Wize High School Grade 9 Math Textbook > Rational Numbers

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}=54​+31​=   17/5  

4×35×3+1×53×5\displaystyle \frac{4\orange{\times3}}{5\orange{\times3}}+\frac{1\orange{\times5}}{3\orange{\times5}}5×34×3​+3×51×5​

=1215+515\displaystyle =\frac{12}{15}+\frac{5}{15}=1512​+155​

=12+515\displaystyle =\frac{12+5}{15}=1512+5​

=1715\displaystyle =\frac{17}{15}=1517​
You can also rewrite this as  a mixed fraction12151\frac{2}{15}1152​.

See video for a picture explanation.

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Example 2
115−512=\displaystyle \frac{1}{15}-\frac{5}{12}=151​−125​=    - 21/60 
  

1×415×4−5×512×5\displaystyle \frac{1\orange{\times4}}{15\orange{\times4}}-\frac{5\orange{\times5}}{12\orange{\times5}}15×41×4​−12×55×5​

=460−2560\displaystyle =\frac{4}{60}-\frac{25}{60}=604​−6025​

=4−2560\displaystyle =\frac{4-25}{60}=604−25​

=−2160\displaystyle =-\frac{21}{60}=−6021​

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}}15×121×12​−12×155×15​

=12180−75180\displaystyle =\frac{12}{180}-\frac{75}{180}=18012​−18075​

=12−75180\displaystyle =\frac{12-75}{180}=18012−75​

=−63180\displaystyle =-\frac{63}{180}=−18063​ (÷\div÷ numerator and denominator by 3)

=−2160\displaystyle =-\frac{21}{60}=−6021​

Practice: Adding & Subtracting Fractions
Evaluate the following without using a calculator.

a) 47−25=\dfrac{4}{7}-\dfrac{2}{5}=74​−52​=

b) 112+58=\dfrac{1}{12}+\dfrac{5}{8}=121​+85​=

c) −710−(−35)=-\dfrac{7}{10}-\left(-\dfrac{3}{5}\right)=−107​−(−53​)=

I don't know

Practice: Adding & Subtracting Fractions
Evaluate 56+(−14)−(−13)\dfrac{5}{6}+\left(-\dfrac{1}{4}\right)-\left(-\dfrac{1}{3}\right)65​+(−41​)−(−31​).

Answer

I don't know

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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)=−53​×(−47​)=   21/20  

−35×(−74)\bcf-\dfrac{3}{5}\times\left(\bcf-\dfrac{7}{4}\right)−53​×(−47​)

=+3×75×4=\bcf+\dfrac{3\times7}{5\times4}=+5×43×7​

=+2120=\bcf+\dfrac{21}{20}=+2021​

We can rewrite this as a mixed fraction 11201 \dfrac{1}{20}1201​.

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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)=53​÷(−47​)=   -12/35 
 

+35÷(−74)\bcf+\dfrac{3}{5}\div\left(\bcf-\dfrac{7}{4}\right)+53​÷(−47​)

*We use our "flip & multiply" trick:

+35×(−47)\bcf+\dfrac{3}{5}\bcth\times\left(\bcf-\bcth{\dfrac{4}{7}}\right)+53​×(−74​)

=−3×45×7=\bcf-\dfrac{3\times4}{5\times7}=−5×73×4​

=−1235=\bcf-\dfrac{12}{35}=−3512​

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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}64​×1527​

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}}=6​  54​  9​×15  327  2​

=23×95=\dfrac{2}{3}\times\dfrac{9}{5}=32​×59​
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}=3​  12​×59​  3​

=21×35=\dfrac{2}{1}\times\dfrac{3}{5}=12​×53​

=2×31×5=\dfrac{2\times3}{1\times5}=1×52×3​

=65=\boxed{\dfrac{6}{5}}=56​​

Practice: Multiplying & Dividing Fractions
Evaluate the following without using a calculator.

a) 34×67=\dfrac{3}{4}\times\dfrac{6}{7}=43​×76​=

b) 37÷12=\dfrac{3}{7}\div\dfrac{1}{2}=73​÷21​=

c) −27÷(−821)=-\dfrac{2}{7}\div\left(-\dfrac{8}{21}\right)=−72​÷(−218​)=

Enter your answers as improper fractions, reduce your answer to most simplified form.

I don't know

Practice: Multiplying & Dividing Fractions
Simplify the following:

a)  −1215×78÷(−215)-\dfrac{12}{15}\times\dfrac{7}{8}\div\left(-\dfrac{21}{5}\right)−1512​×87​÷(−521​)

b) 310×15\dfrac{3}{10}\times15103​×15

c) 310÷9\dfrac{3}{10}\div9103​÷9

a) Enter your answer as an improper fraction

b) Enter your answer as an improper fraction

c) Enter your answer as an improper fraction

I don't know

Practice: Baking with Fractions
Andrew is baking chocolate chip cookies. Here's the recipe for one batch of (very basic, kind of gross) cookies:

How many cups of total ingredients is needed in this recipe?