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Ratio Proportion

Concept of Ratio Proportion and Examples

Every year 1or 2 questions come directly from this chapter. But this chapter is very useful in solving the problems of Data Interpretations and other chapters. Let us discuss the concept of ratio proportion

Ratio:

The comparison between two quantities in terms of magnitude.

Suppose Ram has 7 balls and Shyam has 9 balls. So, the ratio of the balls between Ram and Shyam is 7 is to 9. It can be expressed as 7 : 9.

For ratio the order should be maintained. 

For the above mentioned example. If we have written it as 9 : 7 then it would have been wrong.

So, the ratio of any two quantities a and b in the same units can be expressed as

In ratio a is called first term or antecedent and b is called second term or consequent.

Rule of Ratio:

The comparison of two quantities is meaningless if they are not of the same kind or in the same units.

We cannot compare 10 girls’ with 7 boys, 5 km with 12 centimeters. 6 cows with 13 goats. Therefore to find the ratio of two quantities we need to express them in the same units.

Properties of Ratios:

1. The value of a ratio does not change when the numerator and denominator both are multiplied by same number.

e.g.

All have the same ratio.

2. The value of a ratio does not change when the numerator and denominator both are divided by the same quantities

e.g.

All have the same ratio.

3. Compounded Ratio:

When two or more than two ratios are multiplied with each other. Then it is called Compounded Ratio.

This is called Compounded Ratio.

4. When the ratio is compounded with itself, then it is called duplicate, triplicate ratio, etc.

Proportion:

The equality of two ratios is called proportion.

If a : b = c : d or $\frac{\mathbf{a}}{\mathbf{b}}\mathbf{=}\frac{\mathbf{c}}{\mathbf{d}}$ then we can say that a, b, c, d are in proportion. We write it as a : b :: c : d, where the symbol ‘::’ indicates proportion and we read it as ‘a is to b as c is to d’.

Here a, and d are called extremes, and b and c are called means.

Properties of proportion:

If four numbers are in proportion then we can say that product of the extremes is equal to the product of means.

If a : b :: c:d

Then a x d = b x c

#  Fourth Proportional:

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

#  Third proportion or continued proportion:

If a : b = b : c then these numbers a, b, c are said to be in continued proportion or simply in proportion.

Here b is said to be the mean proportional between a and c, and c is said to be the third proportional to a, b

Example:

4 : 8 :: 8 : 16

Here 4 x 16 = 8 x 8

So we can say that 4, 8, 16 are in continued proportion.

#  Invertendo:

#  Alternando:

#  Componendo:

#  Dividendo:

#  Componendo and Dividendo:

#  If a:b and b:c are given then 

a : b : c = a   

b = (a.b) : (b.b) : (b.c)

Examples:

1. Find the ratio of 27 to 36

Solution:

2. Find the ratio of 250 gm. To 1 kg.

Solution:

To get the ratio we need to convert kg. into gm.

We know 1 kg. = 1000 gm.

3. In an office total of 169 employees are working. Out of them 65 are male and the remaining are female. Find the ratio of the number of females to number of males.

Solution:

Total number of employees 169

Out of them 65 are male.

Total female employees = 169 – 65 = 104

So, The ratio of number of female to number of male = 104 : 65 = 8 : 5

4. Divide 4200 RS among a, b, c in the ratio of 6 : 5 : 3.

Solution:

Total amount is 4200 RS

a : b : c = 6 : 5 : 3

a’s share is 6x

b’s share is 5x

c’s share is 3x

Then,

a’s share is 6 X 300 RS = 1800 RS.

b’s share is 5 X 300 RS = 1500 RS.

c’s share is 3 X 300 RS = 900 RS.

Solution:

Short trick:

6. a : b = 2 : 3  b : c = 5 : 6 then find a : b : c.

Solution:

7. A person earns RS 2000 per day. He spends RS 750 per day. Find the ratio of his savings to expenditure?

Solution:

∴ Savings = RS 2000 – RS 750 = 1250 RS

So, the ratio of savings to expenditure

8. The ratio of A: B = 2 : 3, B : C = 4 : 5 C : D = 6 : 7. Find A : B : C : D.

Solution:

A : B = 2 : 3

B : C = 4 : 5

C : D = 6 : 7

9. If a : b = 2 : 3 Then find the value of (6a – 3b) : (9a – 4b)

Solution: 

Simply we need to put a = 2, b = 3, 

10. A bag contains 1 RS, 5 RS, and 10 RS coins in the ratio 5:3: 4, amounting RS 300. Find the number of coins of 1 RS, 5 RS, and 10 RS.
 

Solution:

The ratio of the number of 1 RS., 5 RS., 10 RS. coins are 5 : 3 : 4

So, if the number of 1 RS. coins are 5x then the number of 5 RS. and 10 RS. coins are 3x and 4x respectively

So, the value of 5x piece 1 RS coin is 5x X 1 RS = 5x RS.

The value of 3x piece 5 RS. coin is 3x X 5 RS. = 15x RS.

The Value of 4x piece 10 RS. coin is 4x X 10 RS. = 40 RS.

Total amount in the bag is 300 RS.

So,

So, Number of 1 RS. coin is 5 X 5 = 25

Number of 5 RS.coin is 5 X 3 = 15

Number of 10 RS.coin is 5 X 4 = 20

11. In an office there are 15 male employees and 10 female employees. During a recruitment drive, the same number of male and female is selected and the present ratio of male to female employees is 6 : 5. So, how many new employees got selected?

Solution: 

Let x number of male and female employees selected newly.

So, 15 male and 15 female candidates were selected.

So, Total 30 new employees got selected.

12. The first, second, and fourth terms of a proportion are 5, 25, and 80. Find the third term

Solution:

Let the 3rd term be x

So, 5, 25, x, and 80 are in proportion.

So, 5 : 25 :: x :80

We know Product of Extremes = Product of Means

So, the third term is 16.

13. What is the least possible number which must be subtracted from 11, 14, 17, 22. So, that the resulting number is in continued proportion.

Solution:

Let the number be x.

So,

So, 2 should be subtracted from 11, 14, 17, 22. So, that they will be in proportion.

14. Find the mean proportional between 5 and 320.

Solution:

15. Ratio between the number of men and women in an office is 4 : 7. If the number of women working is 49. Find the number of men working in the office.

Solution:

Suppose the number of men working is x.

So, 28 men are working in that office.