The ratio between the present ages of A, and B is `6 : 5` respectively. After ten years this ratio will be `7 : 6` respectively. What is the difference between the present ages of A and B? (in years)

Options

A. cannot be determined
B. 10
C. 4
D. 8

Answer

To solve the problem step by step, we will denote the present ages of A and B using a variable.

### Step-by-Step Solution:

1. **Define Variables:**
Let the present age of A be \(6x\) and the present age of B be \(5x\), where \(x\) is a common multiplier.

2. **Set Up the Future Age Equation:**
After 10 years, the age of A will be \(6x + 10\) and the age of B will be \(5x + 10\).

3. **Establish the Future Ratio:**
According to the problem, after 10 years, the ratio of the ages of A to B will be \(7:6\). We can set up the equation:
\[ \frac{6x + 10}{5x + 10} = \frac{7}{6} \]

4. **Cross Multiply to Eliminate the Fraction:**
Cross multiplying gives us:
\[ 6(6x + 10) = 7(5x + 10) \]

5. **Expand Both Sides:**
Expanding both sides results in:
\[ 36x + 60 = 35x + 70 \]

6. **Rearrange the Equation:**
To isolate \(x\), we rearrange the equation:
\[ 36x - 35x = 70 - 60 \]
This simplifies to:
\[ x = 10 \]

7. **Calculate Present Ages:**
Now that we have \(x\), we can find the present ages:
- Age of A: \(6x = 6 \times 10 = 60\)
- Age of B: \(5x = 5 \times 10 = 50\)

8. **Find the Difference in Ages:**
The difference between the present ages of A and B is:
\[ 60 - 50 = 10 \]

### Final Answer:
The difference between the present ages of A and B is **10 years**.

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