If a:b is 32:35 and b:c is 21: 32 then a:c is ?

Options

A. 5:3
B. 3:5
C. 1:1
D. 5:7

Answer

To find the ratio \( a:c \) given the ratios \( a:b = 32:35 \) and \( b:c = 21:32 \), we can follow these steps:

### Step 1: Write the ratios in fractional form
We start with the given ratios:
1. \( a:b = 32:35 \) can be written as \( \frac{a}{b} = \frac{32}{35} \)
2. \( b:c = 21:32 \) can be written as \( \frac{b}{c} = \frac{21}{32} \)

### Step 2: Express \( a \) in terms of \( b \)
From the first ratio \( \frac{a}{b} = \frac{32}{35} \), we can express \( a \) as:
\[ a = \frac{32}{35}b \]

### Step 3: Express \( c \) in terms of \( b \)
From the second ratio \( \frac{b}{c} = \frac{21}{32} \), we can express \( c \) as:
\[ c = \frac{32}{21}b \]

### Step 4: Substitute \( b \) into the expression for \( c \)
Now we can substitute the expressions for \( a \) and \( c \) in terms of \( b \):
\[ \frac{a}{c} = \frac{\frac{32}{35}b}{\frac{32}{21}b} \]

### Step 5: Simplify the expression
The \( b \) cancels out:
\[ \frac{a}{c} = \frac{32/35}{32/21} = \frac{32 \times 21}{35 \times 32} \]
This simplifies to:
\[ \frac{a}{c} = \frac{21}{35} \]

### Step 6: Reduce the fraction
Now we can simplify \( \frac{21}{35} \):
\[ \frac{21}{35} = \frac{3}{5} \]

### Conclusion
Thus, the ratio \( a:c \) is:
\[ a:c = 3:5 \]

### Final Answer
The final answer is \( 3:5 \).

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