If there is a reduction in the number of workers in a factory in the ratio `15 : 11` and an increment in their wage in the ratio ` 22 :25`, then the ratio by which the total wage of the workers should be decreased is:

Options

A. `6:5`
B. `5:6`
C. `3:7`
D. `3:5`

Answer

To solve the problem step by step, we will analyze the given ratios and calculate the required ratio of the total wage decrease.

### Step 1: Understand the given ratios
We are given two ratios:
- The reduction in the number of workers is in the ratio of 15:11.
- The increment in their wages is in the ratio of 22:25.

### Step 2: Define variables for clarity
Let:
- \( N_1 = 15 \) (initial number of workers)
- \( N_2 = 11 \) (final number of workers)
- \( W_1 = 22 \) (initial wage)
- \( W_2 = 25 \) (final wage)

### Step 3: Calculate the total wages before and after the changes
The total wage before the changes can be calculated as:
\[ \text{Total Wage}_1 = N_1 \times W_1 = 15 \times 22 \]
Calculating this gives:
\[ \text{Total Wage}_1 = 330 \]

The total wage after the changes can be calculated as:
\[ \text{Total Wage}_2 = N_2 \times W_2 = 11 \times 25 \]
Calculating this gives:
\[ \text{Total Wage}_2 = 275 \]

### Step 4: Find the decrease in total wage
To find the decrease in total wage, we subtract the final total wage from the initial total wage:
\[ \text{Decrease in Total Wage} = \text{Total Wage}_1 - \text{Total Wage}_2 = 330 - 275 = 55 \]

### Step 5: Calculate the ratio of decrease in total wage
Now, we need to find the ratio by which the total wage of the workers should be decreased. This ratio can be expressed as:
\[ \text{Ratio of Decrease} = \frac{\text{Decrease in Total Wage}}{\text{Total Wage}_1} = \frac{55}{330} \]
Simplifying this ratio:
\[ \text{Ratio of Decrease} = \frac{1}{6} \]

### Conclusion
Thus, the ratio by which the total wage of the workers should be decreased is \( 1:6 \).

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