If two fractions each have values strictly between 0 and 1, what can be said about the value of their product?

Introduction / Context:
This conceptual question examines how multiplication behaves for fractions between 0 and 1. Understanding this property is fundamental in number theory and probability, where such fractions often represent probabilities, proportions, or scaling factors that shrink quantities rather than enlarge them.

Given Data / Assumptions:

• We are dealing with two fractions, call them a and b.
• Both fractions satisfy 0 < a < 1 and 0 < b < 1.
• We must determine how the product a × b compares to a and b individually.
• The options describe different possible relationships between the product and the original fractions.

Concept / Approach:
When you multiply a positive number by a fraction less than 1, the result becomes smaller than the original number. This is because a fraction less than 1 represents a part of the whole. If you apply such a reduction twice, the final result must be smaller than both of the original positive fractions. Thus, the product of two numbers between 0 and 1 should always be less than each of them.

Step-by-Step Solution:
Let 0 < a < 1 and 0 < b < 1.First, consider a × b compared to a. Since 0 < b < 1, multiplying a by b gives a × b < a.Next, consider a × b compared to b. Since 0 < a < 1, multiplying b by a gives a × b < b.Therefore, a × b is less than both a and b.Because both a and b are positive, the product is still positive, but it is strictly smaller than each factor.Hence, the product is always less than either of the original fractions.

Verification / Alternative check:
We can verify this with a simple numerical example. Take a = 1/2 and b = 3/4. Both are between 0 and 1. Their product is (1/2) × (3/4) = 3/8 = 0.375. Compare: 0.375 < 0.5 and 0.375 < 0.75. This confirms the general reasoning. Trying other pairs of fractions such as 0.2 and 0.6 gives 0.12, which again is smaller than both 0.2 and 0.6.

Why Other Options Are Wrong:
Option stating that the product is always greater than either fraction contradicts the basic shrinking effect of multiplying by a number less than 1. The option suggesting that the product can sometimes be greater is also incorrect given the strict inequalities. Saying it remains the same or equals one original fraction would require one factor to be 1, which is excluded because both fractions are strictly less than 1. The product certainly does not equal the sum, which would be larger than either fraction.

Common Pitfalls:
A common misunderstanding is to think of multiplication as always making numbers larger, based on experience with whole numbers. This intuition fails for fractions less than 1. Another mistake is confusing multiplication with addition, mistakenly expecting the result to lie between the two fractions as a kind of average. Remember that multiplying by a fraction less than 1 always scales the quantity down, not up.

Final Answer:
The product of two fractions each between 0 and 1 is always less than either of the original fractions.