maths.free › Arithmetic › 4. Fractions › Multiply and Divide Mixed Numbers and Complex Fractions

Multiply and Divide Mixed Numbers and Complex Fractions

Multiply and divide mixed numbers

Multiply and Divide Mixed Numbers

In the previous section, you learned how to multiply and divide fractions. All of the examples there used either proper or improper fractions. What happens when you are asked to multiply or divide mixed numbers? Remember that we can convert a mixed number to an improper fraction. And you learned how to do that in Visualize Fractions.

Example

Try it.

Multiply: \(3\frac{1}{3}\cdot \frac{5}{8}\)

Solution

	\(3\frac{1}{3}\cdot \frac{5}{8}\)
Convert \(3\frac{1}{3}\) to an improper fraction.	\(\frac{10}{3}\cdot \frac{5}{8}\)
Multiply.	\(\frac{10\cdot 5}{3\cdot 8}\)
Look for common factors.	\(\frac{2̸\cdot 5\cdot 5}{3\cdot 2̸\cdot 4}\)
Remove common factors.	\(\frac{5\cdot 5}{3\cdot 4}\)
Simplify.	\(\frac{25}{12}\)

Notice that we left the answer as an improper fraction, \(\frac{25}{12},\) and did not convert it to a mixed number. In algebra, it is preferable to write answers as improper fractions instead of mixed numbers. This avoids any possible confusion between \(2\frac{1}{12}\) and \(2\cdot \frac{1}{12}.\)

Example

Try it.

Multiply, and write your answer in simplified form: \(2\frac{4}{5}\ (-1\frac{7}{8}).\)

Solution

	\(2\frac{4}{5}\ (-1\frac{7}{8})\)
Convert mixed numbers to improper fractions.	\(\frac{14}{5}\ (-\frac{15}{8})\)
Multiply.	\(-\ \frac{14\cdot 15}{5\cdot 8}\)
Look for common factors.	\(-\ \frac{2̸\cdot 7\cdot 5̸\cdot 3}{5̸\cdot 2̸\cdot 4}\)
Remove common factors.	\(-\ \frac{\ 7\cdot 3}{4}\)
Simplify.	\(-\ \frac{\ 21}{4}\)

Example

Try it.

Divide, and write your answer in simplified form: \(3\frac{4}{7}\ \div \ 5.\)

Solution

	\(3\frac{4}{7}\ \div \ 5\)
Convert mixed numbers to improper fractions.	\(\frac{25}{7}\ \div \ \frac{5}{1}\)
Multiply the first fraction by the reciprocal of the second.	\(\frac{25}{7}\cdot \frac{1}{5}\)
Multiply.	\(\frac{25\cdot 1}{7\cdot 5}\)
Look for common factors.	\(\frac{5̸\cdot 5\cdot 1}{7\cdot 5̸}\)
Remove common factors.	\(\frac{5\cdot 1}{7}\)
Simplify.	\(\frac{5}{7}\)

Condensed — the full section is in OpenStax Prealgebra 2e.

Translate Phrases to Expressions with Fractions

The words quotient and ratio are often used to describe fractions. In Subtract Whole Numbers, we defined quotient as the result of division. The quotient of \(a\) and \(b\) is the result you get from dividing \(a\) by \(b,\) or \(\frac{a}{b}.\) Let’s practice translating some phrases into algebraic expressions using these terms.

Example

Try it.

Translate the phrase into an algebraic expression: “the quotient of \(3x\) and \(8.”\)

Solution

The keyword is quotient; it tells us that the operation is division. Look for the words of and and to find the numbers to divide.

\[\text{The quotient}\ \text{of}\ 3x\ \text{and}\ 8.\]

This tells us that we need to divide \(3x\) by \(8.\) \(\frac{3x}{8}\)

Example

Try it.

Translate the phrase into an algebraic expression: the quotient of the difference of \(m\) and \(n,\) and \(p.\)

Solution

We are looking for the quotient of the difference of \(m\) and \(n\), and \(p.\) This means we want to divide the difference of \(m\) and \(n\) by \(p.\)

\[\frac{m-n}{p}\]

Simplify Complex Fractions

Our work with fractions so far has included proper fractions, improper fractions, and mixed numbers. Another kind of fraction is called complex fraction, which is a fraction in which the numerator or the denominator contains a fraction.

Some examples of complex fractions are:

\[\frac{\ \frac{6}{7}\ }{\ 3\ }\ \frac{\ \frac{3}{4}\ }{\ \frac{5}{8}\ }\ \frac{\ \frac{x}{2}\ }{\ \frac{5}{6}\ }\]

To simplify a complex fraction, remember that the fraction bar means division. So the complex fraction \(\frac{\ \frac{3}{4}\ }{\ \frac{5}{8}\ }\) can be written as \(\frac{3}{4}\div \frac{5}{8}.\)

Example

Try it.

Simplify: \(\frac{\ \frac{3}{4}\ }{\ \frac{5}{8}\ }.\)

Solution

	\(\frac{\ \frac{3}{4}\ }{\ \frac{5}{8}\ }\)
Rewrite as division.	\(\frac{3}{4}\div \frac{5}{8}\)
Multiply the first fraction by the reciprocal of the second.	\(\frac{3}{4}\cdot \frac{8}{5}\)
Multiply.	\(\frac{3\cdot 8}{4\cdot 5}\)
Look for common factors.	\(\frac{3\cdot 4̸\cdot 2}{4̸\cdot 5}\)
Remove common factors and simplify.	\(\frac{6}{5}\)

Example

Try it.

Simplify: \(\frac{-\frac{6}{7}}{3}.\)

Solution

	\(\frac{-\frac{6}{7}}{3}\)
Rewrite as division.	\(-\frac{6}{7}\div 3\)
Multiply the first fraction by the reciprocal of the second.	\(-\frac{6}{7}\cdot \frac{1}{3}\)
Multiply; the product will be negative.	\(-\frac{6\cdot 1}{7\cdot 3}\)
Look for common factors.	\(-\frac{3̸\cdot 2\cdot 1}{7\cdot 3̸}\)
Remove common factors and simplify.	\(-\frac{2}{7}\)

Example

Try it.

Simplify: \(\frac{\ \frac{x}{2}\ }{\ \frac{xy}{6}\ }.\)

Solution

	\(\frac{\ \frac{x}{2}\ }{\ \frac{xy}{6}\ }\)
Rewrite as division.	\(\frac{x}{2}\div \frac{xy}{6}\)
Multiply the first fraction by the reciprocal of the second.	\(\frac{x}{2}\cdot \frac{6}{xy}\)
Multiply.	\(\frac{x\cdot 6}{2\cdot xy}\)
Look for common factors.	\(\frac{x̸\cdot 3\cdot 2̸}{2̸\cdot x̸\cdot y}\)
Remove common factors and simplify.	\(\frac{3}{y}\)

Condensed — the full section is in OpenStax Prealgebra 2e.

Simplify Expressions with a Fraction Bar

Where does the negative sign go in a fraction? Usually, the negative sign is placed in front of the fraction, but you will sometimes see a fraction with a negative numerator or denominator. Remember that fractions represent division. The fraction \(-\frac{1}{3}\) could be the result of dividing \(\frac{-1}{3},\) a negative by a positive, or of dividing \(\frac{1}{-3},\) a positive by a negative. When the numerator and denominator have different signs, the quotient is negative.

If both the numerator and denominator are negative, then the fraction itself is positive because we are dividing a negative by a negative.

\[\frac{-1}{-3}=\frac{1}{3}\ \frac{\text{negative}}{\text{negative}}=\text{positive}\]

Example

Try it.

Which of the following fractions are equivalent to \(\frac{7}{-8}?\)

\[\ \frac{-7}{-8},\frac{-7}{8},\frac{7}{8},-\frac{7}{8}\]

Solution

The quotient of a positive and a negative is a negative, so \(\frac{7}{-8}\) is negative. Of the fractions listed, \(\ \frac{-7}{8}\) and \(-\frac{7}{8}\) are also negative.

Fraction bars act as grouping symbols. The expressions above and below the fraction bar should be treated as if they were in parentheses. For example, \(\frac{4+8}{5-3}\) means \((4+8)\div (5-3).\) The order of operations tells us to simplify the numerator and the denominator first—as if there were parentheses—before we divide.

We’ll add fraction bars to our set of grouping symbols from Use the Language of Algebra to have a more complete set here.

Example

Try it.

Simplify: \(\frac{4+8}{5-3}.\)

Solution

	\(\frac{4+8}{5-3}\)
Simplify the expression in the numerator.	\(\frac{12}{5-3}\)
Simplify the expression in the denominator.	\(\frac{12}{2}\)
Simplify the fraction.	6

Example

Try it.

Simplify: \(\frac{4-2(3)}{{2}^{2}+2}.\)

Solution

	\(\frac{4-2(3)}{{2}^{2}+2}\)
Use the order of operations. Multiply in the numerator and use the exponent in the denominator.	\(\frac{4-6}{4+2}\)
Simplify the numerator and the denominator.	\(\frac{-2}{6}\)
Simplify the fraction.	\(-\frac{1}{3}\)

Condensed — the full section is in OpenStax Prealgebra 2e.

Key Concepts

Multiply or divide mixed numbers.
1. Convert the mixed numbers to improper fractions.
2. Follow the rules for fraction multiplication or division.
3. Simplify if possible.
Simplify a complex fraction.
1. Rewrite the complex fraction as a division problem.
2. Follow the rules for dividing fractions.
3. Simplify if possible.
Placement of negative sign in a fraction.
- For any positive numbers \(a\) and \(b\), \(\frac{-a}{b}=\frac{a}{-b}=-\frac{a}{b}\).
Simplify an expression with a fraction bar.
1. Simplify the numerator.
2. Simplify the denominator.
3. Simplify the fraction.

Multiply and Divide Mixed Numbers and Complex Fractions

Multiply and Divide Mixed Numbers

In the following exercises, multiply and write the answer in simplified form.

Try it.

\(4\frac{3}{8}\cdot \frac{7}{10}\)

Try it.

\(2\frac{4}{9}\cdot \frac{6}{7}\)

Solution

\(\frac{44}{21}\)

Try it.

\(\frac{15}{22}\cdot 3\frac{3}{5}\)

Try it.

\(\frac{25}{36}\cdot 6\frac{3}{10}\)

Solution

\(\frac{35}{8}\)

Try it.

\(4\frac{2}{3}\ (-1\frac{1}{8})\)

Try it.

\(2\frac{2}{5}\ (-2\frac{2}{9})\)

Solution

\(-\frac{16}{3}\)

Try it.

\(-4\frac{4}{9}\cdot 5\frac{13}{16}\)

Try it.

\(-1\frac{7}{20}\cdot 2\frac{11}{12}\)

Solution

\(-\frac{63}{16}\)

In the following exercises, divide, and write your answer in simplified form.

Try it.

\(5\frac{1}{3}\div \ 4\)

Try it.

\(13\frac{1}{2}\div \ 9\)

Solution

\(\frac{3}{2}\)

Try it.

\(-12\div \ 3\frac{3}{11}\)

Try it.

\(-7\div \ 5\frac{1}{4}\)

Solution

\(-\frac{4}{3}\)

Try it.

\(6\frac{3}{8}\div \ 2\frac{1}{8}\)

Try it.

\(2\frac{1}{5}\div \ 1\frac{1}{10}\)

Solution

Try it.

\(-9\frac{3}{5}\div \ (-1\frac{3}{5})\)

Try it.

\(-18\frac{3}{4}\div \ (-3\frac{3}{4})\)

Solution

Translate Phrases to Expressions with Fractions

In the following exercises, translate each English phrase into an algebraic expression.

Try it.

the quotient of \(5u\) and \(11\)

Try it.

the quotient of \(7v\) and \(13\)

Solution

\(\frac{7v}{13}\)

Try it.

the quotient of \(p\) and \(q\)

Try it.

the quotient of \(a\) and \(b\)

Solution

\(\frac{a}{b}\)

Try it.

the quotient of \(r\) and the sum of \(s\) and \(10\)

Try it.

the quotient of \(A\) and the difference of \(3\) and \(B\)

Solution

\(\frac{A}{3-B}\)

Simplify Complex Fractions

In the following exercises, simplify the complex fraction.

Try it.

\(\frac{\ \frac{2}{3}\ }{\ \frac{8}{9}\ }\)

Try it.

\(\frac{\ \frac{4}{5}\ }{\ \frac{8}{15}\ }\)

Solution

\(\frac{3}{2}\)

Try it.

\(\frac{-\frac{8}{21}}{\frac{12}{35}}\)

Try it.

\(\frac{-\frac{9}{16}}{\frac{33}{40}}\)

Solution

\(-\frac{15}{22}\)

Try it.

\(\frac{-\frac{4}{5}}{2}\)

Try it.

\(\frac{-\frac{9}{10}}{3}\)

Solution

\(-\frac{3}{10}\)

Try it.

\(\frac{\ \frac{2}{5}\ }{\ 8\ }\)

Try it.

\(\frac{\ \frac{5}{3}\ }{\ 10\ }\)

Solution

\(\frac{1}{6}\)

Try it.

\(\frac{\ \frac{m}{3}\ }{\ \frac{n}{2}\ }\)

Try it.

\(\frac{\ \frac{r}{5}\ }{\ \frac{s}{3}\ }\)

Solution

\(\frac{3r}{5s}\)

Try it.

\(\frac{-\frac{x}{6}}{-\frac{8}{9}}\)

Try it.

\(\frac{-\frac{3}{8}}{-\frac{y}{12}}\)

Solution

\(\frac{9}{2y}\)

Try it.

\(\frac{2\frac{4}{5}}{\frac{1}{10}}\)

Try it.

\(\frac{4\frac{2}{3}}{\frac{1}{6}}\)

Solution

Try it.

\(\frac{\frac{7}{9}}{-2\frac{4}{5}}\)

Try it.

\(\frac{\frac{3}{8}}{-6\frac{3}{4}}\)

Solution

\(-\frac{1}{18}\)

Simplify Expressions with a Fraction Bar

In the following exercises, identify the equivalent fractions.

Try it.

Which of the following fractions are equivalent to \(\frac{5}{-11}?\)
\(\ \frac{-5}{-11},\frac{-5}{11},\frac{5}{11},-\frac{5}{11}\)

Try it.

Which of the following fractions are equivalent to \(\frac{-4}{9}?\)
\(\ \frac{-4}{-9},\frac{-4}{9},\frac{4}{9},-\frac{4}{9}\)

Solution

\(\ \frac{-4}{9},\ -\frac{4}{9}\)

Try it.

Which of the following fractions are equivalent to \(-\frac{11}{3}?\)
\(\ \frac{-11}{3},\frac{11}{3},\frac{-11}{-3},\ \frac{11}{-3}\)

Try it.

Which of the following fractions are equivalent to \(-\frac{13}{6}?\)
\(\ \frac{13}{6},\frac{13}{-6},\frac{-13}{-6},\frac{-13}{6}\\)

Solution

\(\ \frac{13}{-6},\frac{-13}{6}\\)

In the following exercises, simplify.

Try it.

\(\frac{4+11}{8}\)

Try it.

\(\frac{9+3}{7}\)

Solution

\(\frac{12}{7}\)

Try it.

\(\frac{22+3}{10}\)

Try it.

\(\frac{19-4}{6}\)

Solution

\(\frac{5}{2}\)

Try it.

\(\frac{48}{24-15}\)

Try it.

\(\frac{46}{4+4}\)

Solution

\(\frac{23}{4}\)

Try it.

\(\frac{-6+6}{8+4}\)

Try it.

\(\frac{-6+3}{17-8}\)

Solution

\(-\frac{1}{3}\)

Try it.

\(\frac{22-14}{19-13}\)

Try it.

\(\frac{15+9}{18+12}\)

Solution

\(\frac{4}{5}\)

Try it.

\(\frac{5⋅8}{-10}\)

Try it.

\(\frac{3⋅4}{-24}\)

Solution

\(-\frac{1}{2}\)

Try it.

\(\frac{4⋅3}{6⋅6}\)

Try it.

\(\frac{6⋅6}{9⋅2}\)

Solution

Try it.

\(\frac{{4}^{2}-1}{25}\)

Try it.

\(\frac{{7}^{2}+1}{60}\)

Solution

\(\frac{5}{6}\)

Try it.

\(\frac{8⋅3+2⋅9}{14+3}\)

Try it.

\(\frac{9⋅6-4⋅7}{22+3}\)

Solution

\(\frac{26}{25}\)

Try it.

\(\frac{15⋅5-{5}^{2}}{2⋅10}\)

Try it.

\(\frac{12⋅9-{3}^{2}}{3⋅18}\)

Solution

\(\frac{11}{6}\)

Try it.

\(\frac{5⋅6-3⋅4}{4⋅5-2⋅3}\)

Try it.

\(\frac{8⋅9-7⋅6}{5⋅6-9⋅2}\)

Solution

\(\frac{5}{2}\)

Try it.

\(\frac{{5}^{2}-{3}^{2}}{3-5}\)

Try it.

\(\frac{{6}^{2}-{4}^{2}}{4-6}\)

Solution

−10

Try it.

\(\frac{2+4(3)}{-3-{2}^{2}}\)

Try it.

\(\frac{7+3(5)}{-2-{3}^{2}}\)

Solution

−2

Try it.

\(\frac{7⋅4-2(8-5)}{9⋅3-3⋅5}\)

Try it.

\(\frac{9⋅7-3(12-8)}{8⋅7-6⋅6}\)

Solution

\(\frac{51}{20}\)

Try it.

\(\frac{9(8-2)-3(15-7)}{6(7-1)-3(17-9)}\)

Try it.

\(\frac{8(9-2)-4(14-9)}{7(8-3)-3(16-9)}\)

Solution

\(\frac{18}{7}\)

Condensed — the full section is in OpenStax Prealgebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Divide and reduce, if possible: \((4+5)\div (10-7).\)
If you missed this problem, review .

Revelează răspunsul

\(3\)
Multiply and write the answer in simplified form: \(\frac{1}{8}\cdot \frac{2}{3}\).
If you missed this problem, review .

Revelează răspunsul

\(\frac{1}{12}\)
Convert \(2\frac{3}{5}\) into an improper fraction.
If you missed this problem, review .

Revelează răspunsul

\(\frac{13}{5}\)

Multiply: \(3\frac{1}{3}\cdot \frac{5}{8}\)

Revelează răspunsul

	\(3\frac{1}{3}\cdot \frac{5}{8}\)
Convert \(3\frac{1}{3}\) to an improper fraction.	\(\frac{10}{3}\cdot \frac{5}{8}\)
Multiply.	\(\frac{10\cdot 5}{3\cdot 8}\)
Look for common factors.	\(\frac{2̸\cdot 5\cdot 5}{3\cdot 2̸\cdot 4}\)
Remove common factors.	\(\frac{5\cdot 5}{3\cdot 4}\)
Simplify.	\(\frac{25}{12}\)

Multiply, and write your answer in simplified form: \(5\frac{2}{3}\cdot \frac{6}{17}.\)

Revelează răspunsul

2
Multiply, and write your answer in simplified form: \(\frac{3}{7}\cdot 5\frac{1}{4}.\)

Revelează răspunsul

\(\frac{9}{4}\)

Multiply, and write your answer in simplified form: \(2\frac{4}{5}\ (-1\frac{7}{8}).\)

Revelează răspunsul

	\(2\frac{4}{5}\ (-1\frac{7}{8})\)
Convert mixed numbers to improper fractions.	\(\frac{14}{5}\ (-\frac{15}{8})\)
Multiply.	\(-\ \frac{14\cdot 15}{5\cdot 8}\)
Look for common factors.	\(-\ \frac{2̸\cdot 7\cdot 5̸\cdot 3}{5̸\cdot 2̸\cdot 4}\)
Remove common factors.	\(-\ \frac{\ 7\cdot 3}{4}\)
Simplify.	\(-\ \frac{\ 21}{4}\)

Multiply, and write your answer in simplified form. \(5\frac{5}{7}\ (-2\frac{5}{8}).\)

Revelează răspunsul

−15
Multiply, and write your answer in simplified form. \(-3\frac{2}{5}\cdot 4\frac{1}{6}.\)

Revelează răspunsul

\(-\frac{85}{6}\)

Divide, and write your answer in simplified form: \(3\frac{4}{7}\ \div \ 5.\)

Revelează răspunsul

	\(3\frac{4}{7}\ \div \ 5\)
Convert mixed numbers to improper fractions.	\(\frac{25}{7}\ \div \ \frac{5}{1}\)
Multiply the first fraction by the reciprocal of the second.	\(\frac{25}{7}\cdot \frac{1}{5}\)
Multiply.	\(\frac{25\cdot 1}{7\cdot 5}\)
Look for common factors.	\(\frac{5̸\cdot 5\cdot 1}{7\cdot 5̸}\)
Remove common factors.	\(\frac{5\cdot 1}{7}\)
Simplify.	\(\frac{5}{7}\)

Divide, and write your answer in simplified form: \(4\frac{3}{8}\div 7.\)

Revelează răspunsul

\(\frac{5}{8}\)
Divide, and write your answer in simplified form: \(2\frac{5}{8}\div 3.\)

Revelează răspunsul

\(\frac{7}{8}\)

Divide: \(2\frac{1}{2}\div 1\frac{1}{4}.\)

Revelează răspunsul

	\(2\frac{1}{2}\div 1\frac{1}{4}\)
Convert mixed numbers to improper fractions.	\(\frac{5}{2}\div \frac{5}{4}\)
Multiply the first fraction by the reciprocal of the second.	\(\frac{5}{2}\cdot \frac{4}{5}\)
Multiply.	\(\frac{5\cdot 4}{2\cdot 5}\)
Look for common factors.	\(\frac{5̸\cdot 2̸\cdot 2}{2̸\cdot 1\cdot 5̸}\)
Remove common factors.	\(\frac{2}{1}\)
Simplify.	\(2\)

Divide, and write your answer in simplified form: \(2\frac{2}{3}\div 1\frac{1}{3}.\)

Revelează răspunsul

2
Divide, and write your answer in simplified form: \(3\frac{3}{4}\div 1\frac{1}{2}.\)

Revelează răspunsul

\(\frac{5}{2}\)
Translate the phrase into an algebraic expression: “the quotient of \(3x\) and \(8.”\)

Revelează răspunsul

The keyword is quotient; it tells us that the operation is division. Look for the words of and and to find the numbers to divide.
\[\text{The quotient}\ \text{of}\ 3x\ \text{and}\ 8.\]
This tells us that we need to divide \(3x\) by \(8.\) \(\frac{3x}{8}\)
Translate the phrase into an algebraic expression: the quotient of \(9s\) and \(14.\)

Revelează răspunsul

\(\frac{9s}{14}\)
Translate the phrase into an algebraic expression: the quotient of \(5y\) and \(6.\)

Revelează răspunsul

\(\frac{5y}{6}\)
Translate the phrase into an algebraic expression: the quotient of the difference of \(m\) and \(n,\) and \(p.\)

Revelează răspunsul

We are looking for the quotient of the difference of \(m\) and \(n\), and \(p.\) This means we want to divide the difference of \(m\) and \(n\) by \(p.\)
\[\frac{m-n}{p}\]
Translate the phrase into an algebraic expression: the quotient of the difference of \(a\) and \(b,\) and \(cd.\)

Revelează răspunsul

\(\frac{a-b}{cd}\)
Translate the phrase into an algebraic expression: the quotient of the sum of \(p\) and \(q,\) and \(r.\)

Revelează răspunsul

\(\frac{p+q}{r}\)

Simplify: \(\frac{\ \frac{3}{4}\ }{\ \frac{5}{8}\ }.\)

Revelează răspunsul

	\(\frac{\ \frac{3}{4}\ }{\ \frac{5}{8}\ }\)
Rewrite as division.	\(\frac{3}{4}\div \frac{5}{8}\)
Multiply the first fraction by the reciprocal of the second.	\(\frac{3}{4}\cdot \frac{8}{5}\)
Multiply.	\(\frac{3\cdot 8}{4\cdot 5}\)
Look for common factors.	\(\frac{3\cdot 4̸\cdot 2}{4̸\cdot 5}\)
Remove common factors and simplify.	\(\frac{6}{5}\)

Simplify: \(\frac{\ \frac{2}{3}\ }{\ \frac{5}{6}\ }.\)

Revelează răspunsul

\(\frac{4}{5}\)
Simplify: \(\frac{\ \frac{3}{7}\ }{\ \frac{6}{11}\ }.\)

Revelează răspunsul

\(\frac{11}{14}\)

Simplify: \(\frac{-\frac{6}{7}}{3}.\)

Revelează răspunsul

	\(\frac{-\frac{6}{7}}{3}\)
Rewrite as division.	\(-\frac{6}{7}\div 3\)
Multiply the first fraction by the reciprocal of the second.	\(-\frac{6}{7}\cdot \frac{1}{3}\)
Multiply; the product will be negative.	\(-\frac{6\cdot 1}{7\cdot 3}\)
Look for common factors.	\(-\frac{3̸\cdot 2\cdot 1}{7\cdot 3̸}\)
Remove common factors and simplify.	\(-\frac{2}{7}\)

Simplify: \(\frac{-\frac{8}{7}}{4}.\)

Revelează răspunsul

\(-\frac{2}{7}\)
Simplify: \(-\frac{\ 3\ }{\ \frac{9}{10}\ }.\)

Revelează răspunsul

\(-\frac{10}{3}\)

Simplify: \(\frac{\ \frac{x}{2}\ }{\ \frac{xy}{6}\ }.\)

Revelează răspunsul

	\(\frac{\ \frac{x}{2}\ }{\ \frac{xy}{6}\ }\)
Rewrite as division.	\(\frac{x}{2}\div \frac{xy}{6}\)
Multiply the first fraction by the reciprocal of the second.	\(\frac{x}{2}\cdot \frac{6}{xy}\)
Multiply.	\(\frac{x\cdot 6}{2\cdot xy}\)
Look for common factors.	\(\frac{x̸\cdot 3\cdot 2̸}{2̸\cdot x̸\cdot y}\)
Remove common factors and simplify.	\(\frac{3}{y}\)

Simplify: \(\frac{\ \frac{a}{8}\ }{\ \frac{ab}{6}\ }.\)

Revelează răspunsul

\(\frac{3}{4b}\)
Simplify: \(\frac{\ \frac{p}{2}\ }{\ \frac{pq}{8}\ }.\)

Revelează răspunsul

\(\frac{4}{q}\)

Simplify: \(\frac{2\frac{3}{4}}{\frac{1}{8}}.\)

Revelează răspunsul

	\(\frac{2\frac{3}{4}}{\frac{1}{8}}\)
Rewrite as division.	\(2\frac{3}{4}\div \frac{1}{8}\)
Change the mixed number to an improper fraction.	\(\frac{11}{4}\div \frac{1}{8}\)
Multiply the first fraction by the reciprocal of the second.	\(\frac{11}{4}\cdot \frac{8}{1}\)
Multiply.	\(\frac{11\cdot 8}{4\cdot 1}\)
Look for common factors.	\(\frac{11\cdot 4̸\cdot 2}{4̸\cdot 1}\)
Remove common factors and simplify.	\(22\)

Simplify: \(\frac{\frac{5}{7}}{1\frac{2}{5}}.\)

Revelează răspunsul

\(\frac{25}{49}.\)
Simplify: \(\frac{\frac{8}{5}}{3\frac{1}{5}}.\)

Revelează răspunsul

\(\frac{1}{2}\)
Which of the following fractions are equivalent to \(\frac{7}{-8}?\)
\[\ \frac{-7}{-8},\frac{-7}{8},\frac{7}{8},-\frac{7}{8}\]

Revelează răspunsul

The quotient of a positive and a negative is a negative, so \(\frac{7}{-8}\) is negative. Of the fractions listed, \(\ \frac{-7}{8}\) and \(-\frac{7}{8}\) are also negative.
Which of the following fractions are equivalent to \(\frac{-3}{\ 5}?\)
\(\ \frac{-3}{-5},\ \frac{3}{5},-\frac{3}{5},\frac{\ 3}{-5}\)

Revelează răspunsul

\(\ -\frac{3}{5},\frac{\ 3}{-5}\)
Which of the following fractions are equivalent to \(-\frac{2}{7}?\)
\(\ \frac{-2}{-7},\frac{-2}{7},\frac{2}{7},\frac{2}{-7}\)

Revelează răspunsul

\(\ \frac{-2}{7},\frac{2}{-7}\)
Simplify: \(\frac{4+8}{5-3}.\)

Revelează răspunsul

\(\frac{4+8}{5-3}\)
Simplify the expression in the numerator. \(\frac{12}{5-3}\)
Simplify the expression in the denominator. \(\frac{12}{2}\)
Simplify the fraction. 6
Simplify: \(\frac{4+6}{11-2}.\)

Revelează răspunsul

\(\frac{10}{9}\)
Simplify: \(\frac{3+5}{18-2}.\)

Revelează răspunsul

\(\frac{1}{2}\)

Simplify: \(\frac{4-2(3)}{{2}^{2}+2}.\)

Revelează răspunsul

	\(\frac{4-2(3)}{{2}^{2}+2}\)
Use the order of operations. Multiply in the numerator and use the exponent in the denominator.	\(\frac{4-6}{4+2}\)
Simplify the numerator and the denominator.	\(\frac{-2}{6}\)
Simplify the fraction.	\(-\frac{1}{3}\)

Symbols used here

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\approx

approximately equal
Equal to the precision shown, not exactly.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: Multiply and Divide Mixed Numbers and Complex Fractions

Multiply and divide mixed numbers
Translate phrases to expressions with fractions
Simplify complex fractions
Simplify expressions written with a fraction bar
Convert the mixed numbers to improper fractions.
Follow the rules for fraction multiplication or division.
Simplify if possible.
Rewrite the complex fraction as a division problem.

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

Încearcă pe tine.

Parts of this page are adapted from OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

Mai multe în Arithmetic

Order of operations Fractions Percentages Powers and roots