5th Grade Multiplying Fractions and Mixed Numbers Test Solutions and Scoring (PDF)

(a)

Write the whole number 6 as \(\frac{6}{1}\).

Multiply the numerators and multiply the denominators: \(\frac{6}{1} \times \frac{2}{9} = \frac{6 \times 2}{1 \times 9} = \frac{12}{9}\).

Simplify: \(\frac{12}{9} = \frac{4}{3}\).

Write as a mixed number: \(\frac{4}{3} = 1\frac{1}{3}\).

\(1\frac{1}{3}\)

(b) \(\frac{3}{5} \times 15 = \frac{45}{5} = 9\): 9.

Scoring: 1 point for each correct answer in simplest form.

(a)

Multiply the numerators and multiply the denominators: \(\frac{3}{8} \times \frac{2}{3} = \frac{3 \times 2}{8 \times 3} = \frac{6}{24}\).

Simplify: \(\frac{6}{24} = \frac{1}{4}\).

\(\frac{1}{4}\)

(b)

Multiply the numerators and multiply the denominators: \(\frac{5}{6} \times \frac{9}{10} = \frac{5 \times 9}{6 \times 10} = \frac{45}{60}\).

Simplify: \(\frac{45}{60} = \frac{3}{4}\).

\(\frac{3}{4}\)

Scoring: 1 point for each correct product in simplest form.

(a) \(\frac{2}{3} \times \frac{4}{5}\).

(b) \(\frac{2 \times 4}{3 \times 5} = \frac{8}{15}\): \(\frac{8}{15}\).

Scoring: 1 point for the sentence (either order); 1 point for the product.

(a)

Write the mixed numbers as improper fractions: \(1\frac{1}{2} = \frac{3}{2}\) and \(3\frac{1}{3} = \frac{10}{3}\).

Multiply the numerators and multiply the denominators: \(\frac{3}{2} \times \frac{10}{3} = \frac{3 \times 10}{2 \times 3} = \frac{30}{6}\).

Simplify: \(\frac{30}{6} = 5\).

\(5\)

(b)

Write the mixed numbers as improper fractions: \(2\frac{2}{3} = \frac{8}{3}\) and \(1\frac{3}{4} = \frac{7}{4}\).

Multiply the numerators and multiply the denominators: \(\frac{8}{3} \times \frac{7}{4} = \frac{8 \times 7}{3 \times 4} = \frac{56}{12}\).

Simplify: \(\frac{56}{12} = \frac{14}{3}\).

Write as a mixed number: \(\frac{14}{3} = 4\frac{2}{3}\).

\(4\frac{2}{3}\)

Scoring: (a) 1 point. (b) 1 point for correct improper fractions, 1 point for the correct product as a mixed number.

(a) Multiplying by \(\frac{3}{4}\), which is less than 1, makes the product less than \(4\frac{1}{2}\). But \(6\frac{3}{4}\) is greater than \(4\frac{1}{2}\).

(b)

Write the mixed number as improper fraction: \(4\frac{1}{2} = \frac{9}{2}\).

Multiply the numerators and multiply the denominators: \(\frac{9}{2} \times \frac{3}{4} = \frac{9 \times 3}{2 \times 4} = \frac{27}{8}\).

Write as a mixed number: \(\frac{27}{8} = 3\frac{3}{8}\).

\(3\frac{3}{8}\)

Scoring: 1 point for a valid explanation; 1 point for the correct product.

(a) \(\frac{5}{2} \times \frac{3}{4} = \frac{15}{8} = 1\frac{7}{8}\): \(1\frac{7}{8}\) square feet.

(b)

Write the mixed numbers as improper fractions: \(3\frac{1}{3} = \frac{10}{3}\) and \(2\frac{1}{4} = \frac{9}{4}\).

Multiply the numerators and multiply the denominators: \(\frac{10}{3} \times \frac{9}{4} = \frac{10 \times 9}{3 \times 4} = \frac{90}{12}\).

Simplify: \(\frac{90}{12} = \frac{15}{2}\).

Write as a mixed number: \(\frac{15}{2} = 7\frac{1}{2}\).

\(7\frac{1}{2}\)

The second bed has an area of \(7\frac{1}{2}\) square feet.

Scoring: 1 point for each correct area with units.

(a) \(\frac{9}{4} \times \frac{1}{3} = \frac{9}{12} = \frac{3}{4}\): \(\frac{3}{4}\) cup.

(b) \(\frac{9}{4} \times \frac{5}{2} = \frac{45}{8} = 5\frac{5}{8}\): \(5\frac{5}{8}\) cups.

Scoring: 1 point for each correct answer with units.

(a) \(\frac{5}{8} \times 24 = \frac{120}{8} = 15\): 15 students.

(b) \(\frac{2}{5} \times 15 = \frac{30}{5} = 6\): 6 students.

Scoring: 1 point for each correct answer.

(a) \(\frac{1}{3} \times 36 = 12\): 12 cards.

(b) After the brother: \(36 – 12 = 24\). \(\frac{1}{4} \times 24 = 6\): 6 cards.

(c) \(24 – 6 = 18\): 18 cards.

Scoring: 1 point for each correct answer.