Find the fraction which bears the same ratio to `1/27` that `3/7` does to `5/9`

Options

A. `5/9`
B. `1/35`
C. `45/7`
D. `7/45`

Answer

To solve the problem, we need to find a fraction \( x \) such that the ratio of \( x \) to \( \frac{1}{27} \) is the same as the ratio of \( \frac{3}{7} \) to \( \frac{5}{9} \).

### Step-by-Step Solution:

1. **Set Up the Ratios**:
We can express the relationship as:
\[ \frac{x}{\frac{1}{27}} = \frac{\frac{3}{7}}{\frac{5}{9}} \]

2. **Simplify the Right Side**:
To simplify \( \frac{\frac{3}{7}}{\frac{5}{9}} \), we multiply by the reciprocal of the denominator:
\[ \frac{3}{7} \times \frac{9}{5} = \frac{3 \times 9}{7 \times 5} = \frac{27}{35} \]

3. **Substitute Back into the Equation**:
Now our equation looks like this:
\[ \frac{x}{\frac{1}{27}} = \frac{27}{35} \]

4. **Cross Multiply**:
To eliminate the fraction on the left side, we cross multiply:
\[ x \cdot 35 = \frac{27}{35} \cdot \frac{1}{27} \]
This simplifies to:
\[ x \cdot 35 = 27 \cdot \frac{1}{27} = 1 \]

5. **Solve for \( x \)**:
Now, divide both sides by 35 to isolate \( x \):
\[ x = \frac{1}{35} \]

### Final Answer:
The fraction which bears the same ratio to \( \frac{1}{27} \) that \( \frac{3}{7} \) does to \( \frac{5}{9} \) is \( \frac{1}{35} \).