Questions: 17. Determine whether each set of two fractions is equivalent by calculating equivalent fractions with a common denominator. (a) 18/42 and 3/7 (b) 18/49 and 5/14 (c) 9/25 and 140/500 (d) 24/144 and 32/96

$17. Determine whether each set of two fractions is equivalent by calculating equivalent fractions with a common denominator. (a) 18/42 and 3/7 (b) 18/49 and 5/14 (c) 9/25 and 140/500 (d) 24/144 and 32/96$

Transcript text: 17. Determine whether each set of two fractions is equivalent by calculating equivalent fractions with a common denominator. (a) $\frac{18}{42}$ and $\frac{3}{7}$ (b) $\frac{18}{49}$ and $\frac{5}{14}$ (c) $\frac{9}{25}$ and $\frac{140}{500}$ (d) $\frac{24}{144}$ and $\frac{32}{96}$

Solution

Solution Steps

To determine whether each set of two fractions is equivalent, we need to convert each pair of fractions to have a common denominator and then compare their numerators. If the numerators are equal after conversion, the fractions are equivalent.

Step 1: Check Equivalence of $ \frac{18}{42} $ and $ \frac{3}{7} $

To determine if $ \frac{18}{42} $ is equivalent to $ \frac{3}{7} $, we simplify $ \frac{18}{42} $: \[ \frac{18}{42} = \frac{18 \div 6}{42 \div 6} = \frac{3}{7} \] Since both fractions simplify to $ \frac{3}{7} $, they are equivalent. Thus, $ \frac{18}{42} \equiv \frac{3}{7} $ is True.

Step 2: Check Equivalence of $ \frac{18}{49} $ and $ \frac{5}{14} $

Next, we check if $ \frac{18}{49} $ is equivalent to $ \frac{5}{14} $. We find a common denominator: \[ \text{LCD of } 49 \text{ and } 14 = 98 \] Converting both fractions: \[ \frac{18}{49} = \frac{18 \times 2}{49 \times 2} = \frac{36}{98} \] \[ \frac{5}{14} = \frac{5 \times 7}{14 \times 7} = \frac{35}{98} \] Since $ \frac{36}{98} \neq \frac{35}{98} $, $ \frac{18}{49} \equiv \frac{5}{14} $ is False.

Step 3: Check Equivalence of $ \frac{9}{25} $ and $ \frac{140}{500} $

Finally, we check if $ \frac{9}{25} $ is equivalent to $ \frac{140}{500} $. We find a common denominator: \[ \text{LCD of } 25 \text{ and } 500 = 500 \] Converting both fractions: \[ \frac{9}{25} = \frac{9 \times 20}{25 \times 20} = \frac{180}{500} \] Since $ \frac{180}{500} \neq \frac{140}{500} $, $ \frac{9}{25} \equiv \frac{140}{500} $ is False.