## 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?

Answer:

Assume a1 : a2 = 2 : 3

b1 = Vijay’ salary = 1000

b2 = Badri’s salary

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?

Answer:

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.

Answer:

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.

Answer:

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.

Answer:

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.

Answer:

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.

Answer:

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.

Answer:

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