Seven years ago, the ratio of the ages of A and B was 2 : 3. Three years hence, the ratio of their ages will be 5 : 7. The sum of their present ages (in years) is:

Options

A. 114
B. 118
C. 112
D. 110

Answer

To solve the problem step by step, let's denote the present ages of A and B as \( A \) and \( B \) respectively.

### Step 1: Set up the equations based on the given ratios.
Seven years ago, the ages of A and B can be expressed as:
- Age of A: \( A - 7 \)
- Age of B: \( B - 7 \)

According to the problem, the ratio of their ages seven years ago was \( 2 : 3 \). Therefore, we can write the equation:
\[ \frac{A - 7}{B - 7} = \frac{2}{3} \]
Cross-multiplying gives us:
\[ 3(A - 7) = 2(B - 7) \]
Expanding this, we get:
\[ 3A - 21 = 2B - 14 \]
Rearranging the equation, we have:
\[ 3A - 2B = 7 \quad \text{(Equation 1)} \]

### Step 2: Set up the second equation based on future ages.
Three years hence, the ages of A and B will be:
- Age of A: \( A + 3 \)
- Age of B: \( B + 3 \)

The ratio of their ages three years hence will be \( 5 : 7 \). Thus, we can write:
\[ \frac{A + 3}{B + 3} = \frac{5}{7} \]
Cross-multiplying gives us:
\[ 7(A + 3) = 5(B + 3) \]
Expanding this, we get:
\[ 7A + 21 = 5B + 15 \]
Rearranging the equation, we have:
\[ 7A - 5B = -6 \quad \text{(Equation 2)} \]

### Step 3: Solve the system of equations.
Now we have a system of equations:
1. \( 3A - 2B = 7 \)
2. \( 7A - 5B = -6 \)

We can solve these equations using the method of substitution or elimination. Here, we will use elimination.

First, we can multiply Equation 1 by 5 and Equation 2 by 2 to align the coefficients of \( B \):
\[ 15A - 10B = 35 \quad \text{(Equation 3)} \]
\[ 14A - 10B = -12 \quad \text{(Equation 4)} \]

Now, subtract Equation 4 from Equation 3:
\[ (15A - 10B) - (14A - 10B) = 35 - (-12) \]
This simplifies to:
\[ A = 47 \]

### Step 4: Substitute \( A \) back to find \( B \).
Now substitute \( A = 47 \) into Equation 1:
\[ 3(47) - 2B = 7 \]
This simplifies to:
\[ 141 - 2B = 7 \]
Rearranging gives:
\[ 2B = 134 \quad \Rightarrow \quad B = 67 \]

### Step 5: Find the sum of their present ages.
Now we can find the sum of their present ages:
\[ A + B = 47 + 67 = 114 \]

Thus, the sum of their present ages is **114 years**.

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