The ratio of present ages of R and S is 11 : 17. 11 years ago, the ratio of their ages was 11 : 20. What is R's present age (in years)?

Options

A. 51
B. 33
C. 22
D. 40

Answer

To solve the problem step by step, we will follow these instructions:

1. **Define the Present Ages**:
Let R's present age be \( 11x \) and S's present age be \( 17x \), where \( x \) is a common multiplier.

2. **Set Up the Ages 11 Years Ago**:
Eleven years ago, R's age would be \( 11x - 11 \) and S's age would be \( 17x - 11 \).

3. **Set Up the Ratio 11 Years Ago**:
According to the problem, the ratio of their ages 11 years ago was \( 11 : 20 \). Therefore, we can write the equation:
\[ \frac{11x - 11}{17x - 11} = \frac{11}{20} \]

4. **Cross-Multiply to Eliminate the Fraction**:
Cross-multiplying gives us:
\[ 20(11x - 11) = 11(17x - 11) \]

5. **Expand Both Sides**:
Expanding both sides results in:
\[ 220x - 220 = 187x - 121 \]

6. **Rearrange the Equation**:
Rearranging the equation to isolate \( x \):
\[ 220x - 187x = 220 - 121 \]
\[ 33x = 99 \]

7. **Solve for \( x \)**:
Dividing both sides by 33 gives:
\[ x = 3 \]

8. **Calculate R's Present Age**:
Now, substitute \( x \) back to find R's present age:
\[ R's \, present \, age = 11x = 11 \times 3 = 33 \]

Thus, R's present age is **33 years**.