Find out whether the following fractions are equivalent: (1) 14/21, 18/27; (2) 15/20, 20/15; (3) 24/30, 32/40; (4) 10/35, 14/35.

Steps to Solve

1. Simplifying the first pair: $ \frac{14}{21} $ and $ \frac{18}{27} $

To determine if these fractions are equivalent, we simplify each fraction.

For $ \frac{14}{21} $, divide the numerator and denominator by their greatest common divisor (GCD), which is 7:

$$ \frac{14 \div 7}{21 \div 7} = \frac{2}{3} $$

For $ \frac{18}{27} $, the GCD is 9:

$$ \frac{18 \div 9}{27 \div 9} = \frac{2}{3} $$

Both fractions simplify to $ \frac{2}{3} $, thus they are equivalent.

2. Simplifying the second pair: $ \frac{15}{20} $ and $ \frac{20}{15} $

Simplify $ \frac{15}{20} $ by dividing by its GCD, which is 5:

$$ \frac{15 \div 5}{20 \div 5} = \frac{3}{4} $$

Now simplify $ \frac{20}{15} $, where the GCD is 5 as well:

$$ \frac{20 \div 5}{15 \div 5} = \frac{4}{3} $$

The fractions simplify to $ \frac{3}{4} $ and $ \frac{4}{3} $, which are not equivalent.

3. Simplifying the third pair: $ \frac{24}{30} $ and $ \frac{32}{40} $

Simplify $ \frac{24}{30} $. The GCD is 6:

$$ \frac{24 \div 6}{30 \div 6} = \frac{4}{5} $$

For $ \frac{32}{40} $, the GCD is 8:

$$ \frac{32 \div 8}{40 \div 8} = \frac{4}{5} $$

Both fractions simplify to $ \frac{4}{5} $, thus they are equivalent.

4. Simplifying the fourth pair: $ \frac{10}{35} $ and $ \frac{14}{35} $

For $ \frac{10}{35} $, the GCD is 5:

$$ \frac{10 \div 5}{35 \div 5} = \frac{2}{7} $$

Now look at $ \frac{14}{35} $, where the GCD is 7:

$$ \frac{14 \div 7}{35 \div 7} = \frac{2}{5} $$

The fractions simplify to $ \frac{2}{7} $ and $ \frac{2}{5} $, which are not equivalent.