The ratio of daily wage of three workers P,Q & R 'MANREGA' is `21:16:18` respectively . If any of workers work on Sunday ,then gets Rs .125 extra on that day . The ratio of wage of P,Q & R for a weekday and Sunday is `26:21:23` , then find the difference between wage of P & R on a weekday & Sunday (in rs) ?

Options

A. A)64
B. B)75
C. C)90
D. D)125

Answer

To solve the problem step by step, we will follow the information provided in the question and the video transcript.

### Step 1: Understand the Ratios
The daily wages of workers P, Q, and R are given in the ratio of 21:16:18. This means:
- Wage of P = 21x
- Wage of Q = 16x
- Wage of R = 18x

### Step 2: Calculate the Extra Wage on Sunday
When any of the workers work on Sunday, they receive an extra Rs. 125. Therefore, the wages on Sunday will be:
- Wage of P on Sunday = 21x + 125
- Wage of Q on Sunday = 16x + 125
- Wage of R on Sunday = 18x + 125

### Step 3: Understand the Sunday Wage Ratio
The ratio of the wages of P, Q, and R for a weekday and Sunday is given as 26:21:23. This means:
- Wage of P on Sunday = 26y
- Wage of Q on Sunday = 21y
- Wage of R on Sunday = 23y

### Step 4: Set Up Equations
From the previous steps, we can set up the following equations based on the Sunday wages:
1. \( 21x + 125 = 26y \) (1)
2. \( 16x + 125 = 21y \) (2)
3. \( 18x + 125 = 23y \) (3)

### Step 5: Solve the Equations
We can solve these equations to find the values of x and y.

From equation (1):
\[ 21x + 125 = 26y \]
Rearranging gives:
\[ 21x = 26y - 125 \]
\[ x = \frac{26y - 125}{21} \quad (4) \]

From equation (2):
\[ 16x + 125 = 21y \]
Rearranging gives:
\[ 16x = 21y - 125 \]
\[ x = \frac{21y - 125}{16} \quad (5) \]

Setting equations (4) and (5) equal to each other:
\[ \frac{26y - 125}{21} = \frac{21y - 125}{16} \]

Cross-multiplying:
\[ 16(26y - 125) = 21(21y - 125) \]
Expanding both sides:
\[ 416y - 2000 = 441y - 2625 \]
Rearranging gives:
\[ 441y - 416y = 2625 - 2000 \]
\[ 25y = 625 \]
\[ y = 25 \]

### Step 6: Find x
Substituting y back into equation (4):
\[ x = \frac{26(25) - 125}{21} \]
\[ x = \frac{650 - 125}{21} \]
\[ x = \frac{525}{21} \]
\[ x = 25 \]

### Step 7: Calculate Wages
Now we can find the actual wages:
- Wage of P = \( 21x = 21(25) = 525 \)
- Wage of R = \( 18x = 18(25) = 450 \)

### Step 8: Calculate Sunday Wages
- Wage of P on Sunday = \( 525 + 125 = 650 \)
- Wage of R on Sunday = \( 450 + 125 = 575 \)

### Step 9: Find the Difference
Now we find the difference between the wages of P and R on both weekdays and Sundays:
- Weekday difference = \( 525 - 450 = 75 \)
- Sunday difference = \( 650 - 575 = 75 \)

Thus, the difference between the wage of P and R on a weekday and Sunday is:
\[ \text{Difference} = 75 \]

### Final Answer
The difference between the wage of P and R on a weekday and Sunday is Rs. 75.

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