For each fraction, find an equivalent fraction that has a denominator of 20:
We are tasked with finding equivalent fractions for two given fractions, \(\frac{3}{5}\) and \(\frac{3}{4}\), such that their new denominator is \(20\). Two fractions are considered equivalent if they represent the same value. To create an equivalent fraction, we multiply both the numerator and the denominator of the original fraction by the same non-zero integer \(k\). Our objective is to determine this integer \(k\) for each case, ensuring the denominator becomes \(20\).
For part (a), the fraction is \(\frac{3}{5}\). To reach a denominator of \(20\), we need to find an integer \(k\) such that \(5 \times k = 20\). Dividing \(20\) by \(5\), we find that \(k = 4\). Now, we multiply both the numerator and the denominator by \(4\):
\[\frac{3 \times 4}{5 \times 4} = \frac{12}{20}.\]
Thus, the equivalent fraction for \(\frac{3}{5}\) with a denominator of \(20\) is \(\frac{12}{20}\).
For part (b), the fraction is \(\frac{3}{4}\). To reach a denominator of \(20\), we need to find an integer \(k\) such that \(4 \times k = 20\). Dividing \(20\) by \(4\), we find that \(k = 5\). Now, we multiply both the numerator and the denominator by \(5\):
\[\frac{3 \times 5}{4 \times 5} = \frac{15}{20}.\]
Thus, the equivalent fraction for \(\frac{3}{4}\) with a denominator of \(20\) is \(\frac{15}{20}\).
Equivalent fractions are used frequently in cooking and everyday measurements. Imagine a recipe calls for \(\frac{3}{4}\) of a cup of flour, but your measuring cup only has increments of twentieths. By converting \(\frac{3}{4}\) to \(\frac{15}{20}\), you can easily measure exactly the amount required. Similarly, if a sports team wins \(\frac{3}{5}\) of their games, it is the same as saying they won \(12\) out of every \(20\) games, making it easier to compare their performance over a longer season.