Unit 1 · Lesson 2 of 6
Fractions: simplify, multiply, divide
You'll be able to: simplify a fraction, and multiply and divide fractions (including negative ones) without a calculator.
Step 1 of 5Warm-up
Two quick questions from earlier lessons. Answer from memory first, then check.
\((-6)(-4) \div (-8)\)
Left to right. \((-6)(-4) = 24\) (same signs, positive). Then \(24 \div (-8) = -3\) (different signs, negative). Answer: \(-3\).
In the study plan, how many minutes is the break after each 25-minute focus block?
The plan is 25 minutes on, then a 5-minute break (with a longer break after a few blocks).
Step 2 of 5The idea
A fraction like \(\dfrac{3}{4}\) has a numerator on top and a denominator on the bottom. Today: simplify, multiply, and divide.
Simplify. Divide the top and bottom by their GCF, the biggest number that goes into both: \(\dfrac{12}{18} = \dfrac{12 \div 6}{18 \div 6} = \dfrac{2}{3}\).
Multiply. Top times top, bottom times bottom: \(\dfrac{a}{b} \cdot \dfrac{c}{d} = \dfrac{a \cdot c}{b \cdot d}\). Cancel first: any top and any bottom that share a factor can be divided by it before you multiply.
Divide. Keep the first fraction and multiply by the reciprocal (the flip) of the second: \(\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \cdot \dfrac{d}{c}\).
Negatives. A minus sign on the top, the bottom, or in front all mean the same thing: \(\dfrac{-2}{3} = \dfrac{2}{-3} = -\dfrac{2}{3}\). The sign rules from Lesson 2 decide the sign of the answer.
Cancelling first keeps the numbers small, so there is almost nothing left to simplify at the end.
Why this shows up on the placement test: fractions sit inside equations, slopes, and rational expressions, and answers are expected in lowest terms.
Step 3 of 5Worked examples
Example 1: \(\dfrac{4}{9} \cdot \dfrac{15}{8}\)
- The top 4 and the bottom 8 share a factor of 4: 4 becomes 1 and 8 becomes 2. (A top and a bottom can cancel, even in different fractions.)
- The top 15 and the bottom 9 share a factor of 3: 15 becomes 5 and 9 becomes 3.
- Multiply what is left: \(\dfrac{1 \cdot 5}{3 \cdot 2} = \dfrac{5}{6}\).
- Check the long way: \(\dfrac{4 \cdot 15}{9 \cdot 8} = \dfrac{60}{72}\), and dividing top and bottom by 12 gives \(\dfrac{5}{6}\). Same answer.
Example 2: \(-\dfrac{5}{6} \div \dfrac{10}{9}\)
- Keep the first fraction and flip the second: \(-\dfrac{5}{6} \cdot \dfrac{9}{10}\).
- Cancel 5 and 10 (both divide by 5): they become 1 and 2. Cancel 9 and 6 (both divide by 3): they become 3 and 2.
- Multiply: \(-\dfrac{1 \cdot 3}{2 \cdot 2} = -\dfrac{3}{4}\). (One negative and one positive, so the answer is negative.)
Step 4 of 5Practice
Type your answer as a fraction in lowest terms, like 3/4 or -5/6, and press Check. If you miss twice, open the worked solution. Then mark how it went.
Which fraction is \(\dfrac{36}{48}\) written in lowest terms?
- \(\dfrac{18}{24}\)
- \(\dfrac{9}{12}\)
- \(\dfrac{6}{8}\)
- \(\dfrac{3}{4}\)
The GCF of 36 and 48 is 12: \(\dfrac{36 \div 12}{48 \div 12} = \dfrac{3}{4}\). Answer: (d).
All four fractions are equal, but (a), (b), and (c) are not finished: 18 and 24 still share 6, 9 and 12 share 3, and 6 and 8 share 2.
\(\dfrac{2}{3} \cdot \dfrac{9}{10}\)
Cancel 2 and 10 (divide by 2): 1 and 5. Cancel 9 and 3 (divide by 3): 3 and 1. Multiply: \(\dfrac{1 \cdot 3}{1 \cdot 5} = \dfrac{3}{5}\).
\(\dfrac{7}{12} \cdot \dfrac{8}{21}\)
Cancel 7 and 21 (divide by 7): 1 and 3. Cancel 8 and 12 (divide by 4): 2 and 3. Multiply: \(\dfrac{1 \cdot 2}{3 \cdot 3} = \dfrac{2}{9}\).
\(-\dfrac{3}{4} \cdot \dfrac{10}{9}\)
Cancel 3 and 9 (divide by 3): 1 and 3. Cancel 10 and 4 (divide by 2): 5 and 2. Multiply: \(\dfrac{1 \cdot 5}{2 \cdot 3} = \dfrac{5}{6}\). One negative, so the answer is \(-\dfrac{5}{6}\).
\(\dfrac{5}{8} \div \dfrac{15}{4}\)
Flip the second: \(\dfrac{5}{8} \cdot \dfrac{4}{15}\). Cancel 5 and 15 (divide by 5): 1 and 3. Cancel 4 and 8 (divide by 4): 1 and 2. Multiply: \(\dfrac{1 \cdot 1}{2 \cdot 3} = \dfrac{1}{6}\).
\(\dfrac{-6}{7} \div \left(-\dfrac{9}{14}\right)\)
Flip the second: \(\dfrac{-6}{7} \cdot \left(-\dfrac{14}{9}\right)\). Two negatives, so the answer is positive. Cancel 6 and 9 (divide by 3): 2 and 3. Cancel 14 and 7 (divide by 7): 2 and 1. Multiply: \(\dfrac{2 \cdot 2}{1 \cdot 3} = \dfrac{4}{3}\).
One serving of rice uses \(\dfrac{3}{4}\) cup. How many servings can you make from 6 cups?
"How many \(\dfrac{3}{4}\)s fit in 6" means \(6 \div \dfrac{3}{4}\). Write 6 as \(\dfrac{6}{1}\) and flip: \(\dfrac{6}{1} \cdot \dfrac{4}{3}\). Cancel 6 and 3 (divide by 3): 2 and 1. So \(2 \cdot 4 = 8\) servings.
\(\dfrac{4}{15} \cdot \left(-\dfrac{5}{6}\right) \div \dfrac{2}{9}\)
Flip the last fraction: \(\dfrac{4}{15} \cdot \left(-\dfrac{5}{6}\right) \cdot \dfrac{9}{2}\). Exactly one negative, so the answer is negative.
Tops: 4, 5, 9. Bottoms: 15, 6, 2. Cancel 4 and 2: 2 and 1. Cancel 5 and 15: 1 and 3. Cancel 9 and 6: 3 and 2.
Now \(\dfrac{2 \cdot 1 \cdot 3}{3 \cdot 2 \cdot 1} = \dfrac{6}{6} = 1\), so the answer is \(-1\).
Step 5 of 5Wrap-up
You can now simplify a fraction with the GCF, multiply fractions by cancelling first, and divide by flipping the second fraction.
Read or watch next: OpenStax Elementary Algebra 2e, 1.5 Visualize Fractions (free textbook, with more practice at the end of the section).
Also good: Khan Academy: Fractions (short videos). Quick rules: Number skills cheat sheet.
Something unclear? Ask your teacher bot a follow-up question, for example: "Why am I allowed to cancel a top with the bottom of a different fraction?" It knows this course and can explain things another way.