If a : b = 7 : 9 and b : c =15 : 7, then what is a : c?

Options

A. `5:3`
B. `3:5`
C. `7:21`
D. `7:15`

Answer

To solve the problem, we need to find the ratio \( a : c \) given the ratios \( a : b = 7 : 9 \) and \( b : c = 15 : 7 \).

### Step-by-step Solution:

1. **Write down the given ratios:**
\[ a : b = 7 : 9 \quad \text{(1)} \]
\[ b : c = 15 : 7 \quad \text{(2)} \]

2. **Express \( a \) and \( c \) in terms of \( b \):**
From ratio (1), we can express \( a \) in terms of \( b \):
\[ a = \frac{7}{9}b \]
From ratio (2), we can express \( c \) in terms of \( b \):
\[ c = \frac{7}{15}b \]

3. **Substitute the expressions for \( a \) and \( c \) into the ratio \( a : c \):**
Now we can write the ratio \( a : c \):
\[ a : c = \frac{7}{9}b : \frac{7}{15}b \]

4. **Simplify the ratio:**
Since \( b \) is common in both terms, we can cancel it out:
\[ a : c = \frac{7}{9} : \frac{7}{15} \]
This simplifies to:
\[ a : c = \frac{7}{9} \div \frac{7}{15} = \frac{7}{9} \times \frac{15}{7} \]
The \( 7 \) cancels out:
\[ a : c = \frac{15}{9} \]

5. **Further simplify \( \frac{15}{9} \):**
We can simplify \( \frac{15}{9} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 3:
\[ a : c = \frac{15 \div 3}{9 \div 3} = \frac{5}{3} \]

6. **Final answer:**
Therefore, the ratio \( a : c \) is:
\[ a : c = 5 : 3 \]