# Tips & Tricks To Solve Ratio & Proportion – Advance Level

Ratio & Proportion is one of the most basic & easiest topics of the SSC exam. Questions from this chapter are usually asked in conjunction with the other concepts. About 3-4 questions are usually asked in preliminary examination that involves the usage of concepts of ratio & proportion.

Ratio & Proportion basics are used in almost every chapter of arithmetic portion I.e, Profit & loss, average, partnership, percentage, etc.

Let’s start with the basic definition of Ratio & Proportion.

**What is Ratio?**

#### It tells us about the relation between the two numbers.

**Example: **The ratio of A to B is 3:5.

This means Value of A/ Value of B = 3/5

**What is Proportion? **

#### It is the equality of two ratios. Many students often remain confused about the basic definition of Ratio & Proportion. Here we will try to clear that confusion.

A/B = 3/5 (This is the ratio of A to B)

Here, A is called the antecedent & B is called the consequent.

A/B = C/D (This is Proportion)

Proportion can also be written like,

a:b:: c:d in which Product of means= Product of extremes.

⇒ a×d = b×c

**Compounded Ratio**

If two or more ratios are given, then the ratio of the product of all antecedents to the product of all consequents is called compounded ratio.

Compounded Ratio of a:b, c:d, e:f will be, ace: bdf

### Different Types of Proportion

**Mean Proportion**

a:M :: M:b, here M is the mean proportion of a & b.

M= √ab

**Third Proportion**

a:b :: b:T, here T is the third proportion of a & b.

T= b^{2}/a

**Fourth Proportion**

a:b :: c:F, here F is the fourth proportion

F= (b×c)/a

Now let us try to understand some *basic fundamentals* of ratio that are usually used in questions asked in SSC exam through questions. We will also do some mixed problems of different topics involving the usage of ratio & proportion.

**Illustration 1.** If A:B = 3:4, B:C = 7:8 then find A:B:C?

**Solution:** B is common to both, so make B equal. Multiply 1 with 7 & 2 with 4.

⇒ A:B=21:28, B:C= 28:32

Hence, A:B:C = 21:28:32

**Shortcut= ** A : B (B)

(A) B : C

Write, the ratio in this format & then fill the blank spaces (mentioned with numbers in bracket) with the nearest number. Then, multiply vertically & you will get the answer.

**Solution using shortcut=** 3 : 4 (4)

(7) 7 : 8

Answer is, 21 :28: 32

**Illustration 2.** If A:B=3:4, B:C= 5:6, C:D= 7:8. Find A:B:C:D?

**Solution:** Now, if we approach this question using the basic method, it will take a lot of time. Use the shortcut described above to solve this problem.

A : B : C : D

3 : 4

5 : 6

7 : 8

We have written all the given ratios in this format.

Now fill the empty spaces with the nearest number

A : B : C : D

3 : 4 (4) (4)

(5) 5 : 6 (6)

(7) (7) 7 : 8

Hence the ratio = 105: 140: 168: 192

**Illustration 3.** If ratio is given in reciprocals, A:B:C = 1/4 : 1/7 : 1/12. Convert it

**Solution.** Multiply the numerator with 4×7×12, we get

A : B : C = 7× 12 : 4× 12: 4×7 = 84: 48: 28 = 21: 12: 7

**Illustration 4.** If 2 numbers are in the ratio of 4:7. If 9 is added to both the numbers, the ratio becomes 7:10. Find the sum of the original numbers?

**Solution:** When 9 is added to both the numbers, 4 becomes 7 (in antecedent) & 7 becomes 10 (in consequent). It means the addition of 9 is equivalent to 3 units.

⇒ 1 unit= 3

Now we have to find some of the original numbers, i.e (4+7)×3 = 33

**Illustration 5.** If the Ratio of A:B in a mixture is 4:9 and after adding 9 litres of liquid B the ratio now becomes 1:3. Find the original quantity if liquid A in the mixture?

**Solution.** A:B is 4:9, after adding 9 litres of new, the ratio is 1:3.

As only B is added, the quantity of A should remain the same. Making the quantity of A equal in both the ratios by multiplying second by 4, we get 4: 12.

Now, compare both the ratios for B, 12-9 units = 9 litres

⇒ 1 unit = 3 litres. Hence, 4 units, i.e quantity of liquid A is 12 litres.

**Illustration 6.** The mean proportion of 9 & 25 is M. Third proportion of M & 15 is T. Find the fourth proportion of T, 18 & 30

**Solution.** M = √9×25 = 15

Third proportion of M & 15 = (15×15)/ 15 = 15 = T

Fourth proportion of T, 18 & 30 = (18 × 30)/ 15 = 36

**Illustration 7.** There are 3 containers with equal capacity. The ratio of water to oil in the first container is 4:3, in the second container is 9:5 & in the last container is 13:8. If these containers are mixed together. Find the ratio of water to oil in the resultant mixture.

**Solution. **As capacity is equal, so, making the capacity equal first & then adding.

Water: oil = 77:49

**Illustration 8.** 150 litres of a mixture contains milk & water in a ratio of 13:2. After the addition of some more milk to it, the ratio of milk to water now becomes 9:1. The quantity of milk that was added to the mixture was?

**Solution.** The concept to catch here is that water remains the same in both, so equate that in both ratios. We get, x = 5 units. 15 units = 150 L, so the water added is 50 litres.

**Illustration 9.** A container has a mixture of 2 liquids in a ratio of 7:5. When 9 litres of the mixture are drawn off & filled with B, the ratio now becomes 7: 8. Find the quantity of the initial mixture?

**Solution.** When we took 9 litres out of the mixture the first time, it will not bring any change in the ratio. Then 9 litres of B is added, which gives us 3 units = 9 L, i.e 1 unit = 3 Litres. Now, the total quantity of the mid mixture is, (7+5)× 3= 36 litre. So total of initial mixtures = 36+9 = 45 litres.

**Illustration 10. **The annual income of Varsha & Pooja is in the ratio of 3:2, while the ratio of their expenditure is 4:3. If they save 2000 & 1000 respectively. Then Varsha’s income is?

**Solution. **This is a shortcut to solving questions like this. First write both the ratios, then savings. Cross multiply & take the difference as shown below. It gives us 1 unit = 2000. So, Varsha’s income is 3 × 2000 = 6000

**Illustration 11.** A man divides his property among his 3 sons in the ratio of 7: 4: 2 If the middle son gets, 18000 Rs more than the younger son. Find out how much *more the elder* son got than the combined money of the middle & the younger son.

**Solution.** Given 4x – 2x = 18000, x= 9000

Now asked, 7x – 6x= x = 9000

**Illustration 12:** 564 Rs was to be divided among A, B & C in the ratio of 1/3 : 1/4 : 1/5. But by mistake, it was divided into 3:4:5. The amount in excess received by C was?

*Solution.* The excess amount from the picture below is 235-144= 91

**Illustration 13.** A man covers a long journey by train, bus & *rikshaw* in the ratio 4:5:2. The ratio of the fair* is* 2:3:1. The total expenditure was 500. Then the expenditure on the train was?

**Solution.** As shown in the picture below, this question is based on a compounded ratio. Expenditure on train= 160 Rs

**Illustration 14.** If we increase the price of the movie ticket in the ratio 10:11. The number of tickets sold decreases in the ratio of 10:9. Then the loss in the revenue of the theatre is?

**Solution.**

**Illustration 15.** The ratio of the sum of salaries of Raju & Mohan to the difference in their salary is 10:1. Also, the ratio of the sum of salaries of Mohan to Isha is 7:2. If the sum of their salaries is 16400. Find the salary of Mohan?

**Solution.** Now, R+M+I = 16400 = 25

Mohan salary is (9 × 16400)/25 =164 * 36 = 5904

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