**Ratio & proportion concepts**

Here, You will learn ratio & proportion concepts. A ratio is a comparison or simplified form of two quantities of the same kind. This relation shows how many times one quantity is equal to the other or in other words, a ratio is a number, which expresses one quantity as a fraction of the other.

The Ratio of x to y” is x : y.

**Example:** 2 : 3

The first term of a ratio is called the **antecedent **and the second the **consequent**.

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

**Some important types of ratio:**

1).**Compounded Ratio**: Ratio is compounded by multiplying together the antecedents for new antecedents, and the consequents for a new consequent.

**Example**: Find the ratio compounded of the four ratios:

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

**Solution**: the required ratio = *4 x 9 x 26 x 2*3 x 13 x 5 x 15 = 16/15 =16:15.

**Note**: When the ratio 4:3 is compounded with itself, the resulting ratio is

4^{2}: 3^{2} it duplicate ratio of 4:3.

4^{3}: 3^{3} is triplicate ratio of 4:3.

√4 : √3 is subduplicate ratio of 4:3.

a^{1/3}: b^{1/3} is subtriplicate ratio of 4:3.

2). **Inverse Ratio**: If 2:3 be the given ratio, then ½: ⅓ or 3:2 is called its inverse or reciprocal ratio.

**Some important problems :**

**Example**: 5600 rupees is divided among 3 persons three persons A,B and C in such a way that A receives 2/5th of the total share of B and C find :

a). Share of A.

b). Share of B and C total.

c). Share of B.

d). Share of C.

**Solution**: A:(B+C) = 2:5.

a). 5600/7*2 = 1600.

b). 5600/7*5 = 4000.

c). Can’t be determined.

d). Can’t be determined.

**Example**: 2575 rupees is divided among 3 persons A, B and C in such a way that if each of their shares reduced by 25 rupees; the ratio of their remainders would be 2:3: 5, find the share of B?

**Solution**: 2575 – (reduced by all shares of A+B+C)

So, 2575- (25+25+25)

= 2500

In question, given the ratio of A:B:C = 2:3:5.

So, Share of B 2500/10*3 = 750.

Share of B = 750+25 = 775.

### Proportion

**A statement of equality of two ratios is called proportion.**

Proportion is represented by the symbol ‘= ‘or ‘:: ‘

Four numbers, a, b, c, d, are said to be in proportion…. when the ratio of the first two, a and b, is equal to the ratio of the last two, c and d.

i.e. a/b=c/d

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

e.g. 2/3=6/9

or 2 : 3

**Basics of proportion:**

1). If four quantities be in proportion, the product of the extreme is equal to the product of the means.

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

We have 3/4 = 9/12, (Multiply each ratio by 4*12)

Hence, 3/4 *4*12= 9/12 *4*12

Hence, 3*12= 4*9.

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 to the third.

The second quality is called the mean proportional between the first and the third, and the third quantity is called the **third proportional**.

a : b = c : d, then c is called the third proportion to a and b.

**Example:** If 2, 5, x, 30 are in proportion, find the third proportional “x”.

**Solution:**

Here x is third proportional.

According to the concept

*2*5 = *x*30

5x = 60 → x = 12.

**FOURTH PROPORTIONAL:**

If a : b = c : d, then d is called the fourth proportional to a, b, c.

**Example:** Find the fourth proportional of the numbers 12, 48, 16.

**Solution:** Let the fourth proportional is x. Now as per the concept above the product of extremes should be equal to the product of the means → 12/48 = 16/x → x = 64.

**MEAN PROPORTIONAL:**

Mean proportional between a and b is √ab .

**Example**: Find the mean proportional between 3 and 75.

**Solution**: if X be the required mean proportional,

We have 3 : X :: X : 75

Hence, Square of X> = √3 x 75

X = 15.

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