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Question

Carol is three times older than Andrew. Brad is two years older than Andrew. In six years, the sum of Andrew’s and Brad’s ages will be the same as Carol’s age. How old is Carol?

Answer

To determine Carol's age, we need to set up and solve a system of equations based on the information given. Let's define the variables first:

  • Let AA be Andrew's current age.
  • Let BB be Brad's current age.
  • Let CC be Carol's current age.

From the problem, we have the following relationships:

  1. Carol is three times older than Andrew:

    C=3AC = 3A
  2. Brad is two years older than Andrew:

    B=A+2B = A + 2
  3. In six years, the sum of Andrew’s and Brad’s ages will be the same as Carol’s age:

    (A+6)+(B+6)=C+6(A + 6) + (B + 6) = C + 6

First, substitute the expressions for BB and CC from the first two equations into the third equation:

(A+6)+((A+2)+6)=3A+6(A + 6) + ((A + 2) + 6) = 3A + 6

Simplify the left-hand side:

A+6+A+2+6=3A+6A + 6 + A + 2 + 6 = 3A + 6

Combine like terms:

2A+14=3A+62A + 14 = 3A + 6

Subtract 2A2A from both sides to isolate AA:

14=A+614 = A + 6

Subtract 6 from both sides to solve for AA:

A=8A = 8

Now that we know Andrew's age, we can find Carol's age using the first relationship:

C=3A=3×8=24C = 3A = 3 \times 8 = 24

Thus, Carol is 24\boxed{24} years old.

Key Concept

The key math concepts in the question include:

  1. Variables and Expressions: Assigning variables to represent the ages of Carol, Andrew, and Brad.

  2. Algebraic Relationships: Establishing relationships based on the information provided:

    • Carol's age as three times Andrew's age.
    • Brad's age as two years older than Andrew.
  3. Future Values: Calculating future ages by adding six years to the current ages.

  4. Equation Building: Formulating an equation based on the relationships and future values provided, specifically the condition that the sum of Andrew's and Brad's future ages equals Carol's future age.

  5. Solving Equations: Finding the values of the variables (ages) by solving the constructed equation.

These concepts collectively help in determining the current age of Carol.

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