Answer:

Given: Ratio of ages of two gi_rls = 5 : 7

Ratio of their ages 8 years ago = 7 : 13

To find: Their present ages

Let: Their present ages be xx and yy

Solution: According to the given question and assumption made,

\frac{x}{y} = \frac{5}{7}

y

x

=

7

5

i.e., 7x - 5y = 07x−5y=0 ...(1)

also, \frac{x \ - \ 8}{y \ - \ 8} = \frac{7}{13}

y − 8

x − 8

=

13

7

i.e., 13(x-8) = 7(y-8)13(x−8)=7(y−8)

⇒ 13x - 104 = 7y - 5613x−104=7y−56

⇒ 13x - 7y = 4813x−7y=48 ...(2)

Using elimination method to solve the equations:

Multiplying equation (1) by 7, we get

49x - 35y = 049x−35y=0 ...(3)

Multiplying equation (2) by 5, we get

65x - 35y = 24065x−35y=240 ...(4)

Subtracting equation (3) from equation (4)

⇒ (65x - 35y) - (49x - 35y) = 240 - 0(65x−35y)−(49x−35y)=240−0

⇒ 65x - 49x - 35y + 35y = 24065x−49x−35y+35y=240

⇒ 16x = 24016x=240

⇒ x = \frac{240}{16} = 15x=

16

240

=15

Putting x=15x=15 in equation (1)

⇒ 7 * 15 - 5y = 07∗15−5y=0

⇒ 5y = 1055y=105

⇒ y = \frac{105}{5} = 21y=

5

105

=21

Hence, present ages of the gi_rls are 15 years and 21 years.