Understanding the Father and Son Age Ratio

This question asks us to find the father's age based on a given ratio between his age and his son's age, and the product of their ages. We need to use the concept of ratios and basic algebra to solve this.

Representing Ages Using the Ratio

The ratio of the father's age to the son's age is given as 5 : 2. We can represent their ages using a common multiplier, let's call it x.

• Father's Age = \(5x\)
• Son's Age = \(2x\)

Here, x is a positive number that scales the ratio to their actual ages.

Calculating Ages from the Product

The problem states that the product of their ages is 160. We can set up an equation using our age representations:

Father's Age \(\times\) Son's Age = 160

Substituting the expressions for their ages:

\((5x) \times (2x) = 160\)

Solving the Algebraic Equation

Let's solve the equation for x:

1. Multiply the terms on the left side: \(5 \times 2 \times x \times x = 10x^2\)
2. The equation becomes: \(10x^2 = 160\)
3. Divide both sides by 10 to isolate \(x^2\): \(x^2 = \frac{160}{10}\)
4. Simplify the division: \(x^2 = 16\)
5. Take the square root of both sides to find x: \(x = \sqrt{16}\)
6. Calculate the square root: \(x = 4\)

We consider the positive square root because age must be a positive value.

Finding the Father's Age

Now that we have found the value of the multiplier x, we can calculate the father's actual age:

Father's Age = \(5x\)

Substitute the value of x: Father's Age = \(5 \times 4\)

Father's Age = 20

Checking the Son's Age and Product

We can also find the son's age to verify our answer:

Son's Age = \(2x = 2 \times 4 = 8\)

Let's check if the product of their calculated ages matches the given product:

Product = Father's Age \(\times\) Son's Age = \(20 \times 8 = 160\)

This matches the information given in the problem, confirming that our calculated ages are correct.

Summary of Ages

Here's a summary of the calculated ages:

The question asks for the age of the father, which we found to be 20.