Unit 2; Part 3: Multiplying and Dividing Fractions

Multiplying fractions is pretty straight forward. Now that we understand the fundamentals of fractions, this should be pretty easy. There are 3 steps:

1. Multiply the top numbers (the numerators).

2. Multiply the bottom numbers (the denominators).

3. Simplify the fraction if needed.

Lets do a few examples, then you can practice.

See Khan Academy on Multiplying Fractions Links to an external site.

Example 1: Multiply the fractions

$\frac{8}{10}\times\frac{5}{16}$ $\frac{8}{10}\times\frac{5}{16}$

$\frac{8}{10}\times\frac{5}{16}=\frac{8\cdot5}{10\cdot16}=\frac{40}{160}\:reduce\:\frac{40}{160}\:\div4=\frac{10}{40}\div10=\frac{1}{4}$ $\frac{8}{10}\times\frac{5}{16}=\frac{8\cdot5}{10\cdot16}=\frac{40}{160}\:reduce\:\frac{40}{160}\:\div4=\frac{10}{40}\div10=\frac{1}{4}$

Example 2: Multiply the mixed numbers

$4\frac{1}{6}\times4\frac{6}{11}$ $4\frac{1}{6}\times4\frac{6}{11}$

Here, we will need to convert the mixed numbers into improper fractions.

$6\times4+1=25\:\:\:\:\:\:\frac{25}{6}\:\:\:\:\:\:\:\:\:\:\:\:\:and\:\:\:\:\:\:\:\:\:\:\:\:\:\:11\times4+6=50\:\:\:\:\:\:\frac{50}{11}$ $6\times4+1=25\:\:\:\:\:\:\frac{25}{6}\:\:\:\:\:\:\:\:\:\:\:\:\:and\:\:\:\:\:\:\:\:\:\:\:\:\:\:11\times4+6=50\:\:\:\:\:\:\frac{50}{11}$

$\frac{25}{6}\times\frac{50}{11}=\frac{25\cdot50}{6\cdot11}=\frac{1,250}{66}$ $\frac{25}{6}\times\frac{50}{11}=\frac{25\cdot50}{6\cdot11}=\frac{1,250}{66}$

Convert back to mixed number.

$1,250\div66=18\:r62$ $1,250\div66=18\:r62$

18 is the whole number, the remainder (62) is the numerator and the denominator remains the same (66).

$18\frac{62}{66}$ $18\frac{62}{66}$ Cannot be reduced.

Practice 2.3.A

Multiply the fractions and/or mixed number, reduce.

1. $10\frac{1}{2}\times1\frac{6}{6}$ $10\frac{1}{2}\times1\frac{6}{6}$

2. $\frac{5}{30}\times200$ $\frac{5}{30}\times200$

3. $\frac{1}{5}\times\frac{3}{7}$ $\frac{1}{5}\times\frac{3}{7}$

4. $\frac{3}{4}\times\frac{1}{12}$ $\frac{3}{4}\times\frac{1}{12}$

5. $\frac{12}{48}\times\frac{1}{2}$ $\frac{12}{48}\times\frac{1}{2}$

Dividing fractions is almost the same as multiplying fractions. To divide any number by a fraction: Multiply the number by the reciprocal of the fraction. The reciprocal of a number is: 1 divided by the number. The reciprocal of 2 is 1/2 (half) and the reciprocal of 10 is 1/10 (=0.1). Every number has a reciprocal except 0. For our fractions, when we divide it by one, it "flips", it turns the fraction upside-down. $the\:reciprocal\:of\:\frac{1}{3}\:is\:\frac{3}{1}.$ $the\:reciprocal\:of\:\frac{1}{3}\:is\:\frac{3}{1}.$

Once you convert the second fraction into it's reciprocal, you multiply across.

Lets divide: $\frac{5}{8}\div\frac{1}{4}$ $\frac{5}{8}\div\frac{1}{4}$

$\frac{5}{8}\div\frac{1}{4}=\frac{5}{8}\times\frac{4}{1}=\frac{5\cdot4}{8\cdot1}=\frac{20}{8}$ $\frac{5}{8}\div\frac{1}{4}=\frac{5}{8}\times\frac{4}{1}=\frac{5\cdot4}{8\cdot1}=\frac{20}{8}$

Convert the improper fraction into a mixed number.

$20\div8=2\:r4\:\:\:\:\:\:2\frac{4}{20}$ $20\div8=2\:r4\:\:\:\:\:\:2\frac{4}{20}$

Simplify the resulting fraction if possible.

$4\div4=1\:\:\:\:\:\:\:\:\:\:\:\:\:20\div4=5$ $4\div4=1\:\:\:\:\:\:\:\:\:\:\:\:\:20\div4=5$

$2\frac{4}{20}=2\frac{1}{5}$ $2\frac{4}{20}=2\frac{1}{5}$

Example 1: Divide the following fractions and/or mixed numbers

$2\frac{4}{9}\div\frac{16}{19}$ $2\frac{4}{9}\div\frac{16}{19}$

First, convert mixed number into improper fraction.

$9\times2+4=22\:\:\:\:\frac{22}{9}$ $9\times2+4=22\:\:\:\:\frac{22}{9}$

Find the reciprocal of the second fraction and solve.

$\frac{22}{9}\div\frac{16}{19}=\frac{22}{9}\times\frac{19}{16}=\frac{22\cdot19}{9\cdot16}=\frac{418}{144}$ $\frac{22}{9}\div\frac{16}{19}=\frac{22}{9}\times\frac{19}{16}=\frac{22\cdot19}{9\cdot16}=\frac{418}{144}$

Convert improper fraction to a mixed number.

$2\frac{130}{144}\:\:\:\left(\div2\right)=\:2\:\:\frac{65}{72}$ $2\frac{130}{144}\:\:\:\left(\div2\right)=\:2\:\:\frac{65}{72}$

Practice 2.3.B

Divide the fractions and/or mixed number, reduce.

1. $\frac{2}{3}\div\frac{3}{4}$ $\frac{2}{3}\div\frac{3}{4}$

2. $\frac{2}{200}\div\frac{1}{125}$ $\frac{2}{200}\div\frac{1}{125}$

3. $\frac{1}{75}\div\frac{1}{80}$ $\frac{1}{75}\div\frac{1}{80}$

4. $\frac{1}{4}\div\frac{5}{8}$ $\frac{1}{4}\div\frac{5}{8}$

5. $\frac{1}{4}\div\frac{5}{8}$ $\frac{1}{4}\div\frac{5}{8}$

Submit practice assignments HERE