Question

A is two years older than B who is twice as old as C. If the total of the ages of A, B and C be 27, then how old is B?

• A. \(7 \)
• B. \(8 \)
• C. \(9 \)
• D. \(10 \)

Hint:

Always carefully check who the comparison pronoun refers to! In the phrase "A is two years older than B who is twice...", the word "who" refers directly to B. Defining the youngest person as \(x\) avoids handling negative values or fractions.

Answer 1

The Correct Option is D

Concept: Problems concerning ages are solved by translating verbal real-world statements into systematic linear algebraic equations. To prevent dealing with fractional coefficients, it is best practice to define the independent reference person with the lowest age variable as our primary baseline unknown.

Step 1: Set up the algebraic variables representing individual ages.
Let the current ages of individuals A, B, and C be denoted by the variables \(A\), \(B\), and \(C\) respectively. The problem states that "B is twice as old as C". Let us select the age of C as our primary variable base: \[ \text{Let the age of C} = x \text{ years} \] From the given condition, B's age is directly twice that of C: \[ \text{Age of B} = 2x \text{ years} \] Next, the problem mentions that "A is two years older than B". This implies we add 2 years directly to the expression representing B's current age: \[ \text{Age of A} = B + 2 = 2x + 2 \text{ years} \]

Step 2: Utilize the total age condition to set up a linear equation.
We are told that the cumulative sum of the ages of all three individuals is exactly 27 years: \[ A + B + C = 27 \] Substitute the algebraic expressions formulated in terms of \(x\) into this linear relationship: \[ (2x + 2) + (2x) + (x) = 27 \] Group all the like variable terms containing \(x\) together on the left-hand side: \[ 2x + 2x + x + 2 = 27 \] \[ 5x + 2 = 27 \]

Step 3: Isolate the variable \(x\).
Subtract 2 from both sides of the equation to isolate the variable term: \[ 5x = 27 - 2 \] \[ 5x = 25 \] Divide both sides by 5 to find the baseline value: \[ x = \frac{25}{5} = 5 \] Hence, the current age of C is \( x = 5 \) years.

Step 4: Compute the explicit age of B.
We are required to determine the age of B. From our initial setup, B's age is explicitly defined as \( 2x \). Substituting \( x = 5 \): \[ \text{Age of B} = 2 \times 5 = 10 \text{ years} \] To verify, let's look at A's age as well: \( 2(5) + 2 = 12 \) years. Checking the total sum: \( 12 + 10 + 5 = 27 \), which matches perfectly. Therefore, B is 10 years old, corresponding to Option (D).