Question

Richard's present age in years is 50 percent of Joseph's.In another few years,Richard's age will be 2/3rd of Joseph's.By what percent will Joseph's age increase during this period?
A. 20
B. 40
C. 50
D. 60

Answer 1

Given:

Richard's present age in years is 50 percent of Joseph's present age

After few years, Richard's age will be 2/3rd of Joseph's

To find:

The percentage increase in Joseph's age during this period

Solution:

Let' assume,

"R" = the present age of Richard (in terms of years)

"J" = the present age of Joseph (in terms of years)

Richard's age at present is 50% of Joseph's age, so we have

R = 50% of J

⇒ R = $\frac{50}{100}$ × J

⇒ R = ½ × J

⇒ R = $\frac{J}{2}$ ....... (i)

Let "x" years be the time after which Richard's age will be 2/3rd of Joseph's age.

After x years,

Richard's age = [R + x] years

Joseph's age = [J + x] years

So, according to the question, we have the equation as

$[R + x] = \frac{2}{3} * [J+x]$

⇒ 3 * [R + x] = 2 * [J + x]

⇒ 3R + 3x = 2J + 2x

⇒ 3x - 2x = 2J - 3R

⇒ x = 2J - 3R

substituting the R = $\frac{J}{2}$ from (i)

⇒ x = 2J - 3( $\frac{J}{2}$)

⇒ x = $\frac{4J\: -\: 3J}{2}$

⇒ x = $\frac{J}{2}$ years

∴ Joseph's age after x = $\frac{J}{2}$ years will be = J + x = $J + \frac{J}{2}$ = $\frac{2J + J}{2} = \frac{3J}{2}$ years

∴ The percentage increase in Joseph's age during "x" years will be,

=  $\frac{[Change\:in\:Joseph's\:age]}{[Present\:age\:of\:Joseph]} * 100$

= $\frac{\frac{3J}{2} - J}{J} * 100$

= $\frac{\frac{3J \:- \:2J}{2}}{J} * 100$

= $\frac{\frac{J}{2}}{J} * 100$

= $\frac{1}{2} * 100$

= 50% ← option (C)

Thus, Joseph's age will increase by 50% during this period.

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