4. Multiplying and Dividing Fractions

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Learning Objective:

Multiply fractions by multiplying the numerators and multiplying the denominators. Use the connection between division and multiplying by the reciprocal to divide fractions.

Video:
Interactive Activity

PDF Notes Link:

Multiplying And Dividing FractionsDownload

Notes:

Multiplying fractions

To multiply two fractions, simply multiply the numerators and multiply the denominators: a/b * c/d = ac/bd. For example, 2/3 * 5/7 = (2×5)/(3×7) = 10/21

Multiplying fractions

Read 2/3 x 5/7 as two-thirds of five-sevenths. Take a unit square, mark off 7 columns and block off 5 of them to represent five-sevenths. Then mark off 3 rows and shade the pieces inside 2 rows within the 5 blocked off columns to represent two-thirds of five-sevenths. There are 2 x 5 = 10 shaded pieces, which is the numerator of the answer. There are 3 x 7 = 21 total pieces, which is the denominator of the answer. So, 2/3 x 5/7 = 10/21.

Another example

2/3 x 5/8. Multiplying the numerators and denominators we get (2×5)/(3×8) = 10/24. We would typically then reduce this to lowest terms using the common factor 2: 10/24 = 5/12. Alternatively, we can cancel this common factor before multiplying: 2/3 x 5/8 = (1×5)/(3×4) = 5/12. This technique of canceling common factors is a useful one to remember as it can save a lot of time with more complicated calculations.

Dividing fractions

Consider the connection between dividing whole numbers and multiplying fractions. One way to divide a number, a, by another number, b, is to multiply a by the reciprocal of b. Algebraically, a ÷ b = a x (1/b). For example, 3 ÷ 4 = 3 x 1/4 = 3/4 or 0.75.

We can do something similar when dividing fractions. To divide a fraction, a/b, by another fraction, c/d, we multiply a/b by the reciprocal of c/d. Algebraically, a/b ÷ c/d = a/b x d/c = ad/bc. For example, 2/3 ÷ 5/7 = 2/3 x 7/5 = (2×7)/(3×5) = 14/15.

More examples

\frac{2}{3} \div \frac{5}{8} = \frac{2}{3} \times \frac{8}{5} = \frac{16}{15} = 1\frac{1}{15}

\frac{5}{8} \div \frac{2}{3} = \frac{5}{8} \times \frac{3}{2} = \frac{15}{16}

\frac{5}{8} \div \frac{3}{4} = \frac{5}{8} \times \frac{4}{3} = \frac{20}{24} = \frac{5}{6}

\frac{5}{8} \div \frac{3}{4} = \frac{5}{8} \times \frac{4}{3} = \frac{20}{24} = \frac{5}{6}

\text{Quicker: } \frac{5}{8} \div \frac{3}{4} = \frac{5}{8^2} \times \frac{4^1}{3} = \frac{5}{6}

\frac{4}{9} \div \frac{5}{6} = \frac{4}{9^3} \times \frac{6^2}{5} = \frac{8}{15}

\frac{5}{9} \div \frac{4}{15} = \frac{5}{9^3} \times \frac{15^5}{4} = \frac{25}{12} = 2\frac{1}{12}

Transcript:

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Video Link:

https://media.tru.ca/media/Math+04/0_ljh1q1iq

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