Dear Reader,

Below are three age problems involving percentages.

Question 1

There are three brothers x,y and z. If z's age is as much more than y's as y's age is more than x's age, find by how much percentage y is older than x. Hints: Age of z is 20 and y is 5 years older than x.

a. 100% b. 50% c. 25% d. 75%

Answer : b. 50%

Solution.:

Let a,b and c be the ages of x,y and z respectively.

It is given that z's age is as much more than y's as y's age is more than x's age.

Writing the above statement in the form of an equation, we get

c - b = b - a ...(1)

It is given that age of z i.e c = 20

Substituting c = 20 in eq 1 we get,

20 - b = b - a

20 = 2b - a ...(2)

It is also given that y is 5 years older than x. This implies b - a = 5 or b = a + 5...(3)

Substitute b = a + 5 in eq 2 we get,

20 = 2a + 10 - a

20 = a + 10 or a = 10

Substitute a = 10 in eq 3 we get

b = 15

Percentage by which b is more than a is (b - a)/a x 100% = (15 - 10)/10 x 100 = 50%

Question 2

John is 100% older than Johan. 20 years hence, John will be 50% older than Johan. Can you find their ages.

a. 30,20 b. 40,20 c. 60,40 d. 50,10

Answer : b. 40,20

Solution :

Let age of John be b.
Let age of Johan be a.

Currently John is 100% older than Johan. This implies b - a/a = 100%

Or (b - a)/a = 1 ( this is because, 100% = 100/100 = 1)

Or b - a = a
Or b = 2a ...(1)

20 years later ages of John and Johan will be b + 20 and a + 20 respectively

After 20 years John will be older by 50%

i.e (b + 20)-(a+20)/(a+20) = 50% = 1/2

(b - a)/ (a + 20) = 1/2

Or 2b - 2a = a + 20
Or 2b = 3a + 20 ...(2)

Substitute 1 in 2 we get

4a = 3a + 20 or a = 20.

Substitute a = 20 in eq 1 we get

b = 40

John's age is 40 and Johan's 20.

Question 3

Let a,b and c be ages of three brothers. b's age is greater than the sum of ages of a and c. After 7 years, by how much percentage b will be more than the sum of ages of a and c ?

a. (b - a - c - 7) / (a + c + 14) x 100% b. (b - a - c + 7) / (a + c + 14) x 100% c. (b - a - c) / (a + c + 7) x 100% d. (b - a - c - 7) / (a + c + 7) x 100%

Answer : a. (b - a - c - 7) / (a + c + 14) x 100%

Solution :

Percentage by which b's age will be greater than sum of a's and c's ages after 7 years

= Age of b after 7 years - (sum of ages of a and c after 7 years)
------------------------------------------------------------------------------------ x 100%
sum of ages of a and c after 7 years

= b + 7 - (a + 7 + c + 7)
--------------------------------- x 100%
a + 7 + c + 7

= (b - a - c - 7) / (a + c + 14) x 100%