# Shortcuts for Ratio and Proportion – 1

## Shortcut 1: Direct proportion

$\frac{a1}{a2}=\frac{b1}{b2}$

Question:

Ratio between salary of Vijay and Badri is 2:3. If salary of Vijay is Rs. 1000, what is the salary of Badri?

Assume a1 : a2 = 2 : 3

b1 = Vijay’ salary = 1000

So,

2 : 3 = 1000 : b2

2/3 = 1000/b2   b2 = 1500

## Shortcut 2: Inverse Proportion

$\frac{a1}{a2}=\frac{b2}{b1}$

Question:

A can complete a work in 30 days and B can complete the same work in 35 days. In what ratio they should get the salary, if they are working together?

Salary in case of combined work is inversely proportional to time taken by an individual.

Assume a1 : a2 = 30 : 35

So,

b1 : b2 = 35 : 30

Salary ratio if they work together is = 7 : 6

## Shortcut 3: Third proportion

$\begin{array}{l}\frac ab=\frac bc\\b=\sqrt{ac}\end{array}$

Question:

Three numbers are in proportion to each other. If the first two numbers are 2 and 8, find the third number.

Assume a = 2; b = 8

a/b = b/c

2/8 = 8/c

c = 32

Question:

Three numbers are in proportion. If the first number is 2 and the last number is 8, find the second number.

b =√ac

b =√(2 x 8) = 4

## Shortcut 4: Fourth proportion

$$\frac ab=\frac cd$$

Question:

Four numbers are in proportion to each other. If the first three numbers are 30, 35, 36, find the fourth number.

Assume, a = 30; b = 35; c = 36

30/35 = 36/d

d = 42

## Shortcut 5: Find A : C

$$\frac AB\times\frac BC=\frac AC$$

Question:

The ratio between salary of A and B is 4 : 5. The ratio between salary of B and C is 3 : 4. Find the ratio between salary of A and C.

A/B = 4/5

B/C = 3/4

A/C = (4/5) x (3/4)

A/C = 3/5

Ratio between salary of A and B = 3 : 5

## Shortcut 6:  Find A : D

$$\frac AB\times\frac BC\times\frac CD=\frac AD$$

Question:

Ratio between salary of A and B is 3 : 5, B and C is 4 : 5, C and D is 6 : 7. If the salary of A is 7200, find the salary of D.

A/B = 3/5

B/C = 4/5

C/D = 6/7

A/D = (3/5) x (4/5) x (6/7)

A/D = 72/175

7200/D = 72/175

D = 17500

Salary of D is 17,500

## Shortcut 7: Common factor

Substitution of x

Question:

Two numbers are in the ratio 4 : 5. Sum of their squares is 1025. Find the numbers.

Substitute the common factor x to the ratio 4 : 5

Assume the actual numbers as 4x and 5x

Given,

(4x)² + (5x)² = 1025

16x² + 25x² = 1025

41x² = 1025

x² = 25

x = 5

Substitute x = 5 in 4x and 5x to find the numbers

The numbers are,

20, 25

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