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Wize High School Algebra I Textbook (Common Core) > Foundations of Algebra

📝Review: Fractions, & Mixed Numbers

📝Review: Integers, DecimalsPrevious Section
📝Review: Ratios, Rates, and ProportionsNext Section
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Adding & Subtracting Fractions


The quickest way to add and subtract fractions is to come up with a common denominator, then add or subtract the numerators only.

Example 1
45+13=\displaystyle \frac{4}{5}+\frac{1}{3}=
17/5

4×35×3+1×53×5\displaystyle \frac{4\orange{\times3}}{5\orange{\times3}}+\frac{1\orange{\times5}}{3\orange{\times5}}

=1215+515\displaystyle =\frac{12}{15}+\frac{5}{15}

=12+515\displaystyle =\frac{12+5}{15}

=1715\displaystyle =\frac{17}{15}
You can also rewrite this as a mixed fraction12151\frac{2}{15}.

See video for a picture explanation.

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Example 2
115−512=\displaystyle \frac{1}{15}-\frac{5}{12}=
- 21/60

1×415×4−5×512×5\displaystyle \frac{1\orange{\times4}}{15\orange{\times4}}-\frac{5\orange{\times5}}{12\orange{\times5}}

=460−2560\displaystyle =\frac{4}{60}-\frac{25}{60}

=4−2560\displaystyle =\frac{4-25}{60}

=−2160\displaystyle =-\frac{21}{60}

Another method
If you struggle with finding the lowest common denominator (LCD) between two fractions, you can simply multiply both denominators together.

1×1215×12−5×1512×15\displaystyle \frac{1\orange{\times12}}{15\orange{\times12}}-\frac{5\orange{\times15}}{12\orange{\times15}}

=12180−75180\displaystyle =\frac{12}{180}-\frac{75}{180}

=12−75180\displaystyle =\frac{12-75}{180}

=−63180\displaystyle =-\frac{63}{180} (÷\div numerator and denominator by 3)

=−2160\displaystyle =-\frac{21}{60}

Practice: Adding & Subtracting Fractions

Evaluate 56+(−14)−(−13)\dfrac{5}{6}+\left(-\dfrac{1}{4}\right)-\left(-\dfrac{1}{3}\right).
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Multiplying & Dividing Fractions

Multiplying

To multiply fractions, we multiply the numerators (tops) together, and we multiply the denominators (bottoms) together.

Example 1
−35×(−74)=-\frac{3}{5}\times\left(-\frac{7}{4}\right)=
21/20

−35×(−74)\bcf-\dfrac{3}{5}\times\left(\bcf-\dfrac{7}{4}\right)

=+3×75×4=\bcf+\dfrac{3\times7}{5\times4}

=+2120=\bcf+\dfrac{21}{20}

We can rewrite this as a mixed fraction 11201 \dfrac{1}{20}.

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Dividing

To divide fractions, we multiply the first fraction by the "flip" of the second fraction.

Wize Tip
I remember this as "Flip & Multiply".

Example 2
35÷(−74)=\dfrac{3}{5}\div\left(-\dfrac{7}{4}\right)=
-12/35

+35÷(−74)\bcf+\dfrac{3}{5}\div\left(\bcf-\dfrac{7}{4}\right)

*We use our "flip & multiply" trick:

+35×(−47)\bcf+\dfrac{3}{5}\bcth\times\left(\bcf-\bcth{\dfrac{4}{7}}\right)

=−3×45×7=\bcf-\dfrac{3\times4}{5\times7}

=−1235=\bcf-\dfrac{12}{35}

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Tips for Simplifying Your Calculations

Sometimes we can simplify our fractions along the way to make the numbers we are working with smaller.

Wize Tip
Tip #1
Divide out numbers that go into both the numerator and denominator within a fraction

Tip #2
Divide out numbers that go into both the numerator and denominator across fractions

Example 3
46×2715\dfrac{4}{6}\times\dfrac{27}{15}
  • You can divide 2 from the top and bottom of the first fraction
  • You can divide 3 from the top and bottom of the second fraction
=4  96  5×27  215  3=\dfrac{\cancel4^{~~9}}{\cancel6^{~~5}}\times\dfrac{\cancel{27}^{~~2}}{\cancel{15}^{~~3}}

=23×95=\dfrac{2}{3}\times\dfrac{9}{5}
  • You can divide 3 from the bottom of the first fraction and the top of the second fraction
=23  1×9  35=\dfrac{2}{\cancel3^{~~1}}\times\dfrac{\cancel9^{~~3}}{5}

=21×35=\dfrac{2}{1}\times\dfrac{3}{5}

=2×31×5=\dfrac{2\times3}{1\times5}
=65=\boxed{\dfrac{6}{5}}

Practice: Multiplying & Dividing Fractions

Simplify the following:

a) −1215×78÷(−215)-\dfrac{12}{15}\times\dfrac{7}{8}\div\left(-\dfrac{21}{5}\right)

b) 310×15\dfrac{3}{10}\times15

c) 310÷9\dfrac{3}{10}\div9
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Adding & Subtracting Mixed Numbers

There are a few ways to add and subtract mixed numbers.

Method 1 - Whole Number & Fraction Parts

We add the whole number parts and the fraction parts separately, then we combine and simplify the final answer.

Example 1
3512+145=\displaystyle 3\frac{5}{12}+1\frac{4}{5}=
5 and 13/60


Adding the whole number parts:
3+1=4‾3+1=\underline{4}

Adding the fraction parts:
512+45\displaystyle \frac{5}{12}+\frac{4}{5}

=5×512×5+4×125×12\displaystyle =\frac{5\orange{\times5}}{12\orange{\times5}}+\frac{4\orange{\times12}}{5\orange{\times12}}

=2560+4860\displaystyle =\frac{25}{60}+\frac{48}{60}

=7360\displaystyle =\frac{73}{60}

=11360‾\displaystyle =\underline{1\frac{13}{60}}

Then we combine these results:
3512+145\displaystyle 3\frac{5}{12}+1\frac{4}{5}

=4+11360\displaystyle =4+1\frac{13}{60}

Therefore, the answer is 51360\displaystyle \boxed{5\frac{13}{60}}.
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Example 2
3512−145=\displaystyle 3\frac{5}{12}-1\frac{4}{5}=
1 and 37/60

Subtracting the whole number parts:
3−1=2‾3-1=\underline{2}

Subtracting the fraction parts:
512−45\frac{5}{12}-\frac{4}{5}
=5×512×5−4×125×12=\frac{5\times5}{12\times5}-\frac{4\times12}{5\times12}
=2560−4860=\frac{25}{60}-\frac{48}{60}
=−2360‾=\displaystyle \underline{-\frac{23}{60}}

Then we combine these results:
3512−145\displaystyle 3\frac{5}{12}-1\frac{4}{5}

=2+(−2360)=2+\left(-\frac{23}{60}\right)

=1+6060−2360\displaystyle=1+\frac{60}{60}-\frac{23}{60}

=1+3760\displaystyle=1+\frac{37}{60}

Therefore, the answer is 13760\boxed{\displaystyle1\frac{37}{60}}.
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Method 2 - Improper Fractions

First convert all mixed fractions into improper fractions, then we can add/subtract these fractions and simplify the final answer.

Example 3
3512−145=\displaystyle 3\frac{5}{12}-1\frac{4}{5}=
1 and 37/60

Convert both fractions to improper fractions:
4112−95\displaystyle \frac{41}{12}-\frac{9}{5}

=41×512×5−9×125×12=\displaystyle \frac{41\orange{\times5}}{12\orange{\times5}}-\frac{9\orange{\times 12}}{5\orange{\times 12}}

=20560−10860\displaystyle =\frac{205}{60}-\frac{108}{60}

=9760\displaystyle =\frac{97}{60}

Simplify the answer:
=13760\displaystyle=\boxed{1\frac{37}{60}}

Practice: Adding & Subtracting Mixed Numbers

Evaluate 347+2163\dfrac{4}{7}+2\dfrac{1}{6}

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Multiplying & Dividing Mixed Numbers

Improper Fraction Method

The quickest way to multiply and divide mixed numbers is to first convert them to improper fractions.

Example 1
314×2253\dfrac{1}{4}\times2\dfrac{2}{5}

=134×125=\dfrac{13}{4}\times\dfrac{12}{5}

=134 1×12 35=\dfrac{13}{\cancel4^{~1}}\times\dfrac{\cancel{12}^{~3}}{5}

=13×31×5=\dfrac{13\times3}{1\times5}

=395=\dfrac{39}{5} or 7457\dfrac{4}{5}
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Example 2
314÷2253\dfrac{1}{4}\div2\dfrac{2}{5}

=134÷125=\dfrac{13}{4}\div\dfrac{12}{5}

=134×512=\dfrac{13}{4}\times\dfrac{5}{12}

=13×54×12=\dfrac{13\times5}{4\times12}

=6548=\dfrac{65}{48} or 117481\dfrac{17}{48}

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Area Model Method (for Multiplying Mixed Numbers)

We can treat the multiplication of two mixed numbers as the area of a rectangle.

Example 3
314×2253\dfrac{1}{4}\times2\dfrac{2}{5}

First represent this as an area of a rectangle.

Then add up the areas of each piece of the rectangle.


2×3+2×14+25×3+25×14\bcf{2\times3}+\bcth{2\times\dfrac{1}{4}}+\bcfi{\dfrac{2}{5}\times3}+\bct{\dfrac{2}{5}\times\dfrac{1}{4}}

=6+12+65+110=6+\dfrac{1}{2}+\dfrac{6}{5}+\dfrac{1}{10}

We need to find a common denominator:
=6×101×10+1×52×5+6×25×2+110=\dfrac{6\bm{\colorbox{yellow}{$\times10$}}}{1\bm{\colorbox{yellow}{$\times10$}}}+\dfrac{1\bm{\colorbox{yellow}{$\times5$}}}{2\bm{\colorbox{yellow}{$\times5$}}}+\dfrac{6\bm{\colorbox{yellow}{$\times2$}}}{5\bm{\colorbox{yellow}{$\times2$}}}+\dfrac{1}{10}

=6010+510+1210+110=\dfrac{60}{10}+\dfrac{5}{10}+\dfrac{12}{10}+\dfrac{1}{10}

=60+5+12+110=\dfrac{60+5+12+1}{10}

=7810=\dfrac{78}{10}

=395=\dfrac{39}{5} or 7457\dfrac{4}{5}

Practice: Multiplying & Dividing Mixed Numbers

Calculate −335×(−556)-3\dfrac{3}{5}\times\left(-5\dfrac{5}{6}\right)

Practice: Measurements with Mixed Numbers

A rectangular patio has dimensions 101210\dfrac{1}{2}ft by 5135\dfrac{1}{3} ft.



Find the area of the patio.