The fraction `3/5` is found between which pair of fractions on a number line?

Options

A. `7/10` and `3/4`
B. `2/5` and `1/2`
C. `1/3` and `5/13`
D. `2/7` and `8/11`

Answer

To find which pair of fractions the fraction \( \frac{3}{5} \) is located between on a number line, we will compare \( \frac{3}{5} \) with various pairs of fractions.

### Step-by-Step Solution:

1. **Identify the fractions to compare**: We need to check several pairs of fractions to find where \( \frac{3}{5} \) fits. The options given are:
- A) \( \frac{7}{10} \) and \( \frac{3}{4} \)
- B) \( \frac{2}{5} \) and \( \frac{1}{2} \)
- C) \( \frac{1}{3} \) and \( \frac{5}{13} \)
- D) \( \frac{2}{7} \) and \( \frac{8}{11} \)

2. **Compare \( \frac{3}{5} \) with \( \frac{7}{10} \) and \( \frac{3}{4} \)**:
- Convert \( \frac{3}{5} \) to a common denominator with \( \frac{7}{10} \) and \( \frac{3}{4} \). The least common multiple (LCM) of 5, 10, and 4 is 20.
- Convert each fraction:
- \( \frac{3}{5} = \frac{3 \times 4}{5 \times 4} = \frac{12}{20} \)
- \( \frac{7}{10} = \frac{7 \times 2}{10 \times 2} = \frac{14}{20} \)
- \( \frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20} \)
- Now compare: \( \frac{14}{20} < \frac{12}{20} < \frac{15}{20} \) is false, so \( \frac{3}{5} \) is not between \( \frac{7}{10} \) and \( \frac{3}{4} \).

3. **Compare \( \frac{3}{5} \) with \( \frac{2}{5} \) and \( \frac{1}{2} \)**:
- Convert \( \frac{3}{5} \) to a common denominator with \( \frac{2}{5} \) and \( \frac{1}{2} \). The LCM of 5 and 2 is 10.
- Convert each fraction:
- \( \frac{3}{5} = \frac{3 \times 2}{5 \times 2} = \frac{6}{10} \)
- \( \frac{2}{5} = \frac{2 \times 2}{5 \times 2} = \frac{4}{10} \)
- \( \frac{1}{2} = \frac{1 \times 5}{2 \times 5} = \frac{5}{10} \)
- Now compare: \( \frac{4}{10} < \frac{6}{10} < \frac{5}{10} \) is false, so \( \frac{3}{5} \) is not between \( \frac{2}{5} \) and \( \frac{1}{2} \).

4. **Compare \( \frac{3}{5} \) with \( \frac{1}{3} \) and \( \frac{5}{13} \)**:
- The LCM of 3 and 13 is 39.
- Convert each fraction:
- \( \frac{3}{5} = \frac{3 \times 39}{5 \times 39} = \frac{117}{195} \)
- \( \frac{1}{3} = \frac{1 \times 65}{3 \times 65} = \frac{65}{195} \)
- \( \frac{5}{13} = \frac{5 \times 15}{13 \times 15} = \frac{75}{195} \)
- Now compare: \( \frac{65}{195} < \frac{117}{195} < \frac{75}{195} \) is false, so \( \frac{3}{5} \) is not between \( \frac{1}{3} \) and \( \frac{5}{13} \).

5. **Compare \( \frac{3}{5} \) with \( \frac{2}{7} \) and \( \frac{8}{11} \)**:
- The LCM of 5, 7, and 11 is 385.
- Convert each fraction:
- \( \frac{3}{5} = \frac{3 \times 77}{5 \times 77} = \frac{231}{385} \)
- \( \frac{2}{7} = \frac{2 \times 55}{7 \times 55} = \frac{110}{385} \)
- \( \frac{8}{11} = \frac{8 \times 35}{11 \times 35} = \frac{280}{385} \)
- Now compare: \( \frac{110}{385} < \frac{231}{385} < \frac{280}{385} \) is true, so \( \frac{3}{5} \) is between \( \frac{2}{7} \) and \( \frac{8}{11} \).

### Conclusion:
The fraction \( \frac{3}{5} \) is found between \( \frac{2}{7} \) and \( \frac{8}{11} \) on a number line.