To determine which of the given options is true when a number greater than 0 is multiplied by a fraction greater than 0 but less than 1, let's analyze the situation step-by-step:
1. Understanding Fractions and Multiplication:
- A fraction greater than 0 but less than 1 can be represented as [tex]\(\frac{a}{b}\)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are positive integers and [tex]\(a < b\)[/tex].
- If we multiply a positive number [tex]\(n\)[/tex] (where [tex]\(n > 0\)[/tex]) by such a fraction [tex]\(\frac{a}{b}\)[/tex], the product will be [tex]\(\frac{a}{b} \times n\)[/tex].
2. Analyzing the Product:
- Because [tex]\(0 < \frac{a}{b} < 1\)[/tex], multiplying [tex]\(n\)[/tex] by [tex]\(\frac{a}{b}\)[/tex] means you are taking a part of [tex]\(n\)[/tex].
- Mathematically, [tex]\(\frac{a}{b} \times n\)[/tex] will be less than [tex]\(n\)[/tex] since [tex]\(a < b\)[/tex], implying that the fraction [tex]\(\frac{a}{b}\)[/tex] is less than 1.
3. Generalization:
- For any positive number [tex]\(n\)[/tex], the expression [tex]\(n \times \frac{a}{b}\)[/tex] (with [tex]\(0 < \frac{a}{b} < 1\)[/tex]) results in a product that is always less than [tex]\(n\)[/tex].
Thus, the correct conclusion is that the product of any positive number greater than 0 and a fraction greater than 0 but less than 1 is less than the given number.
Therefore, the true statement is:
"The product is always less than the given number."
