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Multiply and Divide Fractions

Simplify fractions

Simplify Fractions

In working with equivalent fractions, you saw that there are many ways to write fractions that have the same value, or represent the same part of the whole. How do you know which one to use? Often, we’ll use the fraction that is in simplified form.

A fraction is considered simplified if there are no common factors, other than \(1,\) in the numerator and denominator. If a fraction does have common factors in the numerator and denominator, we can reduce the fraction to its simplified form by removing the common factors.

For example,

\(\frac{2}{3}\) is simplified because there are no common factors of \(2\) and \(3.\)
\(\frac{10}{15}\) is not simplified because \(5\) is a common factor of \(10\) and \(15.\)

The process of simplifying a fraction is often called reducing the fraction. In the previous section, we used the Equivalent Fractions Property to find equivalent fractions. We can also use the Equivalent Fractions Property in reverse to simplify fractions. We rewrite the property to show both forms together.

Notice that \(c\) is a common factor in the numerator and denominator. Anytime we have a common factor in the numerator and denominator, it can be removed.

Example

Try it.

Simplify: \(\frac{10}{15}.\)

Solution

To simplify the fraction, we look for any common factors in the numerator and the denominator.

Notice that 5 is a factor of both 10 and 15.	\(\frac{10}{15}\)
Factor the numerator and denominator.
Remove the common factors.
Simplify.	\(\frac{2}{3}\)

To simplify a negative fraction, we use the same process as in . Remember to keep the negative sign.

Example

Try it.

Simplify: \(-\frac{18}{24}.\)

Solution

We notice that 18 and 24 both have factors of 6.	\(-\frac{18}{24}\)
Rewrite the numerator and denominator showing the common factor.
Remove common factors.
Simplify.	\(-\frac{3}{4}\)

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

Multiply Fractions

A model may help you understand multiplication of fractions. We will use fraction tiles to model \(\frac{1}{2}\cdot \frac{3}{4}.\) To multiply \(\frac{1}{2}\) and \(\frac{3}{4},\) think \(\frac{1}{2}\) of \(\frac{3}{4}.\)

Start with fraction tiles for three-fourths. To find one-half of three-fourths, we need to divide them into two equal groups. Since we cannot divide the three \(\frac{1}{4}\) tiles evenly into two parts, we exchange them for smaller tiles.

We see \(\frac{6}{8}\) is equivalent to \(\frac{3}{4}.\) Taking half of the six \(\frac{1}{8}\) tiles gives us three \(\frac{1}{8}\) tiles, which is \(\frac{3}{8}.\)

Therefore,

\[\frac{1}{2}\cdot \frac{3}{4}=\frac{3}{8}\]

Example

Try it.

Use a diagram to model \(\frac{1}{2}\cdot \frac{3}{4}.\)

Solution

First shade in \(\frac{3}{4}\) of the rectangle.

We will take \(\frac{1}{2}\) of this \(\frac{3}{4},\) so we heavily shade \(\frac{1}{2}\) of the shaded region.

Notice that \(3\) out of the \(8\) pieces are heavily shaded. This means that \(\frac{3}{8}\) of the rectangle is heavily shaded.

Therefore, \(\frac{1}{2}\) of \(\frac{3}{4}\) is \(\frac{3}{8},\) or \(\frac{1}{2}\cdot \frac{3}{4}=\frac{3}{8}.\)

Look at the result we got from the model in . We found that \(\frac{1}{2}\cdot \frac{3}{4}=\frac{3}{8}.\) Do you notice that we could have gotten the same answer by multiplying the numerators and multiplying the denominators?

	\(\frac{1}{2}\cdot \frac{3}{4}\)
Multiply the numerators, and multiply the denominators.	\(\frac{1}{2}\cdot \frac{3}{4}\)
Simplify.	\(\frac{3}{8}\)

This leads to the definition of fraction multiplication. To multiply fractions, we multiply the numerators and multiply the denominators. Then we write the fraction in simplified form.

Example

Try it.

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

Solution

	\(\frac{3}{4}\cdot \frac{1}{5}\)
Multiply the numerators; multiply the denominators.	\(\frac{3\cdot 1}{4\cdot 5}\)
Simplify.	\(\frac{3}{20}\)

There are no common factors, so the fraction is simplified.

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

Find Reciprocals

The fractions \(\frac{2}{3}\) and \(\frac{3}{2}\) are related to each other in a special way. So are \(-\frac{10}{7}\) and \(-\frac{7}{10}.\) Do you see how? Besides looking like upside-down versions of one another, if we were to multiply these pairs of fractions, the product would be \(1.\)

\[\frac{2}{3}\cdot \frac{3}{2}=1\ \text{and}\ -\frac{10}{7}(-\frac{7}{10})=1\]

Such pairs of numbers are called reciprocals.

To find the reciprocal of a fraction, we invert the fraction. This means that we place the numerator in the denominator and the denominator in the numerator.

To get a positive result when multiplying two numbers, the numbers must have the same sign. So reciprocals must have the same sign.

To find the reciprocal, keep the same sign and invert the fraction. The number zero does not have a reciprocal. Why? A number and its reciprocal multiply to \(1.\) Is there any number \(r\) so that \(0\cdot r=1?\) No. So, the number \(0\) does not have a reciprocal.

Example

Try it.

Find the reciprocal of each number. Then check that the product of each number and its reciprocal is \(1.\)

ⓐ \(\frac{4}{9}\)
ⓑ \(-\frac{1}{6}\)
ⓒ \(-\frac{14}{5}\)
ⓓ \(7\)

Solution

To find the reciprocals, we keep the sign and invert the fractions.

ⓐ
Find the reciprocal of \(\frac{4}{9}\).	The reciprocal of \(\frac{4}{9}\) is \(\frac{9}{4}\).
Check:
Multiply the number and its reciprocal.	\(\frac{4}{9}⋅\frac{9}{4}\)
Multiply numerators and denominators.	\(\frac{36}{36}\)
Simplify.	\(1✓\)

ⓑ
Find the reciprocal of \(-\frac{1}{6}\).	\(-\frac{6}{1}\)
Simplify.	\(-6\)
Check:	\(-\frac{1}{6}⋅(-6)\)
	\(1✓\)

ⓒ
Find the reciprocal of \(-\frac{14}{5}\).	\(-\frac{5}{14}\)
Check:	\(-\frac{14}{5}⋅(-\frac{5}{14})\)
	\(\frac{70}{70}\)
	\(1✓\)

ⓓ
Find the reciprocal of \(7\).
Write \(7\) as a fraction.	\(\frac{7}{1}\)
Write the reciprocal of \(\frac{7}{1}\).	\(\frac{1}{7}\)
Check:	\(7⋅(\frac{1}{7})\)
	\(1✓\)

In a previous chapter, we worked with opposites and absolute values. compares opposites, absolute values, and reciprocals.

Opposite	Absolute Value	Reciprocal
has opposite sign	is never negative	has same sign, fraction inverts

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

Divide Fractions

Why is \(12\div 3=4?\) We previously modeled this with counters. How many groups of \(3\) counters can be made from a group of \(12\) counters?

There are \(4\) groups of \(3\) counters. In other words, there are four \(3\)s in \(12.\) So, \(12\div 3=4.\)

What about dividing fractions? Suppose we want to find the quotient: \(\frac{1}{2}\div \frac{1}{6}.\) We need to figure out how many \(\frac{1}{6}\)s there are in \(\frac{1}{2}.\) We can use fraction tiles to model this division. We start by lining up the half and sixth fraction tiles as shown in . Notice, there are three \(\frac{1}{6}\) tiles in \(\frac{1}{2},\) so \(\frac{1}{2}\div \frac{1}{6}=3.\)

Example

Try it.

Model: \(\frac{1}{4}\div \frac{1}{8}.\)

Solution

We want to determine how many \(\frac{1}{8}\)s are in \(\frac{1}{4}.\) Start with one \(\frac{1}{4}\) tile. Line up \(\frac{1}{8}\) tiles underneath the \(\frac{1}{4}\) tile.

There are two \(\frac{1}{8}\)s in \(\frac{1}{4}.\)

So, \(\frac{1}{4}\div \frac{1}{8}=2.\)

Example

Try it.

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

Solution

We are trying to determine how many \(\frac{1}{4}\)s there are in \(2.\) We can model this as shown.

Because there are eight \(\frac{1}{4}\)s in \(2,2\div \frac{1}{4}=8.\)

Let’s use money to model \(2\div \frac{1}{4}\) in another way. We often read \(\frac{1}{4}\) as a ‘quarter’, and we know that a quarter is one-fourth of a dollar as shown in . So we can think of \(2\div \frac{1}{4}\) as, “How many quarters are there in two dollars?” One dollar is \(4\) quarters, so \(2\) dollars would be \(8\) quarters. So again, \(2\div \frac{1}{4}=8.\)

Using fraction tiles, we showed that \(\frac{1}{2}\div \frac{1}{6}=3.\) Notice that \(\frac{1}{2}\cdot \frac{6}{1}=3\) also. How are \(\frac{1}{6}\) and \(\frac{6}{1}\) related? They are reciprocals. This leads us to the procedure for fraction division.

We need to say \(b\ne 0,c\ne 0\) and \(d\ne 0\) to be sure we don’t divide by zero.

Example

Try it.

Divide, and write the answer in simplified form: \(\frac{2}{5}\div (-\frac{3}{7}).\)

Solution

	\(\frac{2}{5}\div (-\frac{3}{7})\)
Multiply the first fraction by the reciprocal of the second.	\(\frac{2}{5}(-\frac{7}{3})\)
Multiply. The product is negative.	\(-\frac{14}{15}\)

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

Key Concepts

Equivalent Fractions Property
- If \(a,b,c\) are numbers where \(b\ne 0\), \(c\ne 0\), then \(\frac{a}{b}=\frac{a⋅c}{b⋅c}\) and \(\frac{a⋅c}{b⋅c}=\frac{a}{b}\).
Simplify a fraction.
1. Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers.
2. Simplify, using the equivalent fractions property, by removing common factors.
3. Multiply any remaining factors.
Fraction Multiplication
- If \(a,b,c,\) and \(d\) are numbers where \(b\ne 0\)and \(d\ne 0\), then \(\frac{a}{b}⋅\frac{c}{d}=\frac{ac}{bd}\).
Reciprocal
- A number and its reciprocal have a product of \(1\). \(\frac{a}{b}⋅\frac{b}{a}=1\)
- Opposite Absolute Value Reciprocal
  has opposite sign is never negative has same sign, fraction inverts
Fraction Division
- If \(a,b,c,\) and \(d\) are numbers where \(b\ne 0\), \(c\ne 0\) and \(d\ne 0\) , then
  \[\frac{a}{b}\div \frac{c}{d}=\frac{a}{b}⋅\frac{d}{c}\]
- To divide fractions, multiply the first fraction by the reciprocal of the second.

Multiply and Divide Fractions

Simplify Fractions

In the following exercises, simplify each fraction. Do not convert any improper fractions to mixed numbers.

Try it.

\(\frac{7}{21}\)

Solution

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

Try it.

\(\frac{8}{24}\)

Try it.

\(\frac{15}{20}\)

Solution

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

Try it.

\(\frac{12}{18}\)

Try it.

\(-\frac{40}{88}\)

Solution

\(-\frac{5}{11}\)

Try it.

\(-\frac{63}{99}\)

Try it.

\(-\frac{108}{63}\)

Solution

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

Try it.

\(-\frac{104}{48}\)

Try it.

\(\frac{120}{252}\)

Solution

\(\frac{10}{21}\)

Try it.

\(\frac{182}{294}\)

Try it.

\(-\frac{168}{192}\)

Solution

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

Try it.

\(-\frac{140}{224}\)

Try it.

\(\frac{11x}{11y}\)

Solution

\(\frac{x}{y}\)

Try it.

\(\frac{15a}{15b}\)

Try it.

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

Solution

\(-\frac{x}{4y}\)

Try it.

\(-\frac{4x}{32y}\)

Try it.

\(\frac{14{x}^{2}}{21y}\)

Solution

\(\frac{2{x}^{2}}{3y}\)

Try it.

\(\frac{24a}{32{b}^{2}}\)

Multiply Fractions

In the following exercises, use a diagram to model.

Try it.

\(\frac{1}{2}\cdot \frac{2}{3}\)

Solution

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

Try it.

\(\frac{1}{2}\cdot \frac{5}{8}\)

Try it.

\(\frac{1}{3}\cdot \frac{5}{6}\)

Solution

\(\frac{5}{18}\)

Try it.

\(\frac{1}{3}\cdot \frac{2}{5}\)

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

Try it.

\(\frac{2}{5}\cdot \frac{1}{3}\)

Solution

\(\frac{2}{15}\)

Try it.

\(\frac{1}{2}\cdot \frac{3}{8}\)

Try it.

\(\frac{3}{4}\cdot \frac{9}{10}\)

Solution

\(\frac{27}{40}\)

Try it.

\(\frac{4}{5}\cdot \frac{2}{7}\)

Try it.

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

Solution

\(\frac{1}{4}\)

Try it.

\(-\frac{3}{4}(-\frac{4}{9})\)

Try it.

\(-\frac{5}{9}\cdot \frac{3}{10}\)

Solution

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

Try it.

\(-\frac{3}{8}\cdot \frac{4}{15}\)

Try it.

\(\frac{7}{12}(-\frac{8}{21})\)

Solution

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

Try it.

\(\frac{5}{12}(-\frac{8}{15})\)

Try it.

\((-\frac{14}{15})(\frac{9}{20})\)

Solution

\(-\frac{21}{50}\)

Try it.

\((-\frac{9}{10})(\frac{25}{33})\)

Try it.

\((-\frac{63}{84})(-\frac{44}{90})\)

Solution

\(\frac{11}{30}\)

Try it.

\((-\frac{33}{60})(-\frac{40}{88})\)

Try it.

\(4\cdot \frac{5}{11}\)

Solution

\(\frac{20}{11}\)

Try it.

\(5\cdot \frac{8}{3}\)

Try it.

\(\frac{3}{7}\cdot 21n\)

Solution

Try it.

\(\frac{5}{6}\cdot 30m\)

Try it.

\(-28p(-\frac{1}{4})\)

Solution

Try it.

\(-51q(-\frac{1}{3})\)

Try it.

\(-8(\frac{17}{4})\)

Solution

−34

Try it.

\(\frac{14}{5}(-15)\)

Try it.

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

Solution

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

Try it.

\((-1)(-\frac{6}{7})\)

Try it.

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

Solution

\(\frac{8}{27}\)

Try it.

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

Try it.

\({(\frac{6}{5})}^{4}\)

Solution

\(\frac{1296}{625}\)

Try it.

\({(\frac{4}{7})}^{4}\)

Find Reciprocals

In the following exercises, find the reciprocal.

Try it.

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

Solution

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

Try it.

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

Try it.

\(-\frac{5}{17}\)

Solution

\(-\frac{17}{5}\)

Try it.

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

Try it.

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

Solution

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

Try it.

\(-13\)

Try it.

\(-19\)

Solution

\(-\frac{1}{19}\)

Try it.

\(-1\)

Try it.

\(1\)

Solution

Try it.

Fill in the chart.

	Opposite	Absolute Value	Reciprocal
\(-\frac{7}{11}\)
\(\frac{4}{5}\)
\(\frac{10}{7}\)
\(-8\)

Solution

Try it.

Fill in the chart.

	Opposite	Absolute Value	Reciprocal
\(-\frac{3}{13}\)
\(\frac{9}{14}\)
\(\frac{15}{7}\)
\(-9\)

Solution

Divide Fractions

In the following exercises, model each fraction division.

Try it.

\(\frac{1}{2}\div \frac{1}{4}\)

Try it.

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

Solution

Try it.

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

Try it.

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

Solution

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

Try it.

\(\frac{1}{2}\div \frac{1}{4}\)

Try it.

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

Solution

Try it.

\(\frac{3}{4}\div \frac{2}{3}\)

Try it.

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

Solution

\(\frac{16}{15}\)

Try it.

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

Try it.

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

Solution

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

Try it.

\(-\frac{7}{9}\div (-\frac{7}{9})\)

Try it.

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

Solution

Try it.

\(\frac{3}{4}\div \frac{x}{11}\)

Try it.

\(\frac{2}{5}\div \frac{y}{9}\)

Solution

\(\frac{18}{5y}\)

Try it.

\(\frac{5}{8}\div \frac{a}{10}\)

Try it.

\(\frac{5}{6}\div \frac{c}{15}\)

Solution

\(\frac{25}{2c}\)

Try it.

\(\frac{5}{18}\div (-\frac{15}{24})\)

Try it.

\(\frac{7}{18}\div (-\frac{14}{27})\)

Solution

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

Try it.

\(\frac{7p}{12}\div \frac{21p}{8}\)

Try it.

\(\frac{5q}{12}\div \frac{15q}{8}\)

Solution

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

Try it.

\(\frac{8u}{15}\div \frac{12v}{25}\)

Try it.

\(\frac{12r}{25}\div \frac{18s}{35}\)

Solution

\(\frac{14r}{15s}\)

Try it.

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

Try it.

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

Solution

−12

Try it.

\(\frac{3}{4}\div (-12)\)

Try it.

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

Solution

\(-\frac{1}{25}\)

Try it.

\(-18\div (-\frac{9}{2})\)

Try it.

\(-15\div (-\frac{5}{3})\)

Solution

Try it.

\(\frac{1}{2}\div (-\frac{3}{4})\div \frac{7}{8}\)

Try it.

\(\frac{11}{2}\div \frac{7}{8}\cdot \frac{2}{11}\)

Solution

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

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.

Find the prime factorization of \(48.\)
If you missed this problem, review .

Առաջարկել պատասխանը

\(2⋅2⋅2⋅2⋅3\)
Draw a model of the fraction \(\frac{3}{4}.\)
If you missed this problem, review .

Առաջարկել պատասխանը
Find two fractions equivalent to \(\frac{5}{6}.\)
If you missed this problem, review .

Առաջարկել պատասխանը

Answers may vary. Acceptable answers include \(\frac{10}{12},\frac{15}{18},\frac{50}{60},\) etc.
Simplify: \(\frac{10}{15}.\)

Առաջարկել պատասխանը

To simplify the fraction, we look for any common factors in the numerator and the denominator.
Notice that 5 is a factor of both 10 and 15. \(\frac{10}{15}\)
Factor the numerator and denominator.
Remove the common factors.
Simplify. \(\frac{2}{3}\)
Simplify: \(\frac{8}{12}\).

Առաջարկել պատասխանը

\(\frac{2}{3}\)
Simplify: \(\frac{12}{16}\).

Առաջարկել պատասխանը

\(\frac{3}{4}\)
Simplify: \(-\frac{18}{24}.\)

Առաջարկել պատասխանը

We notice that 18 and 24 both have factors of 6. \(-\frac{18}{24}\)
Rewrite the numerator and denominator showing the common factor.
Remove common factors.
Simplify. \(-\frac{3}{4}\)
Simplify: \(-\frac{21}{28}.\)

Առաջարկել պատասխանը

\(-\frac{3}{4}\)
Simplify: \(-\frac{16}{24}.\)

Առաջարկել պատասխանը

\(-\frac{2}{3}\)
Simplify: \(-\frac{56}{32}.\)

Առաջարկել պատասխանը

\(-\frac{56}{32}\)
Rewrite the numerator and denominator, showing the common factors, 8.
Remove common factors.
Simplify. \(-\frac{7}{4}\)
Simplify: \(-\frac{54}{42}.\)

Առաջարկել պատասխանը

\(-\frac{9}{7}\)
Simplify: \(-\frac{81}{45}.\)

Առաջարկել պատասխանը

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

Simplify: \(\frac{210}{385}.\)

Առաջարկել պատասխանը

Use factor trees to factor the numerator and denominator.	\(\frac{210}{385}\)
Rewrite the numerator and denominator as the product of the primes.	\(\frac{210}{385}=\frac{2⋅3⋅5⋅7}{5⋅7⋅11}\)
Remove the common factors.
Simplify.	\(\frac{2⋅3}{11}\)
Multiply any remaining factors.	\(\frac{6}{11}\)

Simplify: \(\frac{69}{120}.\)

Առաջարկել պատասխանը

\(\frac{23}{40}\)
Simplify: \(\frac{120}{192}.\)

Առաջարկել պատասխանը

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

Simplify: \(\frac{5xy}{15x}.\)

Առաջարկել պատասխանը

	\(\frac{5xy}{15x}\)
Rewrite numerator and denominator showing common factors.	\(\frac{5\cdot x\cdot y}{3\cdot 5\cdot x}\)
Remove common factors.	\(\frac{5\cdot x\cdot y}{3\cdot 5\cdot x}\)
Simplify.	\(\frac{y}{3}\)

Simplify: \(\frac{7x}{7y}.\)

Առաջարկել պատասխանը

\(\frac{x}{y}\)
Simplify: \(\frac{9a}{9b}.\)

Առաջարկել պատասխանը

\(\frac{a}{b}\)
Use a diagram to model \(\frac{1}{2}\cdot \frac{3}{4}.\)

Առաջարկել պատասխանը

First shade in \(\frac{3}{4}\) of the rectangle.
We will take \(\frac{1}{2}\) of this \(\frac{3}{4},\) so we heavily shade \(\frac{1}{2}\) of the shaded region.
Notice that \(3\) out of the \(8\) pieces are heavily shaded. This means that \(\frac{3}{8}\) of the rectangle is heavily shaded.
Therefore, \(\frac{1}{2}\) of \(\frac{3}{4}\) is \(\frac{3}{8},\) or \(\frac{1}{2}\cdot \frac{3}{4}=\frac{3}{8}.\)
Use a diagram to model: \(\frac{1}{2}\cdot \frac{3}{5}.\)

Առաջարկել պատասխանը

\(\frac{3}{10}\)
Use a diagram to model: \(\frac{1}{2}\cdot \frac{5}{6}.\)

Առաջարկել պատասխանը

\(\frac{5}{12}\)
Multiply, and write the answer in simplified form: \(\frac{3}{4}\cdot \frac{1}{5}.\)

Առաջարկել պատասխանը

\(\frac{3}{4}\cdot \frac{1}{5}\)
Multiply the numerators; multiply the denominators. \(\frac{3\cdot 1}{4\cdot 5}\)
Simplify. \(\frac{3}{20}\)
There are no common factors, so the fraction is simplified.
Multiply, and write the answer in simplified form: \(\frac{1}{3}\cdot \frac{2}{5}.\)

Առաջարկել պատասխանը

\(\frac{2}{15}\)
Multiply, and write the answer in simplified form: \(\frac{3}{5}\cdot \frac{7}{8}.\)

Առաջարկել պատասխանը

\(\frac{21}{40}\)

Multiply, and write the answer in simplified form: \(-\frac{5}{8}(-\frac{2}{3}).\)

Առաջարկել պատասխանը

	\(-\frac{5}{8}(-\frac{2}{3})\)
The signs are the same, so the product is positive. Multiply the numerators, multiply the denominators.	\(\frac{5⋅2}{8⋅3}\)
Simplify.	\(\frac{10}{24}\)
Look for common factors in the numerator and denominator. Rewrite showing common factors.
Remove common factors.	\(\frac{5}{12}\)

Another way to find this product involves removing common factors earlier.

	\(-\frac{5}{8}(-\frac{2}{3})\)
Determine the sign of the product. Multiply.	\(\frac{5⋅2}{8⋅3}\)
Show common factors and then remove them.
Multiply remaining factors.	\(\frac{5}{12}\)

We get the same result.

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

Առաջարկել պատասխանը

\(\frac{5}{14}\)
Multiply, and write the answer in simplified form: \(-\frac{7}{12}(-\frac{8}{9}).\)

Առաջարկել պատասխանը

\(\frac{14}{27}\)

Multiply, and write the answer in simplified form: \(-\frac{14}{15}\cdot \frac{20}{21}.\)

Առաջարկել պատասխանը

	\(-\frac{14}{15}\cdot \frac{20}{21}\)
Determine the sign of the product; multiply.	\(-\frac{14}{15}\cdot \frac{20}{21}\)
Are there any common factors in the numerator and the denominator? \(\\)We know that 7 is a factor of 14 and 21, and 5 is a factor of 20 and 15.
Rewrite showing common factors.
Remove the common factors.	\(-\frac{2\cdot 4}{3\cdot 3}\)
Multiply the remaining factors.	\(-\frac{8}{9}\)

Multiply, and write the answer in simplified form: \(-\frac{10}{28}\cdot \frac{8}{15}.\)

Առաջարկել պատասխանը

\(-\frac{4}{21}\)
Multiply, and write the answer in simplified form: \(-\frac{9}{20}\cdot \frac{5}{12}.\)

Առաջարկել պատասխանը

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

Multiply, and write the answer in simplified form:

ⓐ \(\frac{1}{7}\cdot 56\)

ⓑ \(\frac{12}{5}(-20x)\)

Առաջարկել պատասխանը

ⓐ
	\(\frac{1}{7}\cdot 56\)
Write 56 as a fraction.	\(\frac{1}{7}\cdot \frac{56}{1}\)
Determine the sign of the product; multiply.	\(\frac{56}{7}\)
Simplify.	\(8\)

ⓑ
	\(\frac{12}{5}(-20x)\)
Write −20x as a fraction.	\(\frac{12}{5}(\frac{-20x}{1})\)
Determine the sign of the product; multiply.	\(-\frac{12\cdot 20\cdot x}{5\cdot 1}\)
Show common factors and then remove them.
Multiply remaining factors; simplify.	−48x

Multiply, and write the answer in simplified form:
1. ⓐ \(\ \frac{1}{8}\cdot 72\\)
2. ⓑ \(\ \frac{11}{3}(-9a)\)
Առաջարկել պատասխանը
1. ⓐ 9
2. ⓑ −33a
Multiply, and write the answer in simplified form:
1. ⓐ \(\frac{3}{8}\cdot 64\)
2. ⓑ \(16x\cdot \frac{11}{12}\)
Առաջարկել պատասխանը
1. ⓐ \(24\)
2. ⓑ \(\frac{44x}{3}\)

Find the reciprocal of each number. Then check that the product of each number and its reciprocal is \(1.\)

ⓐ \(\frac{4}{9}\)
ⓑ \(-\frac{1}{6}\)
ⓓ \(7\)

Առաջարկել պատասխանը

To find the reciprocals, we keep the sign and invert the fractions.

ⓐ
Find the reciprocal of \(\frac{4}{9}\).	The reciprocal of \(\frac{4}{9}\) is \(\frac{9}{4}\).
Check:
Multiply the number and its reciprocal.	\(\frac{4}{9}⋅\frac{9}{4}\)
Multiply numerators and denominators.	\(\frac{36}{36}\)
Simplify.	\(1✓\)

ⓑ
Find the reciprocal of \(-\frac{1}{6}\).	\(-\frac{6}{1}\)
Simplify.	\(-6\)
Check:	\(-\frac{1}{6}⋅(-6)\)
	\(1✓\)

ⓒ
Find the reciprocal of \(-\frac{14}{5}\).	\(-\frac{5}{14}\)
Check:	\(-\frac{14}{5}⋅(-\frac{5}{14})\)
	\(\frac{70}{70}\)
	\(1✓\)

ⓓ
Find the reciprocal of \(7\).
Write \(7\) as a fraction.	\(\frac{7}{1}\)
Write the reciprocal of \(\frac{7}{1}\).	\(\frac{1}{7}\)
Check:	\(7⋅(\frac{1}{7})\)
	\(1✓\)

Find the reciprocal:
1. ⓐ \(\frac{5}{7}\)
2. ⓑ \(-\frac{1}{8}\)
3. ⓒ \(-\frac{11}{4}\)
4. ⓓ \(14\)
Առաջարկել պատասխանը
1. ⓐ \(\frac{7}{5}\)
2. ⓑ \(-8\)
3. ⓒ \(-\frac{4}{11}\)
4. ⓓ \(\frac{1}{14}\)
Find the reciprocal:
1. ⓐ \(\frac{3}{7}\)
2. ⓑ \(-\frac{1}{12}\)
3. ⓒ \(-\frac{14}{9}\)
4. ⓓ \(21\)
Առաջարկել պատասխանը
1. ⓐ \(\frac{7}{3}\)
2. ⓑ \(-12\)
3. ⓒ \(-\frac{9}{14}\)
4. ⓓ \(\frac{1}{21}\)

Fill in the chart for each fraction in the left column:

Number	Opposite	Absolute Value	Reciprocal
\(-\frac{3}{8}\)
\(\frac{1}{2}\)
\(\frac{9}{5}\)
\(-5\)

Առաջարկել պատասխանը

To find the opposite, change the sign. To find the absolute value, leave the positive numbers the same, but take the opposite of the negative numbers. To find the reciprocal, keep the sign the same and invert the fraction.

Number	Opposite	Absolute Value	Reciprocal
\(-\frac{3}{8}\)	\(\frac{3}{8}\)	\(\frac{3}{8}\)	\(-\frac{8}{3}\)
\(\frac{1}{2}\)	\(-\frac{1}{2}\)	\(\frac{1}{2}\)	\(2\)
\(\frac{9}{5}\)	\(-\frac{9}{5}\)	\(\frac{9}{5}\)	\(\frac{5}{9}\)
\(-5\)	\(5\)	\(5\)	\(-\frac{1}{5}\)

Fill in the chart for each number given:
Number Opposite Absolute Value Reciprocal
\(-\frac{5}{8}\)
\(\frac{1}{4}\)
\(\frac{8}{3}\)
\(-8\)

Առաջարկել պատասխանը
Fill in the chart for each number given:
Number Opposite Absolute Value Reciprocal
\(-\frac{4}{7}\)
\(\frac{1}{8}\)
\(\frac{9}{4}\)
\(-1\)

Առաջարկել պատասխանը
Model: \(\frac{1}{4}\div \frac{1}{8}.\)

Առաջարկել պատասխանը

We want to determine how many \(\frac{1}{8}\)s are in \(\frac{1}{4}.\) Start with one \(\frac{1}{4}\) tile. Line up \(\frac{1}{8}\) tiles underneath the \(\frac{1}{4}\) tile.
There are two \(\frac{1}{8}\)s in \(\frac{1}{4}.\)
So, \(\frac{1}{4}\div \frac{1}{8}=2.\)

Symbols used here

f'(x),\ \frac{dy}{dx}

derivative
Instantaneous rate of change; slope of the graph.

\neq

not equal
The two sides are different.

\pm

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

\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 Fractions

Simplify fractions
Multiply fractions
Find reciprocals
Divide fractions
Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers.
Simplify, using the equivalent fractions property, by removing common factors.
Multiply any remaining factors.
Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers.

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.

Փորձեք ինքներդ

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

Ցուցադրել Arithmetic

Order of operations Fractions Percentages Powers and roots

Opposite	Absolute Value	Reciprocal
has opposite sign	is never negative	has same sign, fraction inverts

	\(-\frac{56}{32}\)
Rewrite the numerator and denominator, showing the common factors, 8.
Remove common factors.
Simplify.	\(-\frac{7}{4}\)