Pand Q earn in the ratio 2:1. They spend in the ratio 5: 3 and save in the ratio 4: 1. Find the monthly income of each if the total monthly savings of both P and Q together is Rs 5000?

Options

A. `14000,7000`
B. `10000,5000`
C. `2000,1000`
D. `8000,4000`

Answer

To solve the problem, we need to find the monthly incomes of P and Q based on the ratios of their earnings, spending, and savings, given that their total monthly savings is Rs 5000.

### Step-by-Step Solution:

1. **Define the Ratios**:
- Let the earnings of P and Q be in the ratio of 2:1.
- Let P's income = 2x
- Let Q's income = x

2. **Define Spending Ratios**:
- The spending of P and Q is in the ratio of 5:3.
- Let P's spending = 5y
- Let Q's spending = 3y

3. **Define Savings Ratios**:
- The savings of P and Q are in the ratio of 4:1.
- Let P's savings = 4z
- Let Q's savings = z

4. **Relate Income, Spending, and Savings**:
- We know that:
- Income = Spending + Savings
- For P:
- \( 2x = 5y + 4z \) (1)
- For Q:
- \( x = 3y + z \) (2)

5. **Total Savings**:
- The total savings of P and Q is given as Rs 5000:
- \( 4z + z = 5000 \)
- \( 5z = 5000 \)
- \( z = 1000 \)

6. **Substitute z back into the savings**:
- P's savings = \( 4z = 4 \times 1000 = 4000 \)
- Q's savings = \( z = 1000 \)

7. **Substitute z into the equations to find y**:
- From equation (2):
- \( x = 3y + 1000 \)
- Substitute \( z = 1000 \) into equation (1):
- \( 2x = 5y + 4000 \)
- Now we have two equations:
- \( x = 3y + 1000 \) (3)
- \( 2x = 5y + 4000 \) (4)

8. **Substituting equation (3) into equation (4)**:
- Substitute \( x \) from equation (3) into equation (4):
- \( 2(3y + 1000) = 5y + 4000 \)
- \( 6y + 2000 = 5y + 4000 \)
- \( 6y - 5y = 4000 - 2000 \)
- \( y = 2000 \)

9. **Find x using y**:
- Substitute \( y = 2000 \) back into equation (3):
- \( x = 3(2000) + 1000 \)
- \( x = 6000 + 1000 = 7000 \)

10. **Find P's and Q's incomes**:
- P's income = \( 2x = 2 \times 7000 = 14000 \)
- Q's income = \( x = 7000 \)

### Final Answer:
- P's monthly income = Rs 14,000
- Q's monthly income = Rs 7,000

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