RATIO and PROPORTION
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)
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
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.
₹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.
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