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Visualize Fractions

Find equivalent fractions

Find Equivalent Fractions

Fractions are a way to represent parts of a whole. The fraction \(\frac{1}{3}\) means that one whole has been divided into 3 equal parts and each part is one of the three equal parts. See . The fraction \(\frac{2}{3}\) represents two of three equal parts. In the fraction \(\frac{2}{3},\) the 2 is called the numerator and the 3 is called the denominator.

If a whole pie has been cut into 6 pieces and we eat all 6 pieces, we ate \(\frac{6}{6}\) pieces, or, in other words, one whole pie.

So \(\frac{6}{6}=1.\) This leads us to the property of one that tells us that any number, except zero, divided by itself is 1.

If a pie was cut in \(6\) pieces and we ate all 6, we ate \(\frac{6}{6}\) pieces, or, in other words, one whole pie. If the pie was cut into 8 pieces and we ate all 8, we ate \(\frac{8}{8}\) pieces, or one whole pie. We ate the same amount—one whole pie.

The fractions \(\frac{6}{6}\) and \(\frac{8}{8}\) have the same value, 1, and so they are called equivalent fractions. Equivalent fractions are fractions that have the same value.

Let’s think of pizzas this time. shows two images: a single pizza on the left, cut into two equal pieces, and a second pizza of the same size, cut into eight pieces on the right. This is a way to show that \(\frac{1}{2}\) is equivalent to \(\frac{4}{8}.\) In other words, they are equivalent fractions.

How can we use mathematics to change \(\frac{1}{2}\) into \(\frac{4}{8}?\) How could we take a pizza that is cut into 2 pieces and cut it into 8 pieces? We could cut each of the 2 larger pieces into 4 smaller pieces! The whole pizza would then be cut into \(8\) pieces instead of just 2. Mathematically, what we’ve described could be written like this as \(\frac{1\cdot 4}{2\cdot 4}=\frac{4}{8}.\) See .

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Simplify Fractions

A fraction is considered simplified if there are no common factors, other than 1, in its numerator and denominator.

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 phrase reduce a fraction means to simplify the fraction. We simplify, or reduce, a fraction by removing the common factors of the numerator and denominator. A fraction is not simplified until all common factors have been removed. If an expression has fractions, it is not completely simplified until the fractions are simplified.

In , we used the equivalent fractions property to find equivalent fractions. Now we’ll use the equivalent fractions property in reverse to simplify fractions. We can rewrite the property to show both forms together.

Example

Try it.

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

Solution

	\(-\ \frac{32}{56}\)
Rewrite the numerator and denominator showing the common factors.
Simplify using the equivalent fractions property.	\(-\ \frac{4}{7}\)

Notice that the fraction \(-\ \frac{4}{7}\) is simplified because there are no more common factors.

Sometimes it may not be easy to find common factors of the numerator and denominator. When this happens, a good idea is to factor the numerator and the denominator into prime numbers. Then divide out the common factors using the equivalent fractions property.

How to Simplify a Fraction

Try it.

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

Solution

We now summarize the steps you should follow to simplify fractions.

Example

Try it.

Simplify: \(\frac{5x}{5y}.\)

Solution

	\(\frac{5x}{5y}\)
Rewrite showing the common factors, then divide out the common factors.
Simplify.	\(\frac{x}{y}\)

Multiply Fractions

Many people find multiplying and dividing fractions easier than adding and subtracting fractions. So we will start with fraction multiplication.

We’ll use a model to show you how to multiply two fractions and to help you remember the procedure. Let’s start with \(\frac{3}{4}.\)

Now we’ll take \(\frac{1}{2}\) of \(\frac{3}{4}.\)

Notice that now, the whole is divided into 8 equal parts. So \(\frac{1}{2}\cdot \frac{3}{4}=\frac{3}{8}.\)

To multiply fractions, we multiply the numerators and multiply the denominators.

When multiplying fractions, the properties of positive and negative numbers still apply, of course. It is a good idea to determine the sign of the product as the first step. In , we will multiply negative and a positive, so the product will be negative.

Example

Try it.

Multiply: \(-\ \frac{11}{12}\cdot \frac{5}{7}.\)

Solution

The first step is to find the sign of the product. Since the signs are the different, the product is negative.

	\(-\ \frac{11}{12}\cdot \frac{5}{7}\)
Determine the sign of the product; multiply.	\(-\ \frac{11\cdot 5}{12\cdot 7}\)
Are there any common factors in the numerator and the demoninator? No.	\(-\ \frac{55}{84}\)

When multiplying a fraction by an integer, it may be helpful to write the integer as a fraction. Any integer, a, can be written as \(\frac{a}{1}.\) So, for example, \(3=\frac{3}{1}.\)

Example

Try it.

Multiply: \(-\ \frac{12}{5}(-20x).\)

Solution

Determine the sign of the product. The signs are the same, so the product is positive.

	\(-\ \frac{12}{5}(-20x)\)
Write \(20x\) as a fraction.	\(\frac{12}{5}(\frac{20x}{1})\)
Multiply.
Rewrite 20 to show the common factor 5 and divide it out.
Simplify.	\(48x\)

Divide Fractions

Now that we know how to multiply fractions, we are almost ready to divide. Before we can do that, we need some vocabulary.

The reciprocal of a fraction is found by inverting the fraction, placing the numerator in the denominator and the denominator in the numerator. The reciprocal of \(\frac{2}{3}\) is \(\frac{3}{2}.\)

Notice that \(\frac{2}{3}\cdot \frac{3}{2}=1.\) A number and its reciprocal multiply to 1.

To get a product of positive 1 when multiplying two numbers, the numbers must have the same sign. So reciprocals must have the same sign.

The reciprocal of \(-\ \frac{10}{7}\) is \(-\ \frac{7}{10},\) since \(-\ \frac{10}{7}(-\ \frac{7}{10})=1.\)

To divide fractions, we multiply the first fraction by the reciprocal of the second.

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

Example

Try it.

Divide: \(-\ \frac{2}{3}\div \frac{n}{5}.\)

Solution

	\(-\ \frac{2}{3}\div \frac{n}{5}\)
To divide, multiply the first fraction by the reciprocal of the second.	\(-\ \frac{2}{3}\cdot \frac{5}{n}\)
Multiply.	\(-\ \frac{10}{3n}\)

Example

Try it.

Find the quotient: \(-\ \frac{7}{18}\div (-\ \frac{14}{27}).\)

Solution

	\(-\ \frac{7}{18}\div (-\ \frac{14}{27})\)
To divide, multiply the first fraction by the reciprocal of the second.	\(-\ \frac{7}{18}⋅-\ \frac{27}{14}\)
Determine the sign of the product, and then multiply..	\(\frac{7⋅27}{18⋅14}\)
Rewrite showing common factors.
Remove common factors.	\(\frac{3}{2⋅2}\)
Simplify.	\(\frac{3}{4}\)

“To multiply fractions, multiply the numerators and multiply the denominators.”
“To divide fractions, multiply the first fraction by the reciprocal of the second.”

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

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Simplify Expressions with a Fraction Bar

The line that separates the numerator from the denominator in a fraction is called a fraction bar. A fraction bar acts as grouping symbol. The order of operations then tells us to simplify the numerator and then the denominator. Then we divide.

To simplify the expression \(\frac{5-3}{7+1},\) we first simplify the numerator and the denominator separately. Then we divide.

\[\frac{5-3}{7+1}\]\[\frac{2}{8}\]\[\frac{1}{4}\]

Example

Try it.

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

Solution

	\(\frac{4-2(3)}{{2}^{2}+2}\)
Use the order of operations to simpliy the numerator and the denominator.	\(\frac{4-6}{4+2}\)
Simplify the numerator and the denominator.	\(\frac{-2}{6}\)
Simplify. A negative divided by a positive is negative.	\(-\ \frac{1}{3}\)

Where does the negative sign go in a fraction? Usually the negative sign is in front of the fraction, but you will sometimes see a fraction with a negative numerator, or sometimes with a negative denominator. Remember that fractions represent division. When the numerator and denominator have different signs, the quotient is negative.

\[\begin{array}{llllll}\frac{-1}{3}=-\ \frac{1}{3} & & & & & \frac{\text{negative}}{\text{positive}}=\text{negative} \\ \frac{1}{-3}=-\ \frac{1}{3} & & & & & \frac{\text{positive}}{\text{negative}}=\text{negative}\end{array}\]

Example

Try it.

Simplify: \(\frac{4(-3)+6(-2)}{-3(2)-2}.\)

Solution

	\(\frac{4(-3)+6(-2)}{-3(2)-2}\)
Multiply.	\(\frac{-12+(-12)}{-6-2}\)
Simplify.	\(\frac{-24}{-8}\)
Divide.	\(3\)

Translate Phrases to Expressions with Fractions

Now that we have done some work with fractions, we are ready to translate phrases that would result in expressions with fractions.

The English words quotient and ratio are often used to describe fractions. Remember that “quotient” means division. The quotient of \(a\) and \(b\) is the result we get from dividing \(a\) by \(b,\) or \(\frac{a}{b}.\)

Example

Try it.

Translate the English 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 \(m\ \text{and}\ n\ \text{by}\ p.\)

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

Key Concepts

Equivalent Fractions Property: If \(a,b,c\) are numbers where \(b\ne 0,c\ne 0,\) then
\(\frac{a}{b}=\frac{a\cdot c}{b\cdot c}\) and \(\frac{a\cdot c}{b\cdot c}=\frac{a}{b}.\)
Fraction Division: If \(a,b,c\ \text{and}\ d\) are numbers where \(b\ne 0,c\ne 0,\ \text{and}\ d\ne 0,\) then \(\frac{a}{b}\div \frac{c}{d}=\frac{a}{b}\cdot \frac{d}{c}.\) To divide fractions, multiply the first fraction by the reciprocal of the second.
Fraction Multiplication: If \(a,b,c\ \text{and}\ d\) are numbers where \(b\ne 0,\ \text{and}\ d\ne 0,\) then \(\frac{a}{b}\cdot \frac{c}{d}=\frac{ac}{bd}.\) To multiply fractions, multiply the numerators and multiply the denominators.
Placement of Negative Sign in a Fraction: For any positive numbers \(a\ \text{and}\ b,\) \(\frac{-a}{b}=\frac{a}{-b}=-\ \frac{a}{b}.\)
Property of One: \(\frac{a}{a}=1;\) Any number, except zero, divided by itself is one.
Simplify a Fraction
1. Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers first.
2. Simplify using the equivalent fractions property by dividing out common factors.
3. Multiply any remaining factors.
Simplify an Expression with a Fraction Bar
1. Simplify the expression in the numerator. Simplify the expression in the denominator.
2. Simplify the fraction.

Visualize Fractions

Find Equivalent Fractions

In the following exercises, find three fractions equivalent to the given fraction. Show your work, using figures or algebra.

Try it.

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

Solution

\(\frac{6}{16},\frac{9}{24},\frac{12}{32}\) answers may vary

Try it.

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

Try it.

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

Solution

\(\frac{10}{18},\frac{15}{27},\frac{20}{36}\) answers may vary

Try it.

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

Simplify Fractions

In the following exercises, simplify.

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.

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

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, multiply.

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.

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

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.

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

Solution

\(-34\)

Try it.

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

Divide Fractions

In the following exercises, divide.

Try it.

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

Solution

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

Try it.

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

Try it.

\(-\ \frac{7}{9}\div (-\ \frac{7}{4})\)

Solution

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

Try it.

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

Try it.

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

Solution

\(\frac{33}{4x}\)

Try it.

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

Try it.

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

Solution

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

Try it.

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

Try it.

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

Solution

\(\frac{10u}{9v}\)

Try it.

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

Try it.

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

Solution

\(-10\)

Try it.

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

Try it.

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

Solution

\(-\ \frac{1}{16}\)

Try it.

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

In the following exercises, simplify.

Try it.

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

Solution

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

Try it.

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

Try it.

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

Solution

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

Try it.

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

Try it.

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

Solution

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

Try it.

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

Simplify Expressions Written with a Fraction Bar

In the following exercises, simplify.

Try it.

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

Solution

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

Try it.

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

Try it.

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

Solution

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

Try it.

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

Try it.

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

Solution

Try it.

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

Try it.

\(\frac{4\cdot 3}{6\cdot 6}\)

Solution

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

Try it.

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

Try it.

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

Solution

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

Try it.

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

Try it.

\(\frac{8\cdot 3+2\cdot 9}{14+3}\)

Solution

\(2\frac{8}{17}\)

Try it.

\(\frac{9\cdot 6-4\cdot 7}{22+3}\)

Try it.

\(\frac{5\cdot 6-3\cdot 4}{4\cdot 5-2\cdot 3}\)

Solution

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

Try it.

\(\frac{8\cdot 9-7\cdot 6}{5\cdot 6-9\cdot 2}\)

Try it.

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

Solution

\(-8\)

Try it.

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

Try it.

\(\frac{7\cdot 4-2(8-5)}{9\cdot 3-3\cdot 5}\)

Solution

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

Try it.

\(\frac{9\cdot 7-3(12-8)}{8\cdot 7-6\cdot 6}\)

Try it.

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

Solution

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

Try it.

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

Translate Phrases to Expressions with Fractions

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

Try it.

the quotient of r and the sum of s and 10

Solution

\(\frac{r}{s+10}\)

Try it.

the quotient of A and the difference of 3 and B

Try it.

the quotient of the difference of \(x\ \text{and}\ y,\ \text{and}\ -3\)

Solution

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

Try it.

the quotient of the sum of \(m\ \text{and}\ n,\ \text{and}\ 4q\)

Condensed — the full section is in OpenStax Elementary Algebra 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 three fractions equivalent to \(\frac{2}{5}.\)

Առաջարկել պատասխանը

To find a fraction equivalent to \(\frac{2}{5},\) we multiply the numerator and denominator by the same number. We can choose any number, except for zero. Let’s multiply them by 2, 3, and then 5.

So, \(\frac{4}{10},\frac{6}{15},\ \text{and}\ \frac{10}{25}\) are equivalent to \(\frac{2}{5}.\)
Find three fractions equivalent to \(\frac{3}{5}.\)

Առաջարկել պատասխանը

\(\frac{6}{10},\frac{9}{15},\frac{12}{20};\) answers may vary
Find three fractions equivalent to \(\frac{4}{5}.\)

Առաջարկել պատասխանը

\(\frac{8}{10},\frac{12}{15},\frac{16}{20};\) answers may vary
Simplify: \(-\ \frac{32}{56}.\)

Առաջարկել պատասխանը

\(-\ \frac{32}{56}\)
Rewrite the numerator and denominator showing the common factors.
Simplify using the equivalent fractions property. \(-\ \frac{4}{7}\)

Notice that the fraction \(-\ \frac{4}{7}\) is simplified because there are no more common factors.
Simplify: \(-\ \frac{42}{54}.\)

Առաջարկել պատասխանը

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

Առաջարկել պատասխանը

\(-\ \frac{5}{9}\)
Simplify: \(-\ \frac{210}{385}.\)
Simplify: \(-\ \frac{69}{120}.\)

Առաջարկել պատասխանը

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

Առաջարկել պատասխանը

\(-\ \frac{5}{8}\)
Simplify: \(\frac{5x}{5y}.\)

Առաջարկել պատասխանը

\(\frac{5x}{5y}\)
Rewrite showing the common factors, then divide out the common factors.
Simplify. \(\frac{x}{y}\)
Simplify: \(\frac{7x}{7y}.\)

Առաջարկել պատասխանը

\(\frac{x}{y}\)
Simplify: \(\frac{3a}{3b}.\)

Առաջարկել պատասխանը

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

Multiply: \(-\ \frac{11}{12}\cdot \frac{5}{7}.\)

Առաջարկել պատասխանը

The first step is to find the sign of the product. Since the signs are the different, the product is negative.

	\(-\ \frac{11}{12}\cdot \frac{5}{7}\)
Determine the sign of the product; multiply.	\(-\ \frac{11\cdot 5}{12\cdot 7}\)
Are there any common factors in the numerator and the demoninator? No.	\(-\ \frac{55}{84}\)

Multiply: \(-\ \frac{10}{28}\cdot \frac{8}{15}.\)

Առաջարկել պատասխանը

\(-\ \frac{4}{21}\)
Multiply: \(-\ \frac{9}{20}\cdot \frac{5}{12}.\)

Առաջարկել պատասխանը

\(-\ \frac{3}{16}\)
Multiply: \(-\ \frac{12}{5}(-20x).\)

Առաջարկել պատասխանը

Determine the sign of the product. The signs are the same, so the product is positive.
\(-\ \frac{12}{5}(-20x)\)
Write \(20x\) as a fraction. \(\frac{12}{5}(\frac{20x}{1})\)
Multiply.
Rewrite 20 to show the common factor 5 and divide it out.
Simplify. \(48x\)
Multiply: \(\frac{11}{3}(-9a).\)

Առաջարկել պատասխանը

\(-33a\)
Multiply: \(\frac{13}{7}(-14b).\)

Առաջարկել պատասխանը

\(-26b\)
Divide: \(-\ \frac{2}{3}\div \frac{n}{5}.\)

Առաջարկել պատասխանը

\(-\ \frac{2}{3}\div \frac{n}{5}\)
To divide, multiply the first fraction by the reciprocal of the second. \(-\ \frac{2}{3}\cdot \frac{5}{n}\)
Multiply. \(-\ \frac{10}{3n}\)
Divide: \(-\ \frac{3}{5}\div \frac{p}{7}.\)

Առաջարկել պատասխանը

\(-\ \frac{21}{5p}\)
Divide: \(-\ \frac{5}{8}\div \frac{q}{3}.\)

Առաջարկել պատասխանը

\(-\ \frac{15}{8q}\)

Find the quotient: \(-\ \frac{7}{18}\div (-\ \frac{14}{27}).\)

Առաջարկել պատասխանը

	\(-\ \frac{7}{18}\div (-\ \frac{14}{27})\)
To divide, multiply the first fraction by the reciprocal of the second.	\(-\ \frac{7}{18}⋅-\ \frac{27}{14}\)
Determine the sign of the product, and then multiply..	\(\frac{7⋅27}{18⋅14}\)
Rewrite showing common factors.
Remove common factors.	\(\frac{3}{2⋅2}\)
Simplify.	\(\frac{3}{4}\)

Find the quotient: \(-\ \frac{7}{27}\div (-\ \frac{35}{36}).\)

Առաջարկել պատասխանը

\(\frac{4}{15}\)
Find the quotient: \(-\ \frac{5}{14}\div (-\ \frac{15}{28}).\)

Առաջարկել պատասխանը

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

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

Առաջարկել պատասխանը

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

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

Առաջարկել պատասխանը

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

Առաջարկել պատասխանը

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

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

Առաջարկել պատասխանը

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

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

Առաջարկել պատասխանը

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

Առաջարկել պատասխանը

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

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

Առաջարկել պատասխանը

	\(\frac{4-2(3)}{{2}^{2}+2}\)
Use the order of operations to simpliy the numerator and the denominator.	\(\frac{4-6}{4+2}\)
Simplify the numerator and the denominator.	\(\frac{-2}{6}\)
Simplify. A negative divided by a positive is negative.	\(-\ \frac{1}{3}\)

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

Առաջարկել պատասխանը

\(-\ \frac{3}{4}\)
Simplify: \(\frac{4-4(6)}{{3}^{2}+3}.\)

Առաջարկել պատասխանը

\(-\ \frac{5}{3}\)
Simplify: \(\frac{4(-3)+6(-2)}{-3(2)-2}.\)

Առաջարկել պատասխանը

\(\frac{4(-3)+6(-2)}{-3(2)-2}\)
Multiply. \(\frac{-12+(-12)}{-6-2}\)
Simplify. \(\frac{-24}{-8}\)
Divide. \(3\)
Simplify: \(\frac{8(-2)+4(-3)}{-5(2)+3}.\)

Առաջարկել պատասխանը

4
Simplify: \(\frac{7(-1)+9(-3)}{-5(3)-2}.\)

Առաջարկել պատասխանը

2
Translate the English phrase into an algebraic expression: the quotient of the difference of m and n, and p.

Առաջարկել պատասխանը

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

Առաջարկել պատասխանը

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

Առաջարկել պատասխանը

\(\frac{p+q}{r}\)
\(\frac{3}{8}\)

Առաջարկել պատասխանը

\(\frac{6}{16},\frac{9}{24},\frac{12}{32}\) answers may vary

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.

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

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

|x|

absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.

imaginary unit
i² = −1.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Visualize Fractions

Find equivalent fractions
Simplify fractions
Multiply fractions
Divide fractions
Simplify expressions written with a fraction bar
Translate phrases to expressions with fractions
Rewrite the numerator and denominator to show the common factors.
Simplify using the equivalent fractions property by dividing out common factors.

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

Փորձեք ինքներդ

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

Ցուցադրել Algebra

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