Question
Out of his monthly salary, Josh spends 37.5% on his bills, 3/16 on groceries, and 1/16 on transportation. If he decides to save 1/5 of the remaining part, what percentage of his salary will he save?
Answers
GPT-4o mini
To determine what percentage of Josh's salary he will save, we first need to establish the fractions of his salary that he spends on each category and what remains after these expenditures.
1. **Spend on Bills**: Josh spends 37.5% of his salary on bills. We convert this to a fraction:
\[
37.5\% = \frac{37.5}{100} = \frac{3}{8}.
\]
2. **Spend on Groceries**: Josh spends \( \frac{3}{16} \) of his salary on groceries.
3. **Spend on Transportation**: Josh spends \( \frac{1}{16} \) of his salary on transportation.
Now we'll calculate the total expenditure:
\[
\text{Total expenditure} = \text{Spend on Bills} + \text{Spend on Groceries} + \text{Spend on Transportation}
\]
Substituting the values in:
\[
\text{Total expenditure} = \frac{3}{8} + \frac{3}{16} + \frac{1}{16}.
\]
To add these fractions, we need a common denominator. The least common multiple of 8 and 16 is 16.
Convert \( \frac{3}{8} \) into sixteenths:
\[
\frac{3}{8} = \frac{3 \cdot 2}{8 \cdot 2} = \frac{6}{16}.
\]
Now we can sum the fractions:
\[
\text{Total expenditure} = \frac{6}{16} + \frac{3}{16} + \frac{1}{16} = \frac{6 + 3 + 1}{16} = \frac{10}{16} = \frac{5}{8}.
\]
4. **Calculate Remaining Salary**: The remaining portion of his salary after the expenditures is:
\[
\text{Remaining Salary} = 1 - \text{Total expenditure} = 1 - \frac{5}{8} = \frac{3}{8}.
\]
5. **Calculate the Amount Saved**: Josh decides to save \( \frac{1}{5} \) of the remaining part of his salary. Thus, the amount saved can be calculated as:
\[
\text{Amount Saved} = \frac{1}{5} \cdot \frac{3}{8} = \frac{3}{40}.
\]
6. **Find the Percentage of Salary Saved**: Now we want this amount as a percentage of his total salary. The percentage of salary saved is:
\[
\text{Percentage of Salary Saved} = \left( \frac{3}{40} \div 1 \right) \cdot 100\% = \frac{3}{40} \cdot 100\%.
\]
Calculating this gives:
\[
\frac{3 \cdot 100}{40} = \frac{300}{40} = 7.5\%.
\]
Thus, the percentage of his salary that Josh will save is:
\[
\boxed{7.5\%}.
\]
1. **Spend on Bills**: Josh spends 37.5% of his salary on bills. We convert this to a fraction:
\[
37.5\% = \frac{37.5}{100} = \frac{3}{8}.
\]
2. **Spend on Groceries**: Josh spends \( \frac{3}{16} \) of his salary on groceries.
3. **Spend on Transportation**: Josh spends \( \frac{1}{16} \) of his salary on transportation.
Now we'll calculate the total expenditure:
\[
\text{Total expenditure} = \text{Spend on Bills} + \text{Spend on Groceries} + \text{Spend on Transportation}
\]
Substituting the values in:
\[
\text{Total expenditure} = \frac{3}{8} + \frac{3}{16} + \frac{1}{16}.
\]
To add these fractions, we need a common denominator. The least common multiple of 8 and 16 is 16.
Convert \( \frac{3}{8} \) into sixteenths:
\[
\frac{3}{8} = \frac{3 \cdot 2}{8 \cdot 2} = \frac{6}{16}.
\]
Now we can sum the fractions:
\[
\text{Total expenditure} = \frac{6}{16} + \frac{3}{16} + \frac{1}{16} = \frac{6 + 3 + 1}{16} = \frac{10}{16} = \frac{5}{8}.
\]
4. **Calculate Remaining Salary**: The remaining portion of his salary after the expenditures is:
\[
\text{Remaining Salary} = 1 - \text{Total expenditure} = 1 - \frac{5}{8} = \frac{3}{8}.
\]
5. **Calculate the Amount Saved**: Josh decides to save \( \frac{1}{5} \) of the remaining part of his salary. Thus, the amount saved can be calculated as:
\[
\text{Amount Saved} = \frac{1}{5} \cdot \frac{3}{8} = \frac{3}{40}.
\]
6. **Find the Percentage of Salary Saved**: Now we want this amount as a percentage of his total salary. The percentage of salary saved is:
\[
\text{Percentage of Salary Saved} = \left( \frac{3}{40} \div 1 \right) \cdot 100\% = \frac{3}{40} \cdot 100\%.
\]
Calculating this gives:
\[
\frac{3 \cdot 100}{40} = \frac{300}{40} = 7.5\%.
\]
Thus, the percentage of his salary that Josh will save is:
\[
\boxed{7.5\%}.
\]