If `x: y : z = 3 : 4 : 5`, then what will be the ratio of 1/x : 1/y : 1/z

Options

A. `5:4:3`
B. `30 : 40 : 27`
C. `45 : 48 : 100`
D. `20 : 15 : 12`

Answer

To solve the problem, we need to find the ratio of \( \frac{1}{x} : \frac{1}{y} : \frac{1}{z} \) given that \( x : y : z = 3 : 4 : 5 \).

### Step-by-Step Solution:

1. **Understand the Given Ratio**:
We have \( x : y : z = 3 : 4 : 5 \). This means we can express \( x \), \( y \), and \( z \) in terms of a common variable \( k \):
\[ x = 3k, \quad y = 4k, \quad z = 5k \]

2. **Find the Reciprocals**:
Next, we need to find \( \frac{1}{x} \), \( \frac{1}{y} \), and \( \frac{1}{z} \):
\[ \frac{1}{x} = \frac{1}{3k}, \quad \frac{1}{y} = \frac{1}{4k}, \quad \frac{1}{z} = \frac{1}{5k} \]

3. **Write the Ratio of Reciprocals**:
Now we can write the ratio:
\[ \frac{1}{x} : \frac{1}{y} : \frac{1}{z} = \frac{1}{3k} : \frac{1}{4k} : \frac{1}{5k} \]

4. **Eliminate the Common Denominator**:
To simplify the ratio, we can multiply each term by \( 60k \) (the least common multiple of the denominators 3, 4, and 5):
\[ 60k \cdot \frac{1}{3k} : 60k \cdot \frac{1}{4k} : 60k \cdot \frac{1}{5k} \]
This simplifies to:
\[ 20 : 15 : 12 \]

5. **Final Ratio**:
Therefore, the ratio \( \frac{1}{x} : \frac{1}{y} : \frac{1}{z} \) is:
\[ 20 : 15 : 12 \]

### Conclusion:
The final answer is \( 20 : 15 : 12 \).

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