Question

My Grandfather was 8 times older to me 16 years ago. He would be 3 times of my age 8 years from now. Eight years ago, what was the ratio of my age to that of my grandfather?
(A). 1:2
(B). 1:5
(C). 13:18
(D). 11:53

Answer 1

Hint: We will assume my age 16 years ago to be x, therefore according to the question, grandfather’s age was $8x$ then according to the given information we will form our equations as present age will be $x+16$ and $8x+16$. Also 8 years from now age will be $x+16+8$ and $8x+16+8$ that is $x+24$ and $8x+24$. We will formulate an equation as per the condition given in the question and then we will solve the obtained equation to get the ratio of age at present.

Complete step-by-step solution -
It is given in the question that grandfather is 8 times the age of me, 16 years ago. Also, Grandfather will be 3 times my age 8 years from now. Then we have to find the ratio of my age to my grandfather's age. Let us assume that 16 years ago, my age was x, then the age of my grandfather will be $8x$. From this assumption we get my present age = $x+16$ and also my grandfather’s present age = $8x+16$.
Now, let us consider the situation 8 years from now. My age after 8 years will be $\left( x+16+8 \right)=\left( x+24 \right)$ and age of my grandfather after 8 years will be $\left( 8x+16+8 \right)=\left( 8x+24 \right)$. But it is given that after 8 years from now, grandfather’s age is three times of my age. Thus we formulate the given situation as follows $3\left( x+24 \right)=8x+24$. Therefore solving the equation further we get $3x+72=8x+24$, grouping the similar terms together, we get $8x-3x=72-24$ or $5x=48$ or $x=\dfrac{48}{5}=9.6$.
Thus, 16 years ago my age was 9.6years and my grandfather’s age was $9.6\times 8=76.8years$. So, my present age will be $9.6+16=25.6years$ and my grandfather’s present age will be $76.8+16=92.8years$. Therefore, 8 years back the ratio of my age to that of my grandfather’s age will be $\dfrac{my\text{ present age - 8}}{my\text{ grandfather }\!\!'\!\!\text{ s age - 8}}$ = $\dfrac{25.6-8}{92.8-8}=\dfrac{11}{53}=11:53$.
Therefore option d) is correct.

Note: Usually students find the ratio of my and grandfather’s age at present age or the ratio 16 years ago of present year. But in question it is given that we have to find the ratio of age 8 years from now. Thus, it is recommended to read and understand the question before solving it. Also students should not try to round off the ages in this question as they might not get the right answer from the given options.