Which of the following numbers occur to the left of 7/5 on the number line ?

Options

A. `7/4`
B. `8/6`
C. `6/4`
D. `5/3`

Answer

To determine which of the given numbers occurs to the left of \( \frac{7}{5} \) on the number line, we need to compare each number with \( \frac{7}{5} \).

### Step-by-Step Solution:

1. **Identify the given number**:
The number we are comparing against is \( \frac{7}{5} \).

2. **List the options**:
The options given are \( \frac{7}{4} \), \( \frac{8}{6} \), \( \frac{4}{3} \), \( \frac{3}{2} \), and \( \frac{5}{3} \).

3. **Convert all fractions to have a common denominator**:
To compare these fractions easily, we will convert them to have a common denominator. The denominators are 4, 6, 3, 2, and 5. The least common multiple (LCM) of these numbers is 60.

4. **Convert \( \frac{7}{5} \)**:
\[ \frac{7}{5} = \frac{7 \times 12}{5 \times 12} = \frac{84}{60} \]

5. **Convert \( \frac{7}{4} \)**:
\[ \frac{7}{4} = \frac{7 \times 15}{4 \times 15} = \frac{105}{60} \]

6. **Convert \( \frac{8}{6} \)**:
\[ \frac{8}{6} = \frac{8 \times 10}{6 \times 10} = \frac{80}{60} \]

7. **Convert \( \frac{4}{3} \)**:
\[ \frac{4}{3} = \frac{4 \times 20}{3 \times 20} = \frac{80}{60} \]

8. **Convert \( \frac{3}{2} \)**:
\[ \frac{3}{2} = \frac{3 \times 30}{2 \times 30} = \frac{90}{60} \]

9. **Convert \( \frac{5}{3} \)**:
\[ \frac{5}{3} = \frac{5 \times 20}{3 \times 20} = \frac{100}{60} \]

10. **Compare the fractions**:
Now we compare all the converted fractions with \( \frac{84}{60} \):
- \( \frac{7}{4} = \frac{105}{60} \) (greater than \( \frac{84}{60} \))
- \( \frac{8}{6} = \frac{80}{60} \) (less than \( \frac{84}{60} \))
- \( \frac{4}{3} = \frac{80}{60} \) (less than \( \frac{84}{60} \))
- \( \frac{3}{2} = \frac{90}{60} \) (greater than \( \frac{84}{60} \))
- \( \frac{5}{3} = \frac{100}{60} \) (greater than \( \frac{84}{60} \))

11. **Conclusion**:
The numbers that are less than \( \frac{7}{5} \) (or \( \frac{84}{60} \)) are \( \frac{8}{6} \) and \( \frac{4}{3} \). Therefore, both of these numbers occur to the left of \( \frac{7}{5} \) on the number line.

### Final Answer:
The numbers that occur to the left of \( \frac{7}{5} \) on the number line are \( \frac{8}{6} \) and \( \frac{4}{3} \).