Question

Adam is 2 years older than Mike. The square of Adam's age is 28 greater than the square of Mike's age in years. What is the sum of Adam's age and Mike's age?

• A. 8
• B. 12
• C. 14
• D. 18
• E. 22

Hint:

Recognizing algebraic identities like the difference of squares (\(a^2 - b^2 = (a-b)(a+b)\)) can significantly shorten your calculation time. Instead of solving for each variable individually, you can often solve directly for the quantity requested.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This is a word problem that can be translated into a system of algebraic equations. We need to solve for the ages and then find their sum.
Step 2: Key Formula or Approach:
Let A be Adam's age and M be Mike's age.
Translate the given sentences into equations:
1. "Adam is 2 years older than Mike": \(A = M + 2\), which can be rewritten as \(A - M = 2\).
2. "The square of Adam's age is 28 greater than the square of Mike's age": \(A^2 = M^2 + 28\), which can be rewritten as \(A^2 - M^2 = 28\).
We can use the difference of squares factorization: \(a^2 - b^2 = (a-b)(a+b)\).
Step 3: Detailed Explanation:
We have the two equations:
1) \(A - M = 2\)
2) \(A^2 - M^2 = 28\)
Apply the difference of squares formula to the second equation:
\[ (A - M)(A + M) = 28 \] We know from the first equation that \(A - M = 2\). Substitute this into the factored second equation:
\[ (2)(A + M) = 28 \] Now, we can solve for the sum of their ages, \(A + M\):
\[ A + M = \frac{28}{2} \] \[ A + M = 14 \] The question asks for the sum of their ages, which we have found to be 14.
Step 4: Final Answer:
The sum of Adam's age and Mike's age is 14.