Question

when is the product not greater than both of its factors
in multiplying a fraction by a whole number

Answer 1

The product of a fraction and a whole number is not greater than both of its factors when that fraction is less than or equal to 1.

To illustrate, let's break this down:

1. **Understanding Factors**: When you multiply a fraction \( \frac{a}{b} \) by a whole number \( c \), the product is \( c \times \frac{a}{b} = \frac{ac}{b} \).

2. **When is the Product Not Greater?**:
- If \( \frac{a}{b} \) is less than 1 (i.e., \( a < b \)), then \( c \times \frac{a}{b} < c \) because you're taking a fraction of \( c \).
- This means that the product \( \frac{ac}{b} \) will not exceed either \( c \) (the whole number factor) or the fraction \( \frac{a}{b} \) (considering that it is less than 1).

3. **When \( \frac{a}{b} = 1 \)**: If the fraction equals 1 (meaning \( a = b \)), then the product equals \( c \). At this point, the product is equal to one of the factors but not greater than both.

4. **Conclusion**: The product \( c \times \frac{a}{b} \) is not greater than both factors when \( \frac{a}{b} \leq 1 \) (or, equivalently, when \( a \leq b \) if we are considering positive integers).

In summary, when multiplying a fraction by a whole number, the product will not be greater than both factors when the fraction is less than or equal to 1.