Question

The ages of Rahul and Haroon are in ratio \[5:7\]. Four years later the sum of their ages will be 56 years. What is their present age?

Answer 1

Hint: Here we take the given ratio and write it in a fraction form. Use the concept of ratio that the ratio remains the same if both numerator and denominator are multiplied by the same value say x and write the value of the present ages of Rahul and Haroon in terms of x. Add 4 to ages of both Rahul and Haroon. Equate the sum of new ages of Rahul and Haroon to 56 and solve for the value of x. Substitute back the value of x to find the present ages of Rahul and Haroon.
* Ratio \[m:n\] can be written in fraction form as \[\dfrac{m}{n}\]

Complete step-by-step answer:
We are given the ratio of ages of Rahul and Haroon is \[5:7\].
Therefore, we can write the ratio in form of a fraction as \[\dfrac{5}{7}\]
Let the present age of Rahul be 5x
Let present age of Haroon be 7x
Then the ratio of their ages is \[\dfrac{{5x}}{{7x}} = \dfrac{5}{7}\]
Now, we know the sum of ages of Rahul and Haroon after 4 years is 56.
Age of Rahul after 4 years is \[5x + 4\]years
Age of Haroon after 4 years is \[7x + 4\]years.
Take the sum of ages after 4 years and equate it to 56
\[ \Rightarrow 5x + 4 + 7x + 4 = 56\]
Add like terms in LHS of the equation
\[ \Rightarrow 12x + 8 = 56\]
Shift the constant value to RHS of the equation.
\[ \Rightarrow 12x = 56 - 8\]
\[ \Rightarrow 12x = 48\]
Divide both sides of the equation by 12
\[ \Rightarrow \dfrac{{12x}}{{12}} = \dfrac{{48}}{{12}}\]
Cancel the same terms from numerator and denominator on both sides of the equation.
\[ \Rightarrow x = 4\]
Substitute the value of x as 4 in assumed ages of Rahul and Haroon i.e. 5x and 7x respectively.
Therefore, present age of Rahul is \[5 \times 4 = 20\] years
Present age of Haroon is \[7 \times 4 = 28\] years.

Therefore, the present ages of Rahul and Haroon are 20 years and 28 years respectively.

Note: Students might make the mistake of writing the fraction equal to a variable x which is a wrong method to solve because then we will have fixed one of the ages and we are taking the second age depending on the first age. In the concept of ratio we have a common factor or a term between the two ages, here we are assuming that term to be x.