Question

The present ages of Peter and Jony are in the ratio of 4 : 3, four years later, their ages will be in the ratio of 6 : 5. What are their present ages?

Answer 1

Hint: The ratio of present ages and ages after four years of Peter and Jony are given. Using this ratio, we can form an equation with the use of a variable, calculate the value of the variable and substitute in the equation to find their present ages.

Complete step-by-step answer:
The ratio of present ages of Peter and Jony are given to be in the ratio of 4 : 3. If Peter’s present age is 4x then the Jony’s present age will be 3x.
 P = 4x
 J = 3x
Then after 4 years, 4 will be added to both their respective ages. After four years, their ages are given as:
 P = 4x + 4
 J = 3x + 4
The ratio of these ages is given as:
 $
  P:J = 4x + 4:3x + 4 \\
   \Rightarrow \dfrac{P}{J} = \dfrac{{4x + 4}}{{3x + 4}}.....(1) \\
  $
The ratio of their ages is given to be 6 : 5, then:
 $
  P:J = 6:5 \\
   \Rightarrow \dfrac{P}{J} = \dfrac{6}{5}.....(2) \\
  $
As (1) and (2) both are equal to $ \dfrac{P}{J} $ , they both will also be equal to each other.
 $ \Rightarrow \dfrac{{4x + 4}}{{3x + 4}} = \dfrac{6}{5} $
Solving this equation by cross multiplication to find the value of x:
 $
  \Rightarrow 20x + 20 = 18x + 24 \\
  \Rightarrow 20x - 18x = 24 - 20 \\
  \Rightarrow 2x = 4 \\
   \Rightarrow x = \dfrac{4}{2} \\
   \Rightarrow x = 2 \;
  $
The value of x is 2. The present ages of Peter and Jony are 4x and 3x respectively, so substituting the value of x to calculate their present ages, we get:
\[
  \Rightarrow P = 4x \\
  \Rightarrow P = 4 \times 2 \\
   \Rightarrow P = 8 \\
  \Rightarrow J = 3x \\
  \Rightarrow J = 3 \times 2 \\
   \Rightarrow J = 6 \;
 \]
Therefore, the present ages of Peter and Jony are 8 and 6 years respectively.
So, the correct answer is “ Present ages of Peter and Jony are 8 and 6 years respectively”.

Note: Ratio of a is to b in fraction can be written as
 $ a:b = \dfrac{a}{b} $ .
When the ratio of their present ages was given to be 4 : 3, we assumed their ages to be as 4x and 3x because we needed a variable to form an equation and for the ratio to be same, if the numerator was multiplied by x, the denominator also gets multiplied by x.
Ratio is generally used to carry out the comparison between two quantities.