Question

Kim is twice as old as Claire. Nick is 3 years older than Claire. Kim is 6 years older than Emily. Their ages combined equal 81. How old is Nick?

• A. 22
• B. 27
• C. 17
• D. 13
• E.

Hint:

In multi-person age problems, identify the person whose age is used as a reference for others. In this case, both Kim's and Nick's ages were given relative to Claire's, making Claire's age (C) the best choice for the primary variable.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This word problem involves multiple age relationships and can be solved by setting up a system of equations. The goal is to express everyone's age in terms of a single base variable and then use the total age to solve for that variable.
Step 2: Key Formula or Approach:
Let K, C, N, and E be the ages of Kim, Claire, Nick, and Emily, respectively. 1. \( K = 2C \) 2. \( N = C + 3 \) 3. \( K = E + 6 \), which implies \( E = K - 6 \) 4. \( K + C + N + E = 81 \) We will express all ages in terms of C.
Step 3: Detailed Explanation:
We have the ages in terms of C: \begin{itemize} \item Claire's age = C \item Kim's age = \(K = 2C\) \item Nick's age = \(N = C + 3\) \item Emily's age = \(E = K - 6 = (2C) - 6 = 2C - 6\) \end{itemize} Now substitute these into the sum equation: \[ (2C) + C + (C + 3) + (2C - 6) = 81 \] Combine all the terms with C: \[ 2C + C + C + 2C = 6C \] Combine the constant terms: \[ 3 - 6 = -3 \] The simplified equation is: \[ 6C - 3 = 81 \] Add 3 to both sides: \[ 6C = 84 \] Divide by 6: \[ C = \frac{84}{6} = 14 \] Claire's age is 14. The question asks for Nick's age: \[ N = C + 3 = 14 + 3 = 17 \] Step 4: Final Answer:
Nick is 17 years old.