Question

Sam is 10 years younger than one-half the age of his aunt. Let a represent his aunt’s age. Which algebraic expression can be used to determine Sam’s age if his aunt is 40?
StartFraction a Over 10 EndFraction minus 2; when a = 40 Sam is 2 years old
StartFraction a Over 10 EndFraction + 2; when a = 40 Sam is 6 years old
StartFraction a Over 2 EndFraction minus 10; when a = 40 Sam is 10 years old
StartFraction a Over 2 EndFraction + 10; when a = 40 Sam is 30 years old abc or d

Answer 1

Let's break down the problem step-by-step.

1. Let \( a \) represent Sam's aunt's age. According to the given information, Sam is 10 years younger than one-half the age of his aunt.

The expression for Sam's age can be written as:
\[
\text{Sam's age} = \frac{a}{2} - 10
\]

2. Now we need to check which option corresponds to this expression when \( a = 40 \).

Substituting \( a = 40 \) into the expression we derived:
\[
\text{Sam's age} = \frac{40}{2} - 10 = 20 - 10 = 10
\]

3. Now let's analyze the choices provided:

- **Option a:** \(\frac{a}{10} - 2\); when \(a = 40\):
\[
\text{Age} = \frac{40}{10} - 2 = 4 - 2 = 2 \text{ (incorrect)}
\]

- **Option b:** \(\frac{a}{10} + 2\); when \(a = 40\):
\[
\text{Age} = \frac{40}{10} + 2 = 4 + 2 = 6 \text{ (incorrect)}
\]

- **Option c:** \(\frac{a}{2} - 10\); when \(a = 40\):
\[
\text{Age} = \frac{40}{2} - 10 = 20 - 10 = 10 \text{ (correct)}
\]

- **Option d:** \(\frac{a}{2} + 10\); when \(a = 40\):
\[
\text{Age} = \frac{40}{2} + 10 = 20 + 10 = 30 \text{ (incorrect)}
\]

Based on the calculations above, the correct algebraic expression to determine Sam's age when his aunt is 40 is:
**Option c:** \(\frac{a}{2} - 10\).