How to Multiply and Divide Fractions

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Fractions describe equal parts of a whole. The top number is the numerator, and the bottom number is the denominator. When you multiply or divide fractions, you are working with parts of parts, groups of parts, or sharing one fractional amount by another.

The good news: multiplying and dividing fractions follows a very dependable pattern. Multiplication goes straight across. Division uses the reciprocal of the second fraction, then becomes multiplication.

Types of fractions:
Proper Fractions: The numerator is less than the denominator, such as \({3 \over 4}\) or \({1 \over 3}\).
Improper Fractions: The numerator is greater than or equal to the denominator, such as \({7 \over 4}\) or \({5 \over 2}\).
Mixed Numbers: A whole number and a fraction written together, such as \(1{1 \over 3}\) or \(2{3 \over 7}\).

Multiplication of Fractions

To multiply fractions, multiply the numerators and multiply the denominators:

\({a \over b}\times {c \over d}={a\times c \over b\times d}={ac \over bd}\)

You may simplify before multiplying by cross-canceling common factors, or you may multiply first and simplify at the end. Cross-canceling often keeps the numbers smaller.

Example: \({3 \over 4}\times {8 \over 9}\). Cross-cancel \(3\) with \(9\), and \(8\) with \(4\): \({3 \over 4}\times {8 \over 9}={1 \over 1}\times {2 \over 3}={2 \over 3}\).

Division of Fractions

To divide by a fraction, keep the first fraction, change division to multiplication, and flip the second fraction. The flipped fraction is called the reciprocal.

\({a \over b}\div {c \over d}={a \over b}\times {d \over c}={ad \over bc}\)

Example: \({5 \over 6}\div {10 \over 9}={5 \over 6}\times {9 \over 10}={45 \over 60}={3 \over 4}\).

Finally, always reduce your answer to simplest form when possible. Learn more

Multiplying and Dividing Fractions

Think of this lesson as more than a rule to memorize. Multiplying and Dividing Fractions is about number sense, equivalent forms, and careful arithmetic. A strong student does not rush to the first formula on the page; they pause, identify the structure of the problem, and then choose the tool that matches that structure. That pause is what prevents most mistakes.

Fractions compare a part to a whole. Keep track of the numerator, denominator, and whether the pieces are the same size before adding, subtracting, or simplifying.

Here is the teacher way to approach the topic. First, read the problem slowly and underline the information that is actually given. Next, name the target: are you finding a value, simplifying an expression, comparing two quantities, solving for a variable, or interpreting a graph? Once the target is clear, the calculation becomes much less mysterious because every step has a job.

Convert mixed numbers if needed.
For division, multiply by the reciprocal.
Cancel common factors before multiplying when possible.
Simplify the final fraction.

A helpful habit is to say what each number represents before using it. For example, if a number is a denominator, a radius, a slope, a common difference, or a coefficient, it should not be treated like an ordinary loose number. Its role tells you where it belongs in the formula. This is especially important on ACT-style questions because many wrong answer choices come from using the right numbers in the wrong places.

Another good habit is to keep the work organized vertically. Write one clean line for substitution, one line for simplifying, and one line for the final answer. If the problem has units, keep the units attached. If the problem has signs, exponents, or parentheses, copy them carefully from one line to the next. Most errors in this topic are not caused by a hard idea; they are caused by dropping a negative sign, combining unlike terms, using the wrong denominator, or skipping a check.

When you finish, ask a quick reasonableness question. Should the answer be positive or negative? Should it be larger or smaller than the original number? Does it fit the graph, table, shape, or equation? Can you plug it back into the original problem? This final check turns the lesson from a procedure into understanding.

On a test, the goal is not to write the longest solution. The goal is to write enough clear work that you can see the structure, avoid traps, and recover quickly if one line goes wrong. Practice the examples below with that mindset: identify the type of problem, choose the matching rule, show the substitution, simplify carefully, and check the answer before moving on.

Free printable Worksheets

Exercises for Multiplying and Dividing Fractions

1) \({2 \over 5}\times {3 \over 7}=\)

2) \({4 \over 9}\times {3 \over 8}=\)

3) \({5 \over 6}\times {9 \over 10}=\)

4) \({7 \over 12}\times {8 \over 21}=\)

5) \({11 \over 15}\times {25 \over 22}=\)

6) \({3 \over 4}\div {2 \over 5}=\)

7) \({5 \over 8}\div {15 \over 16}=\)

8) \({9 \over 10}\div {3 \over 5}=\)

9) \({14 \over 27}\div {7 \over 9}=\)

10) \({6 \over 11}\div {18 \over 55}=\)

11) \(2{1 \over 3}\times {3 \over 7}=\)

12) \(1{2 \over 5}\div {7 \over 10}=\)

13) A recipe uses \({3 \over 4}\) cup of sugar for one batch. How much sugar is needed for \({2 \over 3}\) of a batch?

14) A board is \({5 \over 6}\) yard long. Each small piece is \({1 \over 12}\) yard. How many small pieces can be cut?

15) Simplify \({18 \over 35}\times {14 \over 27}\).

16) Simplify \({16 \over 45}\div {8 \over 15}\).

17) Find \(x\): \({3 \over 5}x={9 \over 20}\).

18) Which expression is larger: \({4 \over 7}\times {21 \over 8}\) or \({5 \over 6}\div {10 \over 9}\)?

19) A hiker walks \(3{1 \over 2}\) miles each hour. How far does the hiker walk in \(2{2 \over 3}\) hours?

20) A tank holds \(7{1 \over 2}\) gallons. If each bottle holds \({3 \over 4}\) gallon, how many bottles can be filled?

Answers

1) Multiply across: \({2 \over 5}\times {3 \over 7}={2\times 3 \over 5\times 7}={6 \over 35}\). Since \(6\) and \(35\) have no common factor except \(1\), it is simplest. Answer: \(\color{red}{{6 \over 35}}\)

2) Multiply: \({4 \over 9}\times {3 \over 8}={4\times 3 \over 9\times 8}={12 \over 72}\). Simplify using GCF(12,72) = 12: \({12 \over 72}={1 \over 6}\). Answer: \(\color{red}{{1 \over 6}}\)

3) Multiply: \({5 \over 6}\times {9 \over 10}={45 \over 60}\). Simplify using GCF(45,60) = 15: \({45 \over 60}={3 \over 4}\). Answer: \(\color{red}{{3 \over 4}}\)

4) Multiply: \({7 \over 12}\times {8 \over 21}={56 \over 252}\). Simplify using GCF(56,252) = 28: \({56 \over 252}={2 \over 9}\). Answer: \(\color{red}{{2 \over 9}}\)

5) Multiply: \({11 \over 15}\times {25 \over 22}={275 \over 330}\). Simplify using GCF(275,330) = 55: \({275 \over 330}={5 \over 6}\). Answer: \(\color{red}{{5 \over 6}}\)

6) Keep, change, flip: \({3 \over 4}\div {2 \over 5}={3 \over 4}\times {5 \over 2}={15 \over 8}\). Answer: \(\color{red}{{15 \over 8}}\)

7) Keep, change, flip: \({5 \over 8}\div {15 \over 16}={5 \over 8}\times {16 \over 15}={80 \over 120}\). Simplify using GCF(80,120) = 40: \({80 \over 120}={2 \over 3}\). Answer: \(\color{red}{{2 \over 3}}\)

8) Divide by multiplying by the reciprocal: \({9 \over 10}\div {3 \over 5}={9 \over 10}\times {5 \over 3}={45 \over 30}\). Simplify using GCF(45,30) = 15: \({45 \over 30}={3 \over 2}\). Answer: \(\color{red}{{3 \over 2}}\)

9) Keep, change, flip: \({14 \over 27}\div {7 \over 9}={14 \over 27}\times {9 \over 7}={126 \over 189}\). Simplify using GCF(126,189) = 63: \({126 \over 189}={2 \over 3}\). Answer: \(\color{red}{{2 \over 3}}\)

10) Keep, change, flip: \({6 \over 11}\div {18 \over 55}={6 \over 11}\times {55 \over 18}={330 \over 198}\). Simplify using GCF(330,198) = 66: \({330 \over 198}={5 \over 3}\). Answer: \(\color{red}{{5 \over 3}}\)

11) Convert the mixed number: \(2{1 \over 3}={7 \over 3}\). Then \({7 \over 3}\times {3 \over 7}={21 \over 21}=1\). Answer: \(\color{red}{1}\)

12) Convert \(1{2 \over 5}\) to \({7 \over 5}\). Then divide: \({7 \over 5}\div {7 \over 10}={7 \over 5}\times {10 \over 7}={70 \over 35}=2\). Answer: \(\color{red}{2}\)

13) This asks for \({2 \over 3}\) of \({3 \over 4}\), so multiply: \({3 \over 4}\times {2 \over 3}={6 \over 12}\). Simplify using GCF(6,12) = 6: \({6 \over 12}={1 \over 2}\). Answer: \(\color{red}{{1 \over 2}\text{ cup}}\)

14) Divide the total length by the piece length: \({5 \over 6}\div {1 \over 12}={5 \over 6}\times {12 \over 1}={60 \over 6}=10\). Answer: \(\color{red}{10\text{ pieces}}\)

15) Multiply: \({18 \over 35}\times {14 \over 27}={252 \over 945}\). Simplify using GCF(252,945) = 63: \({252 \over 945}={4 \over 15}\). Answer: \(\color{red}{{4 \over 15}}\)

16) Keep, change, flip: \({16 \over 45}\div {8 \over 15}={16 \over 45}\times {15 \over 8}={240 \over 360}\). Simplify using GCF(240,360) = 120: \({240 \over 360}={2 \over 3}\). Answer: \(\color{red}{{2 \over 3}}\)

17) Solve \({3 \over 5}x={9 \over 20}\) by dividing by \({3 \over 5}\): \(x={9 \over 20}\div {3 \over 5}={9 \over 20}\times {5 \over 3}={45 \over 60}\). Simplify using GCF(45,60) = 15: \(x={3 \over 4}\). Answer: \(\color{red}{{3 \over 4}}\)

18) First expression: \({4 \over 7}\times {21 \over 8}={84 \over 56}={3 \over 2}\). Second expression: \({5 \over 6}\div {10 \over 9}={5 \over 6}\times {9 \over 10}={45 \over 60}={3 \over 4}\). Since \({3 \over 2}>{3 \over 4}\), the first expression is larger. Answer: \(\color{red}{{4 \over 7}\times {21 \over 8}}\)

19) Convert both mixed numbers: \(3{1 \over 2}={7 \over 2}\) and \(2{2 \over 3}={8 \over 3}\). Multiply: \({7 \over 2}\times {8 \over 3}={56 \over 6}\). Simplify using GCF(56,6) = 2: \({56 \over 6}={28 \over 3}=9{1 \over 3}\). Answer: \(\color{red}{9{1 \over 3}\text{ miles}}\)

20) Divide total gallons by gallons per bottle. Convert \(7{1 \over 2}={15 \over 2}\). Then \({15 \over 2}\div {3 \over 4}={15 \over 2}\times {4 \over 3}={60 \over 6}=10\). Answer: \(\color{red}{10\text{ bottles}}\)

How to Multiply and Divide Fractions