If the ratio of A to B is 3:5 and the ratio of B to C is 2:7, what is the ratio of A to C?

Answer

To find the ratio of A to C given the ratios of A to B and B to C, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given ratios**:
- The ratio of A to B is given as \( A : B = 3 : 5 \).
- The ratio of B to C is given as \( B : C = 2 : 7 \).

2. **Convert the ratios into fractions**:
- From the ratio \( A : B = 3 : 5 \), we can express it as:
\[ \frac{A}{B} = \frac{3}{5} \]
- From the ratio \( B : C = 2 : 7 \), we can express it as:
\[ \frac{B}{C} = \frac{2}{7} \]

3. **Express B in terms of C**:
- Rearranging the second ratio \( \frac{B}{C} = \frac{2}{7} \), we can express B as:
\[ B = \frac{2}{7} C \]

4. **Substitute B in terms of C into the first ratio**:
- Now, substitute \( B = \frac{2}{7} C \) into the first ratio \( \frac{A}{B} = \frac{3}{5} \):
\[ \frac{A}{\frac{2}{7} C} = \frac{3}{5} \]

5. **Cross-multiply to solve for A**:
- Cross-multiplying gives:
\[ 5A = 3 \cdot \frac{2}{7} C \]
- Simplifying this, we get:
\[ 5A = \frac{6}{7} C \]

6. **Solve for A**:
- Dividing both sides by 5 gives:
\[ A = \frac{6}{35} C \]

7. **Find the ratio of A to C**:
- Now, we can express the ratio \( A : C \):
\[ \frac{A}{C} = \frac{\frac{6}{35} C}{C} = \frac{6}{35} \]
- Therefore, the ratio of A to C is:
\[ A : C = 6 : 35 \]

### Final Answer:
The ratio of A to C is \( 6 : 35 \).

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