Free Math Worksheets

- First find the fraction of the original amount left after the first use. - What fraction of that remainder is left after \(\frac{7}{10}\) is used? - In an area model, the overlap represents a fraction of a fraction.

1. After the first use, \(1-\frac{4}{9}=\frac{5}{9}\) of the original sugar remains. 2. After the second use, \(\frac{3}{10}\) of that remaining amount is left. 3. In an area model, one dimension represents \(\frac{5}{9}\) of the original amount and the other represents keeping \(\frac{3}{10}\) of that part. The overlap represents \(\frac{5}{9}\times\frac{3}{10}=\frac{15}{90}=\frac{1}{6}\) of the original amount.

\(\frac{1}{6}\) of the original amount is left. In an area model, the overlap of \(\frac{5}{9}\) in one direction and \(\frac{3}{10}\) in the other represents this fraction of the original amount.

- The word “of” indicates multiplication in these fraction-of-a-quantity problems. - Multiply the numerators and multiply the denominators when multiplying two fractions. - Rewrite the mixed number as an improper fraction before multiplying. - In an area model, the overlap represents the product of the two fractions. - For the real-life examples, think of using or taking only part of an amount that is already a fraction of a whole.

1. For a), the area-model overlap represents \(\frac{4}{5}\times\frac{1}{2}=\frac{2}{5}\), so the amount is \(\frac{2}{5}\,\text{gal}\). One example is drinking \(\frac{4}{5}\) of the juice in a half-gallon container. 2. For b), rewrite \(1 \frac{1}{4}\) as \(\frac{5}{4}\). The overlap represents \(\frac{2}{3}\times\frac{5}{4}=\frac{5}{6}\), so the amount is \(\frac{5}{6}\,\text{lb}\). One example is using \(\frac{2}{3}\) of a \(1 \frac{1}{4}\)-pound bag of flour. 3. For c), the overlap represents \(\frac{3}{10}\times\frac{5}{6}=\frac{1}{4}\), so the length is \(\frac{1}{4}\,\text{yd}\).

a) \(\frac{2}{5}\,\text{gal}\). The area-model overlap represents \(\frac{4}{5}\times\frac{1}{2}\). One example is drinking \(\frac{4}{5}\) of the juice in a half-gallon container. b) \(\frac{5}{6}\,\text{lb}\). The overlap represents \(\frac{2}{3}\times\frac{5}{4}\). One example is using \(\frac{2}{3}\) of a \(1 \frac{1}{4}\)-pound bag of flour. c) \(\frac{1}{4}\,\text{yd}\). The overlap represents \(\frac{3}{10}\times\frac{5}{6}\).

- Write each “fraction of a fraction” as multiplication. - Use the overlap in each area model to represent the product. - Simplify both products before comparing them.

1. For Quantity A, the area-model overlap represents \(\frac{3}{4}\times\frac{2}{5}=\frac{6}{20}=\frac{3}{10}\), so A is \(\frac{3}{10}\,\text{lb}\). 2. For Quantity B, the overlap represents \(\frac{1}{2}\times\frac{3}{5}=\frac{3}{10}\), so B is \(\frac{3}{10}\,\text{lb}\). 3. The quantities are equal.

The quantities are equal; each is \(\frac{3}{10}\,\text{lb}\). In the two area models, the overlaps represent \(\frac{3}{4}\times\frac{2}{5}\) and \(\frac{1}{2}\times\frac{3}{5}\), and both products equal \(\frac{3}{10}\).

- The long jumpers are a fraction of the track-and-field group, which is itself a fraction of all students. - Use the area-model overlap for part a). - For part b), compare finding the overall fraction first with finding the groups step by step.

1. For a), the area-model overlap represents \(\frac{2}{5}\times\frac{1}{2}=\frac{1}{5}\) of all the students. 2. One method for b) is to use the overall fraction: \(\frac{1}{5}\) of \(60\) is \(12\). 3. Another method is to find \(\frac{2}{5}\) of \(60\), which is \(24\), and then take half of \(24\), which is \(12\).

a) \(\frac{1}{5}\). In an area model, the overlap of \(\frac{2}{5}\) in one direction and \(\frac{1}{2}\) in the other represents \(\frac{1}{5}\) of all students. b) \(12\) students. One method is \(\frac{1}{5}\) of \(60\). Another is to find \(\frac{2}{5}\) of \(60\), which is \(24\), and then take \(\frac{1}{2}\) of \(24\).

- The area fraction comes from multiplying the fraction of the width by the fraction of the height. - Use the rectangular grid as an area model for the two fractions. - Count the total cells and the cells in the overlap.

1. The sheep area uses \(\frac{4}{5}\) of the width and \(\frac{2}{3}\) of the height. 2. Use the area model to multiply the fractions: \(\frac{4}{5}\times\frac{2}{3}=\frac{8}{15}\). 3. The grid has \(15\) equal cells, and the overlap covers \(8\) of them, confirming \(\frac{8}{15}\).

\(\frac{8}{15}\) of the pasture's total area