Out of the fractions `(3)/(5), (7)/(9), (6)/(7), (2)/(3)` and `(4)/(11)` what is the sum of the largest and smallest fractions ?

Options

A. `1(17)/(77)`
B. `1(13)/(65)`
C. `1(13)/(77)`
D. `1(17)/(65)`

Answer

To find the sum of the largest and smallest fractions from the given set of fractions: \( \frac{3}{5}, \frac{7}{9}, \frac{6}{7}, \frac{2}{3}, \frac{4}{11} \), we will follow these steps:

### Step 1: Convert each fraction to a decimal
To easily compare the fractions, we can convert each of them into decimal form:

1. \( \frac{3}{5} = 0.6 \)
2. \( \frac{7}{9} \approx 0.777... \) (approximately 0.78)
3. \( \frac{6}{7} \approx 0.857... \) (approximately 0.86)
4. \( \frac{2}{3} \approx 0.666... \) (approximately 0.67)
5. \( \frac{4}{11} \approx 0.363... \) (approximately 0.36)

### Step 2: Identify the largest and smallest fractions
Now we can compare the decimal values:

- \( 0.6 \) (from \( \frac{3}{5} \))
- \( 0.78 \) (from \( \frac{7}{9} \))
- \( 0.86 \) (from \( \frac{6}{7} \))
- \( 0.67 \) (from \( \frac{2}{3} \))
- \( 0.36 \) (from \( \frac{4}{11} \))

From these values, we can see that:
- The largest fraction is \( \frac{6}{7} \) (approximately 0.86).
- The smallest fraction is \( \frac{4}{11} \) (approximately 0.36).

### Step 3: Calculate the sum of the largest and smallest fractions
Now, we will add the largest and smallest fractions together:

\[ \frac{6}{7} + \frac{4}{11} \]

To add these fractions, we need a common denominator. The least common multiple (LCM) of \( 7 \) and \( 11 \) is \( 77 \).

### Step 4: Convert each fraction to have the common denominator
1. Convert \( \frac{6}{7} \):
\[ \frac{6}{7} = \frac{6 \times 11}{7 \times 11} = \frac{66}{77} \]

2. Convert \( \frac{4}{11} \):
\[ \frac{4}{11} = \frac{4 \times 7}{11 \times 7} = \frac{28}{77} \]

### Step 5: Add the converted fractions
Now we can add the two fractions:
\[ \frac{66}{77} + \frac{28}{77} = \frac{66 + 28}{77} = \frac{94}{77} \]

### Step 6: Simplify the result if necessary
The fraction \( \frac{94}{77} \) can be simplified:
\[ \frac{94}{77} = 1 \frac{17}{77} \]

### Final Answer
The sum of the largest and smallest fractions is \( \frac{94}{77} \) or \( 1 \frac{17}{77} \).

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