Fractions

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Quick recall

• Use common denominators for addition or subtraction.
• For division, invert the second fraction; denominators must be nonzero.

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Formula Cards

1. Same denominator

$$\frac{a}{b}+\frac{c}{b}=\frac{a+c}{b}$$

For subtraction, subtract numerators.

2. Different denominators

$$\frac{1}{4}+\frac{1}{6}=\frac{3}{12}+\frac{2}{12}=\frac{5}{12}$$

Use the LCM of the denominators.

3. Multiply

$$\frac{a}{b}\times\frac{c}{d}=\frac{ac}{bd}$$

4. Divide

$$\frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\times\frac{d}{c}$$

The divisor must be nonzero.

5. Compare

$$\frac{a}{b}>\frac{c}{d}\iff ad>bc$$

Cross-multiplication comparison assumes positive denominators.

6. Mixed number

$$a\frac{b}{c}=\frac{ac+b}{c}$$

For a nonnegative mixed number; handle a leading negative sign separately.

Worked Examples

Example 1

Question: Add three-eighths and one-quarter.

Step 1: Use denominator 8.

$$\frac{1}{4}=\frac{2}{8}$$

Step 2: Add the numerators.

$$\frac{3}{8}+\frac{2}{8}=\frac{5}{8}$$

Answer: 5/8.

Example 2

Question: Which is larger: 3/5 or 5/8?

Step 1: Cross-multiply positive denominators.

$$3\times8=24,\qquad5\times5=25$$

Step 2: Compare the products.

$$24<25\quad\Rightarrow\quad\frac{3}{5}<\frac{5}{8}$$

Answer: 5/8 is larger.

Example 3

Question: Convert the mixed number 2¾ to an improper fraction.

Step 1: Multiply the whole number by the denominator and add the numerator.

$$2\times4+3=11$$

Step 2: Keep the denominator.

$$2\frac{3}{4}=\frac{11}{4}$$

Answer: 11/4.

Drill

Try each question before opening its answer.

Question 1

$\dfrac{1}{3}+\dfrac{1}{6}$.

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  1/2

Question 2

$\dfrac{5}{6}-\dfrac{1}{3}$.

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  1/2

Question 3

$\dfrac{3}{4}\times\dfrac{2}{9}$.

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  1/6

Question 4

$\dfrac{7}{8}\div\dfrac{1}{4}$.

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  3.5

Question 5

Which is bigger: 4/7 or 5/9?

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  4/7

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