Question

5 years ago the ratio of the ages of A and B was 5:6. After 5 years the ratio of their ages was 7:8. Find their present ages​

Answer 1

Given:

Two persons -

• A and B

5 years ago their age -

• Ratio = 5:6

After 5 years their age -

• Ratio = 7:8

What To Find:

We have to find

• The present age of A and B.

How To Find:

To find them, we have to,

• Take x as the common multiple of the ratio 5:6 i.e 5x and 6x.
• Form a linear equation on it.
• Solve the equation and find the value of x.
• Substitute the values and find their present ages.

Solution:

• Forming the linear equation.

Let x be the common multiple of the ratio 5:6

⟹ 5x and 6x

Now the age of A after 5 years is,

⟹ 5x + 5

Now the age of B after 5 years is,

⟹ 6x + 5

Therefore the equation is,

$\sf \implies \dfrac{5x + 5}{6x + 5} = \dfrac{7}{8}$

• Solving the linear equation.

$\sf \implies \dfrac{5x + 5}{6x + 5} = \dfrac{7}{8}$

Use cross multiplication,

$\sf \implies 8(5x + 5) = 7(6x + 5)$

Multiply 8 with 5x + 5,

$\sf \implies 40x + 40 = 7(6x + 5)$

Multiply 7 with 6x + 5,

$\sf \implies 40x + 40 = 42x + 35$

Take 40x to RHS,

$\sf \implies 40 = 42x + 35 - 40x$

Take 35 to LHS,

$\sf \implies 40 - 35 = 42x - 40x$

Subtract 35 from 40,

$\sf \implies 5 = 2x$

Take 2 to LHS,

$\sf \implies \dfrac{5}{2} = x$

• Finding the ages of A and B.

• Age of A -

⟹ 5x + 5

Substitute the value of x,

$\sf \implies 5 \times \dfrac{5}{2} + 5$

Multiply 5 with 5,

$\sf \implies \dfrac{25}{2} + 5$

Divide 25 by 2,

$\sf \implies 12.5 + 5$

Add 12.5 and 5,

$\sf \implies 17.5 \: years$

• Age of B

⟹ 6x + 5

Substitute the value of x,

$\sf \implies 6 \times \dfrac{5}{2} + 5$

Multiply 6 with 5,

$\sf \implies \dfrac{30}{2} + 5$

Divide 25 by 2,

$\sf \implies 15 + 5$

Add 12.5 and 5,

$\sf \implies 20 \: years$

Final Answer:

∴ Thus, the present age of A is 17.5 years and B is 20 years.