A man spends 80% of his income with the increase in the cost of living his expenditure increases by `37(1)/(2)%` and his income increases by `16(2)/(3)%`. Find its present per cent saving.

Answer

To solve the problem, we need to determine the present percentage of savings after the man's income and expenditure have changed. Let's break it down step by step.

### Step 1: Define the Initial Income and Expenditure
Let’s assume the initial income of the man is \( I = 100 \) (this is just a convenient number to work with).

Since he spends 80% of his income:
\[ \text{Expenditure} = 80\% \text{ of } I = 0.80 \times 100 = 80 \]

### Step 2: Calculate the Increase in Income
The income increases by \( 16\frac{2}{3}\% \). We can convert this percentage into a fraction:
\[ 16\frac{2}{3}\% = \frac{50}{3}\% \]
To find the new income:
\[ \text{Increase in Income} = \frac{50}{3}\% \text{ of } 100 = \frac{50}{3} \]
\[ \text{New Income} = 100 + \frac{50}{3} = \frac{300}{3} + \frac{50}{3} = \frac{350}{3} \]

### Step 3: Calculate the Increase in Expenditure
The expenditure increases by \( 37\frac{1}{2}\% \). Again, let’s convert this percentage into a fraction:
\[ 37\frac{1}{2}\% = \frac{75}{2}\% \]
To find the new expenditure:
\[ \text{Increase in Expenditure} = \frac{75}{2}\% \text{ of } 80 = \frac{75}{2} \times \frac{80}{100} = \frac{75 \times 80}{200} = \frac{6000}{200} = 30 \]
\[ \text{New Expenditure} = 80 + 30 = 110 \]

### Step 4: Calculate the Present Savings
Now we can find the present savings:
\[ \text{Present Savings} = \text{New Income} - \text{New Expenditure} = \frac{350}{3} - 110 \]
To perform the subtraction, convert 110 into a fraction:
\[ 110 = \frac{330}{3} \]
Now, subtract:
\[ \text{Present Savings} = \frac{350}{3} - \frac{330}{3} = \frac{20}{3} \]

### Step 5: Calculate the Present Percentage of Savings
To find the percentage of savings, we use the formula:
\[ \text{Percentage of Savings} = \left( \frac{\text{Present Savings}}{\text{New Income}} \right) \times 100 \]
Substituting the values we have:
\[ \text{Percentage of Savings} = \left( \frac{\frac{20}{3}}{\frac{350}{3}} \right) \times 100 = \left( \frac{20}{350} \right) \times 100 = \frac{20 \times 100}{350} = \frac{2000}{350} = \frac{200}{35} = \frac{40}{7} \]
Thus, the present percentage of savings is:
\[ \frac{40}{7} \approx 5.71\% \]

### Final Answer
The present percentage of savings is \( 5\frac{5}{7}\% \).

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