If `a : b = 3 : 4` and `b : c = 8 : 9`, then `a : c = ?`

Options

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

Answer

To solve the problem where we need to find the ratio \( a : c \) given \( a : b = 3 : 4 \) and \( b : c = 8 : 9 \), we can follow these steps:

### Step 1: Write the ratios in fractional form
From the given ratios, we can express them as fractions:
- \( \frac{a}{b} = \frac{3}{4} \)
- \( \frac{b}{c} = \frac{8}{9} \)

### Step 2: Express \( a \) in terms of \( b \)
From the first ratio \( \frac{a}{b} = \frac{3}{4} \), we can express \( a \) as:
\[ a = \frac{3}{4}b \]

### Step 3: Express \( c \) in terms of \( b \)
From the second ratio \( \frac{b}{c} = \frac{8}{9} \), we can express \( c \) as:
\[ b = \frac{8}{9}c \quad \Rightarrow \quad c = \frac{9}{8}b \]

### Step 4: Substitute \( b \) in terms of \( c \) into the equation for \( a \)
Now, we substitute \( b \) from the equation for \( c \) into the equation for \( a \):
\[ a = \frac{3}{4}b = \frac{3}{4} \left( \frac{9}{8}c \right) \]

### Step 5: Simplify the expression for \( a \)
Now, simplify the expression:
\[ a = \frac{3 \times 9}{4 \times 8}c = \frac{27}{32}c \]

### Step 6: Write the ratio \( a : c \)
Now we can express the ratio \( a : c \):
\[ a : c = \frac{27}{32}c : c \]
This simplifies to:
\[ a : c = 27 : 32 \]

### Final Answer
Thus, the ratio \( a : c \) is \( 27 : 32 \).

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