The ratio of present ages (in years) of X to Y is equal to the ratio of present ages (in years) of Y to Z. If the present age of Y is 15 years, then which of the following can be the sum of the ages (in years) of X, Y and Z ?
[amp_mcq option1=”35″ option2=”40″ option3=”49″ option4=”55″ correct=”option3″]
This question was previously asked in
UPSC CAPF – 2024
The ratio of present ages of X to Y is equal to the ratio of present ages of Y to Z.
$\frac{X}{Y} = \frac{Y}{Z}$
This implies $Y^2 = XZ$.
Given that the present age of Y is 15 years, so $Y=15$.
$15^2 = XZ \implies 225 = XZ$.
We are looking for a possible sum of the ages $X+Y+Z = X+15+Z$. We need to find integer pairs $(X, Z)$ whose product is 225 and check the sum $X+15+Z$ against the options.
Factors of $225 = 3^2 \times 5^2$: 1, 3, 5, 9, 15, 25, 45, 75, 225.
Possible pairs $(X, Z)$ with $X \le Z$ and $XZ=225$:
– $(1, 225) \implies X+15+Z = 1 + 15 + 225 = 241$. Not an option.
– $(3, 75) \implies X+15+Z = 3 + 15 + 75 = 93$. Not an option.
– $(5, 45) \implies X+15+Z = 5 + 15 + 45 = 65$. Not an option.
– $(9, 25) \implies X+15+Z = 9 + 15 + 25 = 49$. This is option C.
– $(15, 15) \implies X+15+Z = 15 + 15 + 15 = 45$. Not an option. (Note: this case means X=Y=Z=15, which satisfies the ratio condition, although ages are usually assumed different in such problems unless stated).
Since 49 is one of the calculated possible sums (when ages are 9, 15, and 25), it is a valid answer.