Among the following rational numbers, which one is a reducible fraction, that is, not already in its lowest terms?

Introduction / Context:
Fractions can often be simplified by dividing the numerator and the denominator by their greatest common divisor. A fraction that can be simplified is called a reducible fraction, while one that cannot be reduced further is said to be in lowest terms. This question tests your ability to recognize which of the given fractions can still be simplified.

Given Data / Assumptions:

• Given fractions: 79/26, 105/112, 41/17 and 91/15.
• We must identify which fraction is reducible (not in lowest terms).
• All numbers are positive.

Concept / Approach:
A fraction a/b is reducible if the numerator and denominator have a common factor greater than 1. To check this, we find the greatest common divisor (GCD) or simply test divisibility by small primes such as 2, 3, 5, 7, 11, and so on. If no common factor is found, the fraction is already in simplest form.

Step-by-Step Solution:
Step 1: Consider 79/26. 79 is a prime number and 26 = 2 * 13. There is no common factor, so 79/26 is irreducible. Step 2: Consider 105/112. Factor 105: 105 = 3 * 5 * 7. Factor 112: 112 = 2 * 2 * 2 * 2 * 7. Step 3: Both numbers share a common factor 7, so we can simplify. Step 4: Divide numerator and denominator by 7: (105 ÷ 7) / (112 ÷ 7) = 15/16. Step 5: Since 105/112 reduces to 15/16, it is definitely a reducible fraction. Step 6: Check 41/17: 41 is prime and 17 is prime, so there is no common factor. It is in lowest terms. Step 7: Check 91/15: 91 = 7 * 13 and 15 = 3 * 5. No common factors exist, so it is also in lowest terms.

Verification / Alternative Check:
We could also quickly test each fraction by trying to divide both numerator and denominator by small primes. Only 105 and 112 are divisible by 7, confirming that 105/112 simplifies to 15/16. This is a fast mental method in exams.

Why Other Options Are Wrong:
79/26 has no common factor other than 1 between 79 and 26, so it is already simplified.
41/17 uses two prime numbers with no common divisor, so it cannot be reduced.
91/15 has factors 7 * 13 and 3 * 5, which again share no common divisor. Therefore, these three fractions are not reducible, leaving only 105/112 as the correct choice.

Common Pitfalls:
One common mistake is to assume that any fraction with large numbers must be reducible. Another is to miss common factors like 7 or 11 because students usually check only 2, 3 and 5. Always factor or at least test divisibility by a few primes when time permits.

Final Answer:
The fraction that is reducible is 105/112.