Question

Father is 5 times as old as his son. The square of the sum of their ages is 5184. The age of the father is

• A. 67 years
• B. 60 years
• C. 66 years
• D. 70 years

Hint:

Since the father is 5 times as old as his son, his age must be a multiple of 5.
Looking at the options, only \(60\) and \(70\) are multiples of 5.
If Father = \(60\), then Son = \(12\). Sum = \(72\), and \(72^2 = 5184\), which works perfectly!

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This question requires solving a system of algebraic equations based on real-world age relationships involving squares and square roots.

Step 2: Key Formulas and approach:
Let the age of the son be \(S\) and the age of the father be \(F\).
We are given two relationships:
1. Father is 5 times as old as his son:
\[ F = 5S \] 2. The square of the sum of their ages is \(5184\):
\[ (F + S)^2 = 5184 \] We can solve this by taking the square root of both sides and then substituting the first equation into the second.

Step 3: Detailed Explanation:

• Simplify the second equation by taking the square root of both sides:
\[ F + S = \sqrt{5184} \]

• Let us calculate \(\sqrt{5184}\). Since \(70^2 = 4900\) and \(80^2 = 6400\), the value lies between \(70\) and \(80\).

• Since the last digit is \(4\), the number must end in \(2\) or \(8\). Testing \(72\):
\[ 72 \times 72 = 5184 \]

• Therefore, we have:
\[ F + S = 72 \]

• Substitute the first relationship \(F = 5S\) into this equation:
\[ 5S + S = 72 \] \[ 6S = 72 \] \[ S = 12 \]

• Now calculate the father's age \(F\):
\[ F = 5S = 5 \times 12 = 60 \text{ years} \]

Step 4: Final Answer:
The age of the father is \(60\) years, which is Option (B).