From Zero to Engineer

Fractions: multiplying, dividing, adding and simplifying

A review of prime factorization, GCD and LCM, types of fractions, equivalent fractions, simplifying, multiplying, dividing by the reciprocal, and finding a common denominator.

A fraction is one integer divided by another non-zero integer. This lesson covers the types of fractions, how to simplify and expand them, and then how to multiply them, divide by the reciprocal and add them after finding a common denominator.

Review exercises

1) Write each of the following numbers as a product of prime factors

429	3
143	11
13	13
1

429 = 3 \cdot 11 \cdot 13

2992	2
1496	2
748	2
374	2
187	11
17	17
1

2992 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 11 \cdot 17

1820	2
910	2
455	5
91	7
13	13
1

1820 = 2 \cdot 2 \cdot 5 \cdot 7 \cdot 13

3185	5
637	7
91	7
13	13
1

3185 = 5 \cdot 7 \cdot 7 \cdot 13

2) Find the GCD and LCM of the following pairs of numbers

63	3
21	3
7	7
1

63 = 3 \cdot 3 \cdot 7

34	2
17	17
1

34 = 2 \cdot 17

42	2
21	3
7	7
1

42 = 2 \cdot 3 \cdot 7

92	2
46	2
23	23
1

92 = 2 \cdot 2 \cdot 23

\mathrm{GCD}(63;\ 42) = 3 \cdot 7 = 21

\mathrm{LCM}(63;\ 42) = 3 \cdot 3 \cdot 7 \cdot 2 = 126

\mathrm{GCD}(34;\ 92) = 2 = 2

\mathrm{LCM}(34;\ 92) = 2 \cdot 2 \cdot 17 \cdot 23 = 1564

Fractions (rational numbers)

Fractions are written as an integer divided by a non-zero integer and are called rational numbers.

-\tfrac{8}{11}

proper fraction

6\tfrac{2}{3}

mixed number

\tfrac{12}{5}

improper fraction

\tfrac{4}{7}

4 is the numerator, 7 is the denominator

Multiplying fractions

\tfrac{2}{3} \cdot \tfrac{5}{7} = \tfrac{2 \cdot 5}{3 \cdot 7} = \tfrac{10}{21}

\tfrac{1}{3} \cdot \tfrac{2}{5} = \tfrac{2}{15}

one third of two fifths

\tfrac{5}{9} \cdot \tfrac{2}{7} = \tfrac{5 \cdot 2}{9 \cdot 7} = \tfrac{10}{63}

\tfrac{3}{8} \cdot \tfrac{5}{7} = \tfrac{15}{56}

three eighths of five sevenths

Equivalent fractions

\tfrac{4}{5} = \tfrac{4 \cdot 3}{5 \cdot 3} = \tfrac{4}{5} \cdot \tfrac{3}{3} = \tfrac{4}{5} \cdot 1 = \tfrac{4}{5}

\tfrac{4}{5} \cdot \tfrac{3}{3} = \tfrac{12}{15}

\tfrac{4}{5} = \tfrac{12}{15}

\tfrac{7}{5} \cdot \tfrac{4}{4} = \tfrac{28}{20}

\tfrac{7}{5} = \tfrac{28}{20}

Simplifying fractions

\tfrac{16}{96} = \tfrac{16}{6 \cdot 16} = \tfrac{1}{6}

\tfrac{84}{108} = \tfrac{7 \cdot 12}{9 \cdot 12} = \tfrac{7}{9}

Dividing fractions

To divide two fractions, multiply the first fraction by the reciprocal of the second fraction.

\tfrac{2}{3} : \tfrac{5}{7} = \tfrac{2}{3} \cdot \tfrac{7}{5} = \tfrac{14}{15}

\tfrac{7}{13} : \tfrac{3}{4} = \tfrac{7}{13} \cdot \tfrac{4}{3} = \tfrac{28}{39}

Adding and subtracting fractions

\tfrac{2}{7} + \tfrac{3}{7} = \tfrac{2 + 3}{7} = \tfrac{5}{7}

common denominator

\tfrac{2}{3} + \tfrac{1}{5} = \tfrac{2}{3} \cdot \tfrac{5}{5} + \tfrac{1}{5} \cdot \tfrac{3}{3} = \tfrac{10}{15} + \tfrac{3}{15} = \tfrac{13}{15}

no common denominator, so we find one

\tfrac{5}{9} + \tfrac{1}{6} = \tfrac{5}{9} \cdot \tfrac{2}{2} + \tfrac{1}{6} \cdot \tfrac{3}{3} = \tfrac{10}{18} + \tfrac{3}{18} = \tfrac{13}{18}

\tfrac{11}{15} - \tfrac{2}{3} = \tfrac{11}{15} - \tfrac{2}{3} \cdot \tfrac{5}{5} = \tfrac{11}{15} - \tfrac{10}{15} = \tfrac{1}{15}

Frequently asked questions

How do you multiply two fractions?

Multiply numerator by numerator and denominator by denominator; no common denominator is needed. For example 2/3 · 5/7 = 10/21.

How do you divide a fraction by a fraction?

Multiply the first fraction by the reciprocal of the second, that is by the fraction with numerator and denominator swapped. So 2/3 : 5/7 = 2/3 · 7/5 = 14/15.

Why do fractions need a common denominator?

Because you can only add parts of the same size. Only once 2/3 and 1/5 are written as 10/15 and 3/15 may you add the numerators to get 13/15.

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Divisors, prime factorization, GCD and LCM

Ratios and percentages