Which of the following fractions is the largest? \(\frac{5}{7}, \frac{9}{14},…

Concept

To compare fractions with different denominators, rewrite them using a common positive denominator.

Once the denominators are equal, the fraction with the greatest numerator has the greatest value.

Application

1. The denominators are 7, 14, 21, and 42; their least common multiple is 42.

2. Rewrite 5/7 as 30/42 by multiplying its numerator and denominator by 6.

3. Rewrite 9/14 as 27/42 by multiplying its numerator and denominator by 3.

4. Rewrite 16/21 as 32/42 by multiplying its numerator and denominator by 2.

5. The fraction 29/42 already has denominator 42, so its equivalent form remains 29/42.

6. Comparing the common-denominator numerators gives 32 > 30 > 29 > 27; therefore, 16/21 is the largest fraction.

Cross-check

Decimal approximations provide an independent check:

Both methods show that the largest fraction is 16/21.