The difference between the greatest and smallest fraction amongst `(6)/(7), (8)/(9), (9)/(10)` and `(7)/(8)` is

Options

A. `(1)/(72)`
B. `(2)/(63)`
C. `(3)/(70)`
D. `(1)/(56)`

Answer

To find the difference between the greatest and smallest fractions among \( \frac{6}{7}, \frac{8}{9}, \frac{9}{10}, \) and \( \frac{7}{8} \), we will follow these steps:

### Step 1: Identify the fractions
We have the following fractions:
- \( \frac{6}{7} \)
- \( \frac{8}{9} \)
- \( \frac{9}{10} \)
- \( \frac{7}{8} \)

### Step 2: Compare the fractions
To determine the greatest and smallest fractions, we can either convert them to decimal form or compare their values directly.

1. **Convert to decimal:**
- \( \frac{6}{7} \approx 0.857 \)
- \( \frac{8}{9} \approx 0.889 \)
- \( \frac{9}{10} = 0.9 \)
- \( \frac{7}{8} = 0.875 \)

2. **Identify the greatest and smallest:**
- The greatest fraction is \( \frac{9}{10} \) (0.9).
- The smallest fraction is \( \frac{6}{7} \) (approximately 0.857).

### Step 3: Calculate the difference
Now, we need to find the difference between the greatest and smallest fractions:
\[ \text{Difference} = \frac{9}{10} - \frac{6}{7} \]

### Step 4: Find a common denominator
The least common multiple (LCM) of the denominators 10 and 7 is 70. We will convert both fractions to have this common denominator.

- Convert \( \frac{9}{10} \):
\[ \frac{9}{10} = \frac{9 \times 7}{10 \times 7} = \frac{63}{70} \]

- Convert \( \frac{6}{7} \):
\[ \frac{6}{7} = \frac{6 \times 10}{7 \times 10} = \frac{60}{70} \]

### Step 5: Subtract the fractions
Now we can subtract:
\[ \frac{63}{70} - \frac{60}{70} = \frac{63 - 60}{70} = \frac{3}{70} \]

### Final Answer
The difference between the greatest and smallest fraction is \( \frac{3}{70} \).

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