Question

The ages of A and B are in the ratio. Fifteen years hence, the ratio will be 2:1. There present ages, respectively, are
A) 30 years, 10 years
B) 45 years, 15 years
C) 21 years, 7 years
D) 60 years, 20 years

Answer 1

Hint:
Here we have to find the present age of A and B. For that, we will first assume the age of A and B to be any variable and then we will calculate its ratio and we will equate it with the given ratio of their present ages. Then we will add 15 to each of their ages and will again calculate its ratio. We will equate this ratio with the given ratio of their ages after fifteen years. From these two equations, we will calculate the present age of A and B.

Complete step by step solution:
Let the present age of A be x, and present age of B be y.
Now, we will calculate the ratio of present age of A and B.
$\dfrac{\text{Present age of }A}{\text{Present age of }B}=\dfrac{x}{y}$
The ratio of present age of A and B given in the question is 3:1 now we will equate this ratio with the calculated ratio.
$\Rightarrow \dfrac{3}{1}=\dfrac{x}{y}$
On using cross multiplication method here, we get
$\Rightarrow 3y=x$…………… $(1)$
Fifteen years later, the present age of A becomes equal to $x+15$ and the present age of B becomes equal to $y+15$
Now, we will calculate the ratio of age of A and B after 15 years.
$\Rightarrow \dfrac{\text{ age of A after }15\text{ years}}{\text{age of B after 15 years}}=\dfrac{x+15}{y+15}$
The ratio of age of A and B after 15 years, which is given here is 2:1. Now, we will equate this ratio with the calculated ratio.
$\Rightarrow \dfrac{2}{1}=\dfrac{x+15}{y+15}$
On using cross multiplication method here, we get
$\Rightarrow 2y+30=x+15$
Simplifying the equation further, we get
$\Rightarrow 2y+15=x$…………… $(2)$
Now, we will put the value of x from equation $(1)$to equation $(2)$
$\Rightarrow 2y+15=3y$
Subtracting 2y from 3y, we get
$\Rightarrow y=15$
We will put the value of y in equation $(1)$
$\begin{align}
  &\Rightarrow 3\times 15=x \\
 & \therefore x=45 \\
\end{align}$

Therefore, the present age of A is 45 years and the present age of B is 15 years.
So, the correct option is B.

Note:
Since we have applied a cross multiplication method here, let’s understand it deeply.
A cross multiplication method is defined as a method in which we multiply the numerator of the first fraction with the denominator of the second fraction and we multiply the numerator of the second fraction with the denominator of the first fraction.
A cross multiplication method is also used in the addition and subtraction of unlike fractions.