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# Ration and Proportion

Ratio and proportion concept is most important to solve questions related to Ages, Partnership, Mixture and Allegation and when one quantity is measure with respect to another.

Ratio is defined as comparison of two like terms or Quantities in its simplest form and the number of times one quantity contains another quantity of same type.

The ratio between m and n, separated by colon can be represented as m: n or m/n where, m the numerator is called antecedent and n the denominator is called consequent. If both the numerator and denominator is multiplied or divided by the same number then the ratio will remain same.

Important formula related to ratio are explained as follows:-

(1) Duplicate Ratio: – Duplicate ratio is defined as to the ratio of the squares of both the numerator (antecedent) and denominator (consequent) is called duplicate ratio.

Duplicate ratio of m: n = m2:n2

For example: 4:6 = 42:62 = 16:36.

(2) Sub-Duplicate Ratio: – Sub-Duplicate ratio is defined as the ratio of the square roots of both the numerator (antecedent) and denominator (consequent) is called sub-duplicate ratio.

Sub-Duplicate ratio of m: n = √m:√n

For example: 49:64 = √49:√64 = 7:8.

(3) Triplicate Ratio: – Triplicate ratio is defined as the ratio of the cube of both the numerator      (antecedent) and denominator (consequent) is called Triplicate ratio.

Triplicate ratio of m: n = m3:n3

For Example – 4:6 = 43: 63= 64:125

(4) Sub-triplicate Ratio: – Sub-Triplicate ratio is defined as the ratio of the cube roots of both the numerator (antecedent) and denominator (consequent) is called sub-Triplicate ratio.

Sub-triplicate Ratio of m: n = 3m: 3n

For Example: 216:512 = 3216: 3512 = 6:8.

(5) Inverse Ratio: – Inverse Ratio is defined as the ratio in which the position of their antecedent and consequent are interchanged.

Inverse Ratio m: n = n: m.

For Example 6:8 = 8:6.

Proportion: – Proportion is defined as equality of two ratios, represented by double colons, is called the proportion.

If P: Q:: R: S can be written as P:Q = R : S or P/Q = R/S, then we can say that P, Q, R, S are in proportion.

Some important case (Rule) of proportion:-

Case I: – If p: x:: x: r, then x is called second proportional or mean proportional to p and r.

⇒ p : x ∷ x : r

⇒ p/x = x/r

⇒ p× r = x× x

⇒ x² = pr

⇒ x =√pr

Case II: – If p: q:: q: r, then r is called third proportional to p and q.

⇒ p : q ∷ q : r

⇒ p : q = q: r

⇒ p /q = q/ r

Case III- If P: Q:: x : y, then y is called the 4th proportional to P, Q and x, y.

i.e P : Q ∷ x : y

⇒ P: Q = x: y

⇒ P / Q = x / y

⇒ P × y = Qx

Where, P and Y is called extremes term, Q and x is called mean terms.

Some examples are as follow:-

Example: – calculate the mean proportion between 8 and 18?

Explanation: –

Let us suppose x as mean proportion

⇒8: x:: x:18

⇒8/x = x/18

⇒x2 = 8×18

⇒ x =√144

⇒ x = 12

Thus, mean proportion between 8 and 18 is 12.

Example: – Find the third proportion to 12 and 24?

Explanation:-

Let us suppose x as third proportion

⇒ 12:24:: 24: x

⇒ 12/24 = 24/x

⇒ 12 × x = 24 ×24

⇒ x   = 24 × 24/12

⇒ x = 24 ×2

⇒ x = 48

Thus, third proportion to 12 and 24 is 48.

Example:- Find the fourth proportion of 6,4,12 ,?

Explanation: – Let us suppose x as fourth proportion

⇒6:4:12: x

⇒6/4 =12/x

⇒x = 12×4

6

⇒x= 2×4

⇒x=8

Thus, fourth proportion of 6,4,12 is 8.

Example: – if x: y = 5:4, y: z =2:3 then find x: y: z?

Explanation:-

x: y = 5:4

y:z = 2:3

As we see in the above ratio y is common, so multiply the ratio of y in cross i.e.

x:y = (5:4) × 2

y:z = (2:3) × 4

x:y  = 10: 8

y:z  =  8: 12

= 10:8:12

Trick:

In trick we multiply the number of first column (5×2) and last column (4×3) vertically. And the first column second number and second column first number i.e. (2×4) diagonally as shown above.

Therefore, x: y: z =10: 8:12

Example: – The sum of two numbers in ratio of 5:9 is 210. Find out the difference between them?

Explanation:-

Let us suppose the two numbers be 5x and 9x respectively.

According to the question

5x + 9x = 210

14x = 210

x = 210/14

x = 15

The, difference between the two number is 9x – 5x= 4x = 4 ×15 = 60.

Example: – The ratio of monthly income of Neha and Amit is in ratio of 2: 3 and their expenditure is in ratio 7:12.  When each of them saves 3000 Rs, then find their income and expenditure?

Explanation:-

Let income of Neha and Amit be 2x and 3x and their expenditure is 7y and 12y respectively.

Income – Expenditure = saving

2x-7y = 3000……………….. (i)

3x-12y = 3000……………… (ii)

By solving equation (i) and (ii)

(2x – 7y   = 3000) ×3      (by cross multiplying the coefficient of x)

(3x – 12y = 3000) ×2

6x – 21y =9000

6x – 24y = 6000       (By subtracting)

–      –          –

3y =   3000

y = 3000/3

y= 1000

Substitute the value of y in equation (i)

2x – 7×1000 = 3000

2x = 3000 + 7000

x =10000/2

x = 5000

Therefore, Income = 2x, 3x = 2×5000, 3×5000 =10000 Rs, 15000 Rs.

Expenditure = 7y, 12y = 7×1000, 12×1000 =7000 Rs , 12000 Rs.

Example: If two numbers are in ratio 6: 4. Their ratio becomes 7: 5. When 8 is added to both the numbers, Find the lowest number?

Explanation:-

Let us suppose the two numbers be 6x and 4x respectively.

According to the question,

6x + 8 = 7

4x + 8    5

⇒ 5(6x + 8) = 7 (4x + 8)       (By cross multiplying)

⇒ 30x + 40 = 28x + 56

⇒ 30x – 28x = 56 – 40

⇒ 2x    =   16

⇒   x    = 16/2

⇒   x    = 8

Thus, lowest number is 4x = 4×8 = 32.

Example: – Two-third of Neha amount is equal to two-ninth of Reena amount. What will be the ratio between their shares of amount?

Explanation:-

Let us suppose the Neha’s amount is N and Reena’s amount is R.

According to the question,

2/3 N = 2/9 R

⇒ N/R =2/9 × 3/2

⇒ N/R = 1/3

Therefore, the required ration of N: R = 1: 3.

Example: – The angles of triangle are in the ratio of 2:6:4, what are the angles of the triangle?

Explanation: –

Let us suppose the angles of triangle are 2x, 6x, 3x

According to the question,    (we known that sum of all the angles of triangle be 180˚)

2x+6x+4x = 180

12x   = 180

x     = 180/12

x =15

The angles of triangle are 2x =2×15=30˚, 6x = 6×15=90˚, 4x= 4×15=60˚.

Example: – The ratio of number of men and women in an office is 8:7, if the number of men and women increase by 10% and 20% respectively then what is the new ratio?

Explanation: –

Let the number of men and women be 8x and 7x

As we known that,

10 % increase on 100 % means 110% of 8x.

20 % increase on 100 % means 120% of 7x.

⇒110 % of 8x :  120 % of 7x

⇒110/100 ×8x : 120/100 ×7x

⇒ 880x/100:  840x/100   (by cancelling 100)

⇒ 88: 84                 (Dividing by 2)

⇒ 44: 42

⇒ 22:  21

Therefore, the required ratio of men and women be 22:21.

Example: – In a company the ratio of male and female employees is 7:4, if the total number of employees in the company is 880, find the number of female employees?

Explanation:-

Let the number of employees be x, therefore male and female employees be 7x and 4x respectively.

Number of female employee’s ⇒ [(Ratio of female employee’s ÷ Total ratio of male and              female) × total number of employees].

⇒ [4x/ (7x+4x)] ×880

⇒ 4x/11x × 880

⇒ 4 × 80

⇒ 320

Thus, number of female employees is 320.

Example: – What number must be added to each terms of ratio 11:6, so that their ratio will become 5:4?

Explanation: –

Let, us suppose the number added to both the terms be x

11+ x = 5

6 + x    4

By cross multiplication

4× (11+x) = 5× (6+x)

44 + 4x = 30 + 5x

44 – 30 = 5x – 4x

14   = x

Therefore, 14 must be added to both the numbers to get ratio 5:4.

Example:- If 130 pencils where distributed among 60 students in such a way that each girl gets 2 pencils and each boy gets 3 pencils, find out the ratio of number of girls and boys in class?

Explanation: –

Let us suppose number of girls and boys be x and y respectively in class.

According to question

x + y = 60……………….. (i)

2x + 3y =130……………. (ii)

By solving equation (i) and (ii)

x + y =   60………………. (i) ×2 (by cross multiplying the coefficient of x)

2x + 3y =130……………. (ii)×1

2x + 2y = 120

2x + 3y = 130

–         –       –

– y = -10

y = 10

Substituting the value of y in equation (i)

x + y = 60

x = 60 – y

x = 60 – 10

x = 50

Hence, the ratio of number of girls and boys is 50:10 or 5:1.

Example: – The sum of 6 % of x and 5 % of y is 2/3 of the sum of 6% of x and 10 % of y. then find the ratio x: y?

Explanation:-

(6 % × x + 5% × y) = 2/3 (6% × x + 10 % of y)                   (By cross multiplying)

3× (6 % × x + 5% × y) = 2× (6% × x + 10 % of y)

18%x+15%y = 12%x + 20%y

18%x – 12%x = 20%y – 15%y

6%x    = 5 %y

6x       =   5y

100          100         (by cancelling 100)

6x     = 5y

x   = 5

y      6

Hence, the required ratio of x: y =5:6.

Thanks.