Question

Audrey, Penelope and Clementine are all sisters. Penelope is 8 years older than Clementine and 2 years younger than Audrey. If the sum of Penelope and Clementine's age is Audrey's age, how old is Clementine's age?

• A. 4
• B. 2
• C. 8
• D. 3
• E.

Hint:

When setting up equations from word problems, define your variables clearly first. Then, read each phrase carefully to translate it into a mathematical relationship. Often, you can solve the system by expressing all variables in terms of just one variable.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This word problem can be solved by translating the given sentences into a system of linear equations and then solving for the required variable.
Step 2: Key Formula or Approach:
Let A, P, and C represent the ages of Audrey, Penelope, and Clementine, respectively. From the problem statement, we can write the following equations:
1. Penelope is 8 years older than Clementine: \( P = C + 8 \)
2. Penelope is 2 years younger than Audrey: \( P = A - 2 \)
3. The sum of Penelope and Clementine's age is Audrey's age: \( P + C = A \)
Step 3: Detailed Explanation:
We have a system of three equations: \[ P = C + 8 \quad \text{(1)} \] \[ P = A - 2 \quad \text{(2)} \] \[ P + C = A \quad \text{(3)} \] We want to find the value of C. Let's use substitution to solve the system.
From equation (2), we can express A in terms of P: \[ A = P + 2 \] Now, substitute this expression for A into equation (3): \[ P + C = (P + 2) \] We can now solve this equation for C. Subtract P from both sides: \[ C = 2 \] So, Clementine's age is 2.
Let's check this answer with the other equations.
If C = 2, then from equation (1), P = 2 + 8 = 10.
If P = 10, then from equation (2), 10 = A - 2, which means A = 12.
Now check if equation (3) holds: P + C = A \(\Rightarrow\) 10 + 2 = 12. This is true. The solution is consistent.
Step 4: Final Answer:
Clementine's age is 2.