The ratio of Ram's savings to his expenditure is 5:2 and that of Manu is 4:3. If Ram's expenditure is `1/3` rd of Manu's expenditure and the sum of their expenditures is Rs3000, then find the salaries of Ram and Manu, respectively.

Answer

To solve the problem step by step, we will follow the given ratios and relationships between Ram's and Manu's savings and expenditures.

### Step 1: Define the Ratios
The ratio of Ram's savings to his expenditure is given as 5:2. Let Ram's savings be \(5x\) and his expenditure be \(2x\).

The ratio of Manu's savings to his expenditure is given as 4:3. Let Manu's savings be \(4y\) and his expenditure be \(3y\).

### Step 2: Establish the Relationship Between Expenditures
It is given that Ram's expenditure is \( \frac{1}{3} \) of Manu's expenditure. Therefore, we can write:
\[ 2x = \frac{1}{3}(3y) \]
This simplifies to:
\[ 2x = y \]

### Step 3: Sum of Expenditures
The total expenditure of Ram and Manu is given as Rs 3000. Therefore, we can write:
\[ 2x + 3y = 3000 \]

### Step 4: Substitute y in Terms of x
From the relationship \(y = 2x\), we can substitute \(y\) in the expenditure equation:
\[ 2x + 3(2x) = 3000 \]
This simplifies to:
\[ 2x + 6x = 3000 \]
\[ 8x = 3000 \]
\[ x = \frac{3000}{8} = 375 \]

### Step 5: Calculate y
Now that we have the value of \(x\), we can find \(y\):
\[ y = 2x = 2 \times 375 = 750 \]

### Step 6: Calculate Salaries
Now we can find the salaries of Ram and Manu:
- Ram's salary = Expenditure + Savings = \(2x + 5x = 7x\)
\[ \text{Ram's salary} = 7 \times 375 = 2625 \]

- Manu's salary = Expenditure + Savings = \(3y + 4y = 7y\)
\[ \text{Manu's salary} = 7 \times 750 = 5250 \]

### Final Answer
The salaries of Ram and Manu are:
- Ram's salary: Rs 2625
- Manu's salary: Rs 5250

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