The ratio of the ages of Ram and Rahim 10 years ago was 1 : 3. The ratio of their ages five years hence will be 2 : 3. Then the ratio of their present ages is

Options

A. `1:2`
B. 3:5
C. `3:4`
D. `2:5`

Answer

To solve the problem, we will follow these steps:

1. **Define Variables for Ages 10 Years Ago**:
Let the age of Ram 10 years ago be \( X \).
Therefore, the age of Rahim 10 years ago will be \( 3X \) (since the ratio of their ages was 1:3).

2. **Calculate Present Ages**:
- Present age of Ram = \( X + 10 \)
- Present age of Rahim = \( 3X + 10 \)

3. **Set Up the Equation for Ages 5 Years Hence**:
According to the problem, the ratio of their ages 5 years hence will be 2:3.
- Age of Ram in 5 years = \( (X + 10) + 5 = X + 15 \)
- Age of Rahim in 5 years = \( (3X + 10) + 5 = 3X + 15 \)

Setting up the ratio:
\[ \frac{X + 15}{3X + 15} = \frac{2}{3} \]

4. **Cross Multiply to Solve for X**:
Cross multiplying gives:
\[ 3(X + 15) = 2(3X + 15) \]
Expanding both sides:
\[ 3X + 45 = 6X + 30 \]

5. **Rearranging the Equation**:
Rearranging gives:
\[ 45 - 30 = 6X - 3X \]
\[ 15 = 3X \]

6. **Solve for X**:
Dividing both sides by 3:
\[ X = 5 \]

7. **Calculate Present Ages**:
Now substituting \( X \) back to find the present ages:
- Present age of Ram = \( 5 + 10 = 15 \)
- Present age of Rahim = \( 3(5) + 10 = 15 + 10 = 25 \)

8. **Find the Ratio of Present Ages**:
The ratio of their present ages is:
\[ \frac{15}{25} = \frac{3}{5} \]

Thus, the ratio of their present ages is \( 3:5 \).