Which of the fractions given below is not equal to `5/9` ?

Options

A. `105/198`
B. `80/144`
C. `40/72`
D. `25/45`

Answer

To determine which of the given fractions is not equal to \( \frac{5}{9} \), we will evaluate each option one by one.

### Step 1: Evaluate the first fraction \( \frac{105}{198} \)

1. **Simplifying \( \frac{105}{198} \)**:
- Find the greatest common divisor (GCD) of 105 and 198.
- The GCD of 105 and 198 is 3.
- Now, divide both the numerator and the denominator by their GCD:
\[ \frac{105 \div 3}{198 \div 3} = \frac{35}{66} \]
- Next, check if \( \frac{35}{66} \) can be simplified further. The GCD of 35 and 66 is 1, so it cannot be simplified further.

2. **Comparing with \( \frac{5}{9} \)**:
- Since \( \frac{35}{66} \) cannot be simplified to \( \frac{5}{9} \), we can conclude that \( \frac{105}{198} \) is not equal to \( \frac{5}{9} \).

### Step 2: Evaluate the second fraction \( \frac{80}{144} \)

1. **Simplifying \( \frac{80}{144} \)**:
- Find the GCD of 80 and 144, which is 16.
- Divide both the numerator and the denominator by their GCD:
\[ \frac{80 \div 16}{144 \div 16} = \frac{5}{9} \]
- This fraction simplifies to \( \frac{5}{9} \).

### Step 3: Evaluate the third fraction \( \frac{40}{72} \)

1. **Simplifying \( \frac{40}{72} \)**:
- Find the GCD of 40 and 72, which is 8.
- Divide both the numerator and the denominator by their GCD:
\[ \frac{40 \div 8}{72 \div 8} = \frac{5}{9} \]
- This fraction also simplifies to \( \frac{5}{9} \).

### Step 4: Evaluate the fourth fraction \( \frac{25}{45} \)

1. **Simplifying \( \frac{25}{45} \)**:
- Find the GCD of 25 and 45, which is 5.
- Divide both the numerator and the denominator by their GCD:
\[ \frac{25 \div 5}{45 \div 5} = \frac{5}{9} \]
- This fraction simplifies to \( \frac{5}{9} \).

### Conclusion

After evaluating all the fractions, we find that the only fraction that does not equal \( \frac{5}{9} \) is \( \frac{105}{198} \).

Thus, the answer is:
**The fraction that is not equal to \( \frac{5}{9} \) is \( \frac{105}{198} \).**

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