Question

The present ages of A and B are in the ratio 4:5. Ten years ago their ages were in the ratio 3:4. What are their present ages?

Answer 1

Hint: Age of A and B are in the ratio 4:5, so let the present age of A be 4x and the present age of B be 5x. So ten years ago (before) the age of A was $ 4x - 10 $ and the age of B was $ 5x - 10 $ . So find the ratio of the ages of A and B tan years ago and equate it to 3:4 to find the value of x and their present ages.

Complete step-by-step answer:
We are given that the present ages of A and B are in the ratio 4:5 and ten years ago their ages were in the ratio 3:4. We have to find their present ages.
As the present ages of A and B are given in ratio, we can convert the ratio into numbers by multiplying a common variable ‘x’. So the present age of A is 4x years and B is 5x years.
Ten years ago, the present will be 10 years less, so the age of A and B ten years ago was $ 4x - 10 $ and $ 5x - 10 $ respectively.
And the ratio of ages of A and B ten years ago is 3:4.
This means $ \dfrac{{4x - 10}}{{5x - 10}} = \dfrac{3}{4} $
On cross multiplying the above equation, we get $ 4\left( {4x - 10} \right) = 3\left( {5x - 10} \right) $
 $ \Rightarrow 16x - 40 = 15x - 30 $
Putting the terms containing variable ‘x’ one side and the constant terms another side, we get
 $ \Rightarrow 16x - 15x = 40 - 30 $
 $ \therefore x = 10 $
The value of x is 10. Therefore the present age of A is $ 4x = 4 \times 10 = 40 $ years and the present age of B is $ 5x = 5 \times 10 = 50 $ years.

So, the correct answer is “40 years AND 50 years”.

Note: While solving a linear equation, put all the terms containing variables one side and then solve. The process will get easier. A ratio compares values. Ratio of two numbers will result in a fraction with the quotients as numerator and denominator when the numbers are divided by their greatest common divisor. Here for 40 and 50, 10 is the GCD and when they are divided by 10, the quotients are 4 and 5. Therefore it is the ratio.