Find the greatest among 1029/1025 , 1030/1026, 256/255, 1023/1019.

Options

A. 1029/1025
B. 1030/1026
C. 256/255
D. 1023/1019

Answer

To find the greatest among the fractions \( \frac{1029}{1025} \), \( \frac{1030}{1026} \), \( \frac{256}{255} \), and \( \frac{1023}{1019} \), we can simplify each fraction to make comparisons easier.

### Step 1: Simplify each fraction

1. **For \( \frac{1029}{1025} \)**:
\[ \frac{1029}{1025} = 1 + \frac{4}{1025} \]

2. **For \( \frac{1030}{1026} \)**:
\[ \frac{1030}{1026} = 1 + \frac{4}{1026} \]

3. **For \( \frac{256}{255} \)**:
\[ \frac{256}{255} = 1 + \frac{1}{255} \]

4. **For \( \frac{1023}{1019} \)**:
\[ \frac{1023}{1019} = 1 + \frac{4}{1019} \]

### Step 2: Compare the fractional parts

Now we need to compare the fractional parts \( \frac{4}{1025} \), \( \frac{4}{1026} \), \( \frac{1}{255} \), and \( \frac{4}{1019} \).

### Step 3: Finding a common denominator

To compare these fractions, we can find a common denominator or simply evaluate their approximate values.

1. **Evaluate \( \frac{4}{1025} \)**:
\[ \frac{4}{1025} \approx 0.003902439 \]

2. **Evaluate \( \frac{4}{1026} \)**:
\[ \frac{4}{1026} \approx 0.003901573 \]

3. **Evaluate \( \frac{1}{255} \)**:
\[ \frac{1}{255} \approx 0.003921569 \]

4. **Evaluate \( \frac{4}{1019} \)**:
\[ \frac{4}{1019} \approx 0.003923 \]

### Step 4: Compare the values

Now we can compare the calculated values:

- \( \frac{4}{1025} \approx 0.003902439 \)
- \( \frac{4}{1026} \approx 0.003901573 \)
- \( \frac{1}{255} \approx 0.003921569 \)
- \( \frac{4}{1019} \approx 0.003923 \)

### Step 5: Determine the greatest fraction

From the comparisons, we see that:
- \( \frac{4}{1019} \) is the largest among the fractional parts.

Thus, the greatest among the original fractions is:
\[ \frac{1023}{1019} \]

### Final Answer
The greatest among \( \frac{1029}{1025} \), \( \frac{1030}{1026} \), \( \frac{256}{255} \), and \( \frac{1023}{1019} \) is \( \frac{1023}{1019} \).

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