## RATIO and PROPORTION

**Ratio:**

The ratio between two quantities of the same kind is expressed in simple terms as the number of times one quantity contains the other.

The ratio of two quantities a and b is the fraction and is expressed as

*a : b**a : b*can also be expressed as

*a / b*or

*a ÷ b*

Ratio can be

① Duplicate Ratio i.e.

*a² : b²*of

*a : b*

*② Triplicate Ratio i.e.*

*a³ : b³*of

*a : b*

③ Sub -Duplicate Ratio i.e.

*√a : √b*of

*a : b*

*④ Sub - triplicate Ratio i.e.*

*∛a : ∛b*of

*a : b*

⑤ Compound Ratio i.e.

*ab : cd*is compound ratio of

*a : c*and

*b : d*

⑥ Inverse ratio i.e.

*1/a : 1/b*of

*a : b*

**Componendo and Dividendo**

If a/b = c/d then (a+b)/(a-b) = (c+d)/(c-d)

**Proportion**:

If the ratio of two quantities is equal to the ratio of any other two quantities then the four quantities are said to be in proportion.

This is represented as

*a : b : : c : d*

**Variation**

If two quantities a and b are related in such a way that as the quantity a changes it also brings a

change in the second quantity b, then the two quantities are in variation.

**Direct Variation**: If value of a increases or decreases with corresponding increase or decrease of value b

i.e.

*a α b*⇒

*a/b = k*(constant)

**Inverse Variation:**The quantity a varies inversely to b when value of a increases or decreases with corresponding decrease or increase in value of b.

i.e.

*a α 1/b*⇒

*a ✕ b = k*(constant)

**Question 1: Reduce the ratio ₹34 : ₹ 136 to simple ratio.**

Answer:

₹34 : ₹ 136 = 34/136 = 1/4 = 1 : 4

**Question 2: Divide 60 in the ratio of 1 : 3**

Answer: It means divide 60 into 1 part and 3 parts

∴ Total parts = 4

First Part = (1/4) ✕ 60 = 15

Second Part = (3/4) ✕ 60 = 45

**Question 3: A train covers 360 km in 6 hours and another train covers the same distance in 540 minutes. Find the ratio between speeds of two trains.**

Answer:

Speed of first train = 360 / 6 = 60 km/h

Speed of 2nd train = 360/540min = 360/9hrs = 40 km/hr

Ratio of speeds = 60/40 = 6/4 = 3/2 ⇒ 3 : 2

**Question 4: The ratio between two numbers is 3 : 7. If their LCM is 210, find the numbers.**

Answer: Let the numbers be 3p and 7p

LCM = 3 ✕ 7 ✕ p = 210

⇒ p = 210/21 = 10

Thus the numbers are 30 and 70