Maths - Ratio and Proportion

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