If A : B = 3 : 4 and B : C = 12 : 13, then find A : C.

Answer

To solve the problem of finding the ratio A : C given the ratios A : B = 3 : 4 and B : C = 12 : 13, we can follow these steps:

### Step-by-Step Solution:

1. **Write the Ratios in Fraction Form**:
- From the ratio A : B = 3 : 4, we can express it as:
\[ \frac{A}{B} = \frac{3}{4} \]
- From the ratio B : C = 12 : 13, we can express it as:
\[ \frac{B}{C} = \frac{12}{13} \]

2. **Multiply the Two Ratios**:
- To find A : C, we can multiply the two fractions:
\[ \frac{A}{B} \times \frac{B}{C} = \frac{3}{4} \times \frac{12}{13} \]
- This simplifies to:
\[ \frac{A}{C} = \frac{3 \times 12}{4 \times 13} \]

3. **Calculate the Numerator and Denominator**:
- Calculate the numerator:
\[ 3 \times 12 = 36 \]
- Calculate the denominator:
\[ 4 \times 13 = 52 \]
- So, we have:
\[ \frac{A}{C} = \frac{36}{52} \]

4. **Simplify the Fraction**:
- To simplify \(\frac{36}{52}\), we can divide both the numerator and the denominator by their greatest common divisor (GCD), which is 4:
\[ \frac{36 \div 4}{52 \div 4} = \frac{9}{13} \]

5. **Express as a Ratio**:
- Finally, we can express \(\frac{A}{C}\) as a ratio:
\[ A : C = 9 : 13 \]

### Final Answer:
Thus, the ratio A : C is \(9 : 13\).

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