Question : Seven years ago, the ratio of the ages of A and B was 4 : 5. Eight years hence, the ratio of the ages of A and B will be 9 : 10. What is the sum of their present ages in years?
Option 1: 32
Option 2: 82
Option 3: 41
Option 4: 56
Correct Answer: 41
Solution :
Let the present age of A is denoted as $a$ and B as $b$.
Seven years ago, the ratio of the ages of A and B
⇒ $\frac{a - 7}{b - 7} = \frac{4}{5}$
⇒ $5(a - 7) = 4(b - 7)$
⇒ $5a-35=4b-28$
⇒ $5a - 4b = 7$...............................(1)
Eight years hence, the ratio of the ages of A and B
⇒ $\frac{a + 8}{b + 8} = \frac{9}{10}$
⇒ $10(a + 8) = 9(b + 8)$
⇒ $10a+ 80 = 9b + 72$
⇒ $10a-9b = - 8$....................................(2)
Multiplying Equation (1) by 2 and subtracting from Equation (2),
We get $b=22$ and $a = 19$
$\therefore$ Sum of their present ages is $a + b = 19 + 22 = 41$
Hence, the correct answer is 41 years.
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