Suman saves `33 (1)/(3)%` of her income. If her income increases by x% and expenditure increases by 10%, then her savings increase by 22%. What is the value of x ?

Options

A. 10
B. 14
C. 15
D. 12

Answer

To solve the problem step by step, let's break it down clearly:

### Step 1: Understand the Savings Percentage
Suman saves \(33 \frac{1}{3}\%\) of her income. This can be converted to a fraction:
\[ 33 \frac{1}{3}\% = \frac{100}{3}\% \]
This means that if her income is \(I\), her savings \(S\) can be expressed as:
\[ S = \frac{100}{3} \times \frac{I}{100} = \frac{I}{3} \]

### Step 2: Determine Expenditure
If Suman saves \(\frac{I}{3}\), then her expenditure \(E\) can be calculated as:
\[ E = I - S = I - \frac{I}{3} = \frac{2I}{3} \]

### Step 3: Calculate New Income and Expenditure
If her income increases by \(x\%\), the new income \(I'\) will be:
\[ I' = I + \frac{x}{100} \times I = I \left(1 + \frac{x}{100}\right) \]
Her expenditure increases by \(10\%\), so the new expenditure \(E'\) will be:
\[ E' = E + 10\% \text{ of } E = E + 0.1E = 1.1E = 1.1 \times \frac{2I}{3} = \frac{2.2I}{3} \]

### Step 4: Calculate New Savings
The new savings \(S'\) will be:
\[ S' = I' - E' = I \left(1 + \frac{x}{100}\right) - \frac{2.2I}{3} \]
Simplifying this:
\[ S' = I \left(1 + \frac{x}{100}\right) - \frac{2.2I}{3} = I \left(1 + \frac{x}{100} - \frac{2.2}{3}\right) \]

### Step 5: Savings Increase
According to the problem, her savings increase by \(22\%\):
\[ S' = S + 22\% \text{ of } S = \frac{I}{3} + 0.22 \times \frac{I}{3} = \frac{I}{3} \left(1 + 0.22\right) = \frac{I}{3} \times 1.22 \]

### Step 6: Set Up the Equation
Now we set the two expressions for \(S'\) equal to each other:
\[ I \left(1 + \frac{x}{100} - \frac{2.2}{3}\right) = \frac{I}{3} \times 1.22 \]
Dividing both sides by \(I\) (assuming \(I \neq 0\)):
\[ 1 + \frac{x}{100} - \frac{2.2}{3} = \frac{1.22}{3} \]

### Step 7: Solve for \(x\)
Now, we need to solve for \(x\):
1. First, simplify the left side:
\[ 1 - \frac{2.2}{3} = \frac{3}{3} - \frac{2.2}{3} = \frac{0.8}{3} \]
So, we have:
\[ \frac{0.8}{3} + \frac{x}{100} = \frac{1.22}{3} \]
2. Rearranging gives:
\[ \frac{x}{100} = \frac{1.22}{3} - \frac{0.8}{3} = \frac{1.22 - 0.8}{3} = \frac{0.42}{3} \]
3. Multiply both sides by \(100\):
\[ x = \frac{0.42 \times 100}{3} = 14 \]

### Final Answer
Thus, the value of \(x\) is:
\[ \boxed{14} \]