The ratio of the monthly incomes of Ram and Shyam is ` 3 : 4` and the ratio of their monthly expenditures is 4 : 5`. If Shyam saves Rs. 400 per month, which of the following cannot be the savings of Ram? (In Rs./month)

Options

A. 290
B. 280
C. 270
D. 310

Explanation

To solve the problem step by step, we will use the information provided in the question regarding the ratios of incomes and expenditures of Ram and Shyam, and Shyam's savings.

### Step 1: Define the variables for incomes and expenditures
Let the monthly incomes of Ram and Shyam be represented as:
- Ram's income = \(3x\)
- Shyam's income = \(4x\)

Let the monthly expenditures of Ram and Shyam be represented as:
- Ram's expenditure = \(4y\)
- Shyam's expenditure = \(5y\)

### Step 2: Write the equations for savings
The savings of Ram and Shyam can be calculated as:
- Ram's savings = Income - Expenditure = \(3x - 4y\)
- Shyam's savings = Income - Expenditure = \(4x - 5y\)

### Step 3: Use the information about Shyam's savings
We know that Shyam saves Rs. 400 per month. Therefore, we can set up the equation:
\[ 4x - 5y = 400 \]

### Step 4: Solve for one variable in terms of the other
From the equation \(4x - 5y = 400\), we can express \(x\) in terms of \(y\):
\[ 4x = 400 + 5y \implies x = \frac{400 + 5y}{4} \]

### Step 5: Substitute \(x\) back into Ram's savings equation
Now, we substitute \(x\) into Ram's savings equation:
\[ \text{Ram's savings} = 3x - 4y = 3\left(\frac{400 + 5y}{4}\right) - 4y \]
\[ = \frac{1200 + 15y}{4} - 4y \]
\[ = \frac{1200 + 15y - 16y}{4} = \frac{1200 - y}{4} \]

### Step 6: Analyze the possible savings of Ram
Now we have Ram's savings expressed as:
\[ \text{Ram's savings} = \frac{1200 - y}{4} \]
To find the possible values of Ram's savings, we need to consider the value of \(y\). Since \(y\) represents expenditures, it must be a non-negative value.

### Step 7: Determine the range of \(y\)
Since Shyam's expenditure is \(5y\) and his income is \(4x\), we can deduce:
\[ 5y < 4x \implies y < \frac{4x}{5} \]
Substituting \(x\) from the earlier equation:
\[ y < \frac{4\left(\frac{400 + 5y}{4}\right)}{5} \implies y < \frac{400 + 5y}{5} \implies 5y < 400 + 5y \]
This inequality is always true, so we need to find specific values for \(y\) that yield valid savings for Ram.

### Step 8: Calculate possible savings values
To find the possible savings values for Ram, we can check the options given in the question (290, 280, 270, 310) by substituting them back into the savings equation.

1. **For savings = 290**:
\[ 290 = \frac{1200 - y}{4} \implies 1200 - y = 1160 \implies y = 40 \]
(Valid)

2. **For savings = 280**:
\[ 280 = \frac{1200 - y}{4} \implies 1200 - y = 1120 \implies y = 80 \]
(Valid)

3. **For savings = 270**:
\[ 270 = \frac{1200 - y}{4} \implies 1200 - y = 1080 \implies y = 120 \]
(Valid)

4. **For savings = 310**:
\[ 310 = \frac{1200 - y}{4} \implies 1200 - y = 1240 \implies y = -40 \]
(Not valid, as \(y\) cannot be negative)

### Conclusion
The value that cannot be the savings of Ram is **310 Rs/month**.