Question

A man spends $ \frac{2}{5} $ of his salary on rent, $ \frac{1}{4} $ on food, and $ \frac{1}{10} $ on transportation. What fraction of his salary is left?

• A. \( \frac{11}{20} \)
• B. \( \frac{1}{4} \)
• C. \( \frac{9}{20} \)
• D. \( \frac{13}{20} \)

Hint:

Always convert all fractions to have a common denominator when adding or subtracting. Double-check for any errors in options if your math is consistent but no choice matches.

Answer 1

The Correct Option is B

Let's determine the fraction of the salary that is left after a man has covered his expenses. We'll calculate the total fraction spent and subtract it from the whole salary.

First, the fraction spent on rent is \( \frac{2}{5} \), food is \( \frac{1}{4} \), and transportation is \( \frac{1}{10} \). The total fraction spent is:

\[ \frac{2}{5} + \frac{1}{4} + \frac{1}{10} \] 

To add these fractions, we need a common denominator. The least common multiple of 5, 4, and 10 is 20.

Convert each fraction to have a denominator of 20:

• For \( \frac{2}{5} \): Multiply numerator and denominator by 4 → \( \frac{2 \times 4}{5 \times 4} = \frac{8}{20} \)
• For \( \frac{1}{4} \): Multiply numerator and denominator by 5 → \( \frac{1 \times 5}{4 \times 5} = \frac{5}{20} \)
• For \( \frac{1}{10} \): Multiply numerator and denominator by 2 → \( \frac{1 \times 2}{10 \times 2} = \frac{2}{20} \)

Add the fractions:

\[ \frac{8}{20} + \frac{5}{20} + \frac{2}{20} = \frac{15}{20} \]

The total fraction spent is \( \frac{15}{20} \) of the salary. Therefore, the fraction of the salary that is left is the remaining part of the whole:

\[ 1 - \frac{15}{20} = \frac{20}{20} - \frac{15}{20} = \frac{5}{20} \]

To simplify \( \frac{5}{20} \), divide both the numerator and denominator by their greatest common divisor, which is 5:

\[ \frac{5 \div 5}{20 \div 5} = \frac{1}{4} \]

Thus, the fraction of the salary that is left is \( \frac{1}{4} \).

Answer 2

To solve the problem, we need to find the fraction of the salary remaining after the man spends on rent, food, and transportation.

1. Understanding the Concepts:

- Fraction of Salary Spent: Sum of fractions spent on different items.
- Fraction Left: Total salary (1) minus the sum of fractions spent.

2. Given Values:

- Rent = \( \frac{2}{5} \)
- Food = \( \frac{1}{4} \)
- Transportation = \( \frac{1}{10} \)

3. Calculating the Fraction Left:

Sum of fractions spent: \[ \frac{2}{5} + \frac{1}{4} + \frac{1}{10} = \frac{8}{20} + \frac{5}{20} + \frac{2}{20} = \frac{15}{20} = \frac{3}{4} \] Fraction left: \[ 1 - \frac{3}{4} = \frac{1}{4} \]

Final Answer:

The fraction of his salary left is \(\frac{1}{4}\).