Subtracting Fractions and Mixed Numbers

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Subtracting Fractions and Mixed Numbers

Introduction

In order to subtract fractions, we find the highest common factor of the denominator, and subtract one equivalent numerator from the other.

Terms

Fraction - Fractions are used when using numbers to express parts of whole.

Least/Lowest Common Multiple - The lowest multiple of two or more numbers.

Equivalent Fraction - Equivalent fractions are fractions that have the same value.

Lesson

When subtracting fractions, we have to make sure the denominators are the same. So if we are given $dfrac{5}{6} - dfrac{1}{6}$ in this case the denominators are the same, so we subtract 1 from 5 $dfrac{5}{6} - dfrac{1}{6} = dfrac{4}{6}$ , which reduces to $dfrac{2}{3}$

When the denominators are different, we ‘make’ them the same by finding their least common multiple (LCM).

For example, in the problem $dfrac{6}{7} - dfrac{1}{2}$ , the LCM of 7 and 2 is 14

So to make the denominator of 6/7 = x/14, we have to multiply the denominator by 2 (don’t forget to multiply the numerator by 2 as well!). We then get $6*2 / 7*2 = 12/14$ . Similarly for 1/2, we multiply the numerator and denominator by 7 to get $1*7 / 2*7 = 7/14$ . The equivalent fractions of 6/7 (12/14) and 1/2 (7/14) now have the same denominators, so we simply subtract 7 from 12 to get 5/14.

When subtracting mixed fractions, we follow the same procedure but we need to convert the mixed fraction to an improper fraction first. So if we are given 1 2/3 – 1/4, we convert 1 2/3 to the improper fraction 5/3. Then we ‘make’ the denominators the same by finding the LCM, so we end up with $5*4/3*4 - 1*3/4*3 = 20/12 - 3/12$ . Since the denominators are now the same, we subtract 3 from 20 to get 17/12, or 1 5/12 (expressed as a mixed fraction).

Examples

\dfrac{2}{3} - \dfrac{1}{3}

Since these fractions have the same denominator, we can just subtract the numerators

\dfrac{2}{3} - \dfrac{1}{3} = \dfrac{1}{3}

\dfrac{3}{4} - \dfrac{1}{3}

Since these fractions have different denominators, we need to find the least common multiple of the denominators

The least common multiple of 3 and 4 is 12, so we need to multiply to make each of the denominators = 12

\dfrac{3}{4} * \dfrac{3}{3} = \dfrac{9}{12}

-\dfrac{1}{3} * \dfrac{4}{4} = -\dfrac{4}{12}

Since these fractions have the same denominator, we can just subtract the numerators

\dfrac{9}{12} - \dfrac{4}{12} = \dfrac{5}{12}

3\dfrac{5}{6} + 1\dfrac{5}{6}

3\dfrac{5}{6} + 1\dfrac{5}{6}

3 + 1 = 4

Since these fractions have the same denominator, we can just subtract the numerators

\dfrac{5}{6} - \dfrac{1}{6} = \dfrac{4}{6}

\dfrac{4}{6}

can be reduced, since

2

is a factor of both

4

and

6

\dfrac{4}{6} \div \dfrac{2}{2} = \dfrac{2}{3}

The fraction is now in lowest terms

4 + \dfrac{2}{3} = 4\dfrac{2}{3}