Question

The ages of A and B are in the ratio $8:3$. Six years hence, their ages will be in the ratio $9:4$. Find the sum of their present ages.

Answer 1

Hint: Identify the known and unknown ratios and set up the proportion and solve accordingly.
In these ratio types of questions, take any variable as the reference number.

Complete step by step solution:
Given that:
The ages of A and B are in the ratio $8:3$
Let us suppose that, $'x'$ is common in the ages of A and B.
Therefore, Present age of A is equal to $8x$
And the Present age of B is equal to $3x$
Therefore age of A after six years would be $8x + 6$
Therefore age of B after six years would be $3x + 6$
Second given condition is that after six years the ratio of the ages would be $9:4$
$\dfrac{{8x + 6}}{{3x + 6}} = \dfrac{9}{4}$
Doing cross multiplication, and simplifying
$4(8x + 6) = 9(3x + 6)$
$32x + 24 = 27x + 54$
$32x - 27x = 54 - 24$
$5x = 30$
$x = 6$
Present age of A is equal to $8x$
Put, $x = 6$
Therefore,
$\begin{array}{l}
 \Rightarrow 8x = 8 \times 6\\
 \Rightarrow 8x = 48
\end{array}$
And the Present age of B is equal to $3x$
Therefore
$\begin{array}{l}
3x = 3 \times 6\\
3x = 18
\end{array}$
The sum of the present ages of A and B
$\begin{array}{l}
 = 8x + 3x\\
 = 48 + 18\\
 = 66
\end{array}$
Sum of the present ages of A and B $ = 66$ is the required solution.

Note: Always read the question twice and frame the word statement in the mathematical form. Segregate all the known and unknown terms and form the corresponding ratios and the proportions accordingly. Ratio is the comparison between two numbers without any units. Whereas, when two ratios are set equal to each other are called proportion.