If two proper fractions (each greater than 0 and less than 1) are multiplied together, how does the value of their product compare with each of the original fractions?

Introduction / Context:
This is a conceptual question about the behaviour of fractions when multiplied. Specifically, it focuses on proper fractions, which are fractions between 0 and 1. Understanding how products of such numbers behave is fundamental in arithmetic and number theory, and it often appears in competitive exams to test conceptual clarity.

Given Data / Assumptions:

• We are dealing with two proper fractions, say a and b, where 0 < a < 1 and 0 < b < 1.
• We consider the product a × b.
• We must compare a × b with each of the original fractions a and b.

Concept / Approach:
For any number between 0 and 1, multiplying it by a positive number less than 1 makes it smaller. This is because you are taking a fraction of it. Therefore, when you multiply a by b (where b is less than 1), the product is less than a. Similarly, multiplying b by a (where a is less than 1) makes the product less than b. Hence, the product is less than both a and b.

Step-by-Step Solution:
Step 1: Assume 0 < a < 1 and 0 < b < 1. Step 2: Because b is less than 1 and positive, a × b < a. Step 3: Similarly, because a is less than 1 and positive, a × b < b. Step 4: Therefore, a × b is less than both a and b. Step 5: This is true for all positive proper fractions a and b.

Verification / Alternative check:
Take example fractions: a = 1/2 and b = 2/3. Then a × b = (1/2) × (2/3) = 1/3. Notice that 1/3 < 1/2 and 1/3 < 2/3, confirming the general rule. Try other pairs, such as 3/4 and 4/5, and you will observe the same pattern.

Why Other Options Are Wrong:
Always greater than either: This would require the product of numbers less than 1 to be larger, which contradicts basic arithmetic.
Sometimes greater and sometimes less: For positive proper fractions, the product is always less than each factor, so there is no variation in behaviour.
Remains the same: The product equals a only if b = 1, and equals b only if a = 1, but proper fractions are strictly less than 1, so this cannot happen here.
Cannot be determined: In fact, it can be determined using general reasoning as shown.

Common Pitfalls:
Many learners confuse this with multiplication involving numbers greater than 1 or mixing fractions greater and less than 1. It is essential to focus on the condition that both fractions are strictly between 0 and 1. Visualising the fractions on a number line can help build intuition that their product must lie closer to 0 than either factor.

Final Answer:
The product of two proper fractions is always less than either of the original fractions.