Out of the fractions `(5)/( 7) , ( 7)/( 13) , ( 4)/( 7), ( 4)/( 15)` and `( 9 )/( 14)`, which is the third highest ?

Options

A. `(5)/( 7)`
B. `( 7 )/( 13)`
C. `( 4)/( 7)`
D. `( 4)/( 15)`

Answer

To find the third highest fraction among the given fractions \( \frac{5}{7}, \frac{7}{13}, \frac{4}{7}, \frac{4}{15}, \frac{9}{14} \), we will follow these steps:

### Step 1: List the Fractions
We start by writing down the fractions clearly:
- \( \frac{5}{7} \)
- \( \frac{7}{13} \)
- \( \frac{4}{7} \)
- \( \frac{4}{15} \)
- \( \frac{9}{14} \)

### Step 2: Find the LCM of the Denominators
The denominators are \( 7, 13, 7, 15, 14 \). We need to find the LCM of these numbers.

- The prime factorization is:
- \( 7 = 7^1 \)
- \( 13 = 13^1 \)
- \( 15 = 3^1 \times 5^1 \)
- \( 14 = 2^1 \times 7^1 \)

The LCM will be the product of the highest powers of all prime factors:
- LCM = \( 2^1 \times 3^1 \times 5^1 \times 7^1 \times 13^1 = 2 \times 3 \times 5 \times 7 \times 13 \)

Calculating this gives:
- \( 2 \times 3 = 6 \)
- \( 6 \times 5 = 30 \)
- \( 30 \times 7 = 210 \)
- \( 210 \times 13 = 2730 \)

So, the LCM is \( 2730 \).

### Step 3: Convert Each Fraction to Have the Same Denominator
Now we will convert each fraction to have the denominator of \( 2730 \).

1. **For \( \frac{5}{7} \)**:
\[ \frac{5}{7} = \frac{5 \times 390}{7 \times 390} = \frac{1950}{2730} \]

2. **For \( \frac{7}{13} \)**:
\[ \frac{7}{13} = \frac{7 \times 210}{13 \times 210} = \frac{1470}{2730} \]

3. **For \( \frac{4}{7} \)**:
\[ \frac{4}{7} = \frac{4 \times 390}{7 \times 390} = \frac{1560}{2730} \]

4. **For \( \frac{4}{15} \)**:
\[ \frac{4}{15} = \frac{4 \times 182}{15 \times 182} = \frac{728}{2730} \]

5. **For \( \frac{9}{14} \)**:
\[ \frac{9}{14} = \frac{9 \times 195}{14 \times 195} = \frac{1755}{2730} \]

### Step 4: Compare the Numerators
Now we have the fractions with a common denominator:
- \( \frac{1950}{2730} \)
- \( \frac{1470}{2730} \)
- \( \frac{1560}{2730} \)
- \( \frac{728}{2730} \)
- \( \frac{1755}{2730} \)

The numerators are:
- 1950
- 1470
- 1560
- 728
- 1755

### Step 5: Order the Numerators
Now we will order the numerators from highest to lowest:
1. 1950 (from \( \frac{5}{7} \))
2. 1755 (from \( \frac{9}{14} \))
3. 1560 (from \( \frac{4}{7} \))
4. 1470 (from \( \frac{7}{13} \))
5. 728 (from \( \frac{4}{15} \))

### Step 6: Identify the Third Highest
From the ordered list, the third highest numerator is 1560, which corresponds to the fraction \( \frac{4}{7} \).

### Final Answer
Thus, the third highest fraction is:
\[ \frac{4}{7} \]