inequality**, as 4 :3 - If the antecedent < the consequent, the ratio is called the
**ratio of less inequality**, as 3 : 4.

**Ex. Divide 1458 into two parts such that one may be to the other as 2: 7.**

**Solution:**

1^{st} part =

2^{nd} part =

**Proportion**

**Consider the two ratios: **

** **1^{st} ratio 2^{nd} ratio

6 : 18 8 : 24

Since 6 is one-third of 18, and 8 is one –third of 24, the two ratios are equal. The equality of ratio is called **proportion.**

The number 6, 18, 8 and 24 are said to be in **proportion.**

The proportion may be written as

6 : 18 :: 8 : 24 (6 is to 18 as 8 is to 24)

Or, 6 :18 = 8:24 or

The numbers 6, 18, 8 and 24 are called the **terms**. 6 is the **first terms**, 18 the **second,** 8 the **third,** and 24 **the fourth**. The first and fourth terms, i.e. 6 and 24 are called the** extremes **(end terms), and the second and the third terms, i.e., 18 and 8 are called the **means** (middle terms). 24 is called the **fourth** proportional.

**Type 1:** If your quantities be in **proportion**, the product of the extremes is equal to the product of the means.

Let the four quantities 3, 4, 9 and 12 be in proportion.

We have ,

**Type 2:** Three quantities of the same kind are said to be in **continued proportion** when the ratio of the first to the second is equal to the ratio of the second the third.

*The second quantity is called the mean proportional between the first and the third ; and the third quantity is called the third proportional to the first and second.*

Thus, 9, 6 and 4 are in continued proportion for 9 :6 ::6 :4.

Hence, 6 is the mean proportional between 9 and 4, and 4 is the third proportional to 9 and 6.

**Ex1. **Find the fourth proportional to the numbers 6, 8 and 15.

**Solution:**

If x be the fourth proportional, then 6 : 8 = 15:x

**Ex2.** Find the third proportional to 15 and 20.

**Solution**:

Here, we have to find a fourth proportional to 15, 20 and 20. If x be the fourth proportional, we have 15 :20 = 20:x

**Direct Proportion: **consider the following example .

**Ex. 1:** If 5 ball cost Rs 8, what do 15 balls cost?

**Solution:**

It will be seen at once that if the number of balls be increased 2, 3, 4,….times, the price will also be increased 2, 3, 4… times.

Therefore, 5 balls is the same fraction of 15 balls that the cost of balls is of the cost of 15 balls.

5 balls : 15 balls :: Rs8 : required cost

the required cost = Rs=Rs 24

This, example is an illustration of what is called direct proportion. In this case, the two given quantities are so related to each other that if one of them is multiplied (or divided) by any number, the other is also multiplied (or divided) by the same number.

**Inverse Proportion:** Consider the following example

**Ex. 1**: If 15 men can reap a filed in 28 days, in how many days will 10 men reap it?

**Solution:**

Here, it will be seen that if the number of men be increased 2, 3,4,….times, the number of days will be decreased 2, 3, 4…times. Therefore, the inverse ratio of the number of men is equal to the ratio of the corresponding number of days.

Or , 10: 15 :: 28 : the required number of days

the required number of days=

The above example is an illustration of what is called inverse proportion. In this case, the two quantities are so related that if one of them is multiplied by any number, the other is divided by the same number, and *vice versa.*

**Ex 2: **The employer decreases the number of his employees in the ratio 10 :9 and increase their ways in the ratio 11:12. What is the ratio of his two expenditures?

** Solution:**

The required ratio = 10×11 : 9 ×12 = 55 : 54

**Ex3:** A vessel contains liquid A and B in ratio 5 :3. If 16 liters of the mixture are removed and the same quantity of liquid B is added, the ratio becomes 3 :5. What quantity does the vessel hold?

**Solution:**

Quicker Method:

When the ratio is reversed (i.e., 5:3 becomes 3 :5), we can use the formula;

Total quantity = ×Quantity of A in the removed mixture

= liters.

**Question 1:** A : B : C is in the ratio of 3 : 2 : 5. Then how much money will C get out of Rs 500.

(1) Rs 200

(2) Rs 250

(3) Rs 300

(4) Rs 350

**Solution: **

**Short Cut: **

**Question **2: The ratio of income of A and B is 3 : 2 and the ratio of their expenditure is 4 : 3 and their savings are Rs 2,000 and Rs 1,000. Find the income of A and B respectively.

(1)Rs 3000, Rs 4000

(2) Rs 4000, Rs 6000

(3) Rs 5000, Rs 6000

(4) Rs 6000, Rs 4000

**Solution: **

__A__ : __B__

Income = 3(x) : 2(x)

Expenditure = 4 : 3

Income of A = 3x = 6000

Income of B = 4x = 4000

**Question **3: A, B, C alone completed a piece of work in 30, 50 and 40 days. The ratio of the salary of each day is 4: 3 : 2 respectively. If the total income of A Rs 144, find total income of B.

(1) Rs 180

(2) Rs 185

(3) Rs 190

(4) Rs 195

**Solution ** __A____B____C__

Days of working = 30 : 50 : 40

Each day salary = __4 : 3 : 2__

Total income = 120 : 150 : 80

12 : 15 : 8

12 unit = 144

1 unit = 12

Income of B = 12 x 15 = 180

**Question 4:** A bag contains Rs 55 in the denominations of Rs 1, 50 paise and 25 paise coins. The coins re in the ratio 1 : 2 : 3. Find the number of 50 p coins.

(1) Rs 41

(2) Rs 42

(3) Rs 40

(4) Rs 38

**Solution:** __Re.1____50 P____25 P__

**Face Value ** = 1 : 50 : 25

**No. of coins ** = __1 : 2 : 3__

__ x : 2x : 3x__

x = 20

50 paise coins = 2x = 40/-

**Other method :**

4 : 4 : 3 = 11

**If 11 = 55**

1 = 5

4 = 20

50p coins = 20 x 2 = 40

**Question 5: ** Ratio of land and water on “earth is 1 : 2. In northern hemisphere, the ratio is 2 : 3. What is the ratio in Southern hemisphere.

(1) 2 : 11

(2) 4 : 11

(3) 3 : 11

(4) 5 : 11

**Solution: ** **Land : Water**

on earth 1_{x5x2} : 2_{x5x2} = 3_{x5x2}

North hemisph. 2_{x3} : 3_{x3} = 5_{x3}

** L : W**

Earth 10 : 20 = 30

N.H.S. __6 : 9 __ = 15

S.H.S. 4 : 11 = 15