The square of a proper fraction is always __________ than itself

Options

A. Less
B. Greater
C. Both
D. None

Answer

To solve the question "The square of a proper fraction is always __________ than itself," we need to analyze the properties of proper fractions and their squares.

### Step-by-Step Solution:

1. **Understanding Proper Fractions**:
A proper fraction is defined as a fraction where the numerator is less than the denominator. For example, \( \frac{4}{5} \) is a proper fraction because 4 (the numerator) is less than 5 (the denominator).

**Hint**: Remember that proper fractions are always less than 1.

2. **Squaring the Proper Fraction**:
When we square a proper fraction, we multiply the fraction by itself. For example, squaring \( \frac{4}{5} \):
\[ \left( \frac{4}{5} \right)^2 = \frac{4 \times 4}{5 \times 5} = \frac{16}{25} \]

**Hint**: To square a fraction, square both the numerator and the denominator.

3. **Comparing the Square to the Original Fraction**:
Now we need to compare \( \frac{16}{25} \) to \( \frac{4}{5} \). Since both fractions are proper fractions, we can convert them to decimal form for easier comparison:
- \( \frac{4}{5} = 0.8 \)
- \( \frac{16}{25} = 0.64 \)

**Hint**: Converting fractions to decimals can help in comparing their sizes.

4. **Conclusion**:
From our comparison, we see that \( 0.64 < 0.8 \). Therefore, the square of the proper fraction \( \frac{4}{5} \) is less than the proper fraction itself.

**Final Answer**: The square of a proper fraction is always **less** than itself.