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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. 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 \(\frac{1\cdot 4}{2\cdot 4}=\frac{4}{8}.\)
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.”
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
- Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers first.
- Simplify using the equivalent fractions property by dividing out common factors.
- Multiply any remaining factors.
- Simplify an Expression with a Fraction Bar
- Simplify the expression in the numerator. Simplify the expression in the denominator.
- 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
9n
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
0
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.
이제 너 계산기는 이것을 해결하지 않지만, 그 조각은 계산 가능합니다. 아래의 하나를 시도하거나 자신의 것을 입력하십시오.
연습 (40)
먼저 종이에 각각을 시도해 보세요. 확인하기 위해 답을 드러내세요. 확인된 답은 각 단계의 솔버에서 열 수 있습니다.
-
Find three fractions equivalent to \(\frac{2}{5}.\)
답을 드러내세요
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 \(\frac{3}{5}.\)
답을 드러내세요
\(\frac{6}{10},\frac{9}{15},\frac{12}{20};\) answers may vary
-
Find three fractions equivalent to \(\frac{4}{5}.\)
답을 드러내세요
\(\frac{8}{10},\frac{12}{15},\frac{16}{20};\) answers may vary
-
Simplify: \(-\ \frac{32}{56}.\)
답을 드러내세요
\(-\ \frac{32}{56}\) 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: \(-\ \frac{42}{54}.\)
답을 드러내세요
\(-\ \frac{7}{9}\)
-
Simplify: \(-\ \frac{45}{81}.\)
답을 드러내세요
\(-\ \frac{5}{9}\)
-
Simplify: \(-\ \frac{210}{385}.\)
-
Simplify: \(-\ \frac{69}{120}.\)
답을 드러내세요
\(-\ \frac{23}{40}\)
-
Simplify: \(-\ \frac{120}{192}.\)
답을 드러내세요
\(-\ \frac{5}{8}\)
-
Simplify: \(\frac{5x}{5y}.\)
답을 드러내세요
\(\frac{5x}{5y}\) Rewrite showing the common factors, then divide out the common factors. Simplify. \(\frac{x}{y}\) -
Simplify: \(\frac{7x}{7y}.\)
답을 드러내세요
\(\frac{x}{y}\)
-
Simplify: \(\frac{3a}{3b}.\)
답을 드러내세요
\(\frac{a}{b}\)
-
Multiply: \(-\ \frac{11}{12}\cdot \frac{5}{7}.\)
답을 드러내세요
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: \(-\ \frac{10}{28}\cdot \frac{8}{15}.\)
답을 드러내세요
\(-\ \frac{4}{21}\)
-
Multiply: \(-\ \frac{9}{20}\cdot \frac{5}{12}.\)
답을 드러내세요
\(-\ \frac{3}{16}\)
-
Multiply: \(-\ \frac{12}{5}(-20x).\)
답을 드러내세요
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: \(\frac{11}{3}(-9a).\)
답을 드러내세요
\(-33a\)
-
Multiply: \(\frac{13}{7}(-14b).\)
답을 드러내세요
\(-26b\)
-
Divide: \(-\ \frac{2}{3}\div \frac{n}{5}.\)
답을 드러내세요
\(-\ \frac{2}{3}\div \frac{n}{5}\) To divide, multiply the first fraction by the reciprocal of the second. \(-\ \frac{2}{3}\cdot \frac{5}{n}\) Multiply. \(-\ \frac{10}{3n}\) -
Divide: \(-\ \frac{3}{5}\div \frac{p}{7}.\)
답을 드러내세요
\(-\ \frac{21}{5p}\)
-
Divide: \(-\ \frac{5}{8}\div \frac{q}{3}.\)
답을 드러내세요
\(-\ \frac{15}{8q}\)
-
Find the quotient: \(-\ \frac{7}{18}\div (-\ \frac{14}{27}).\)
답을 드러내세요
\(-\ \frac{7}{18}\div (-\ \frac{14}{27})\) 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: \(-\ \frac{7}{27}\div (-\ \frac{35}{36}).\)
답을 드러내세요
\(\frac{4}{15}\)
-
Find the quotient: \(-\ \frac{5}{14}\div (-\ \frac{15}{28}).\)
답을 드러내세요
\(\frac{2}{3}\)
-
Simplify: \(\frac{\frac{3}{4}}{\frac{5}{8}}.\)
답을 드러내세요
\(\frac{\frac{3}{4}}{\frac{5}{8}}\) 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: \(\frac{\frac{2}{3}}{\frac{5}{6}}.\)
답을 드러내세요
\(\frac{4}{5}\)
-
Simplify: \(\frac{\frac{3}{7}}{\frac{6}{11}}.\)
답을 드러내세요
\(\frac{11}{14}\)
-
Simplify: \(\frac{\frac{x}{2}}{\frac{xy}{6}}.\)
답을 드러내세요
\(\frac{\frac{x}{2}}{\frac{xy}{6}}\) 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: \(\frac{\frac{a}{8}}{\frac{ab}{6}}.\)
답을 드러내세요
\(\frac{3}{4b}\)
-
Simplify: \(\frac{\frac{p}{2}}{\frac{pq}{8}}.\)
답을 드러내세요
\(\frac{4}{q}\)
-
Simplify: \(\frac{4-2(3)}{{2}^{2}+2}.\)
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\(\frac{4-2(3)}{{2}^{2}+2}\) 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: \(\frac{6-3(5)}{{3}^{2}+3}.\)
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\(-\ \frac{3}{4}\)
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Simplify: \(\frac{4-4(6)}{{3}^{2}+3}.\)
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\(-\ \frac{5}{3}\)
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Simplify: \(\frac{4(-3)+6(-2)}{-3(2)-2}.\)
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\(\frac{4(-3)+6(-2)}{-3(2)-2}\) Multiply. \(\frac{-12+(-12)}{-6-2}\) Simplify. \(\frac{-24}{-8}\) Divide. \(3\) -
Simplify: \(\frac{8(-2)+4(-3)}{-5(2)+3}.\)
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4
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Simplify: \(\frac{7(-1)+9(-3)}{-5(3)-2}.\)
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2
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Translate the English phrase into an algebraic expression: the quotient of the difference of m and n, and p.
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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.
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\(\frac{a-b}{cd}\)
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Translate the English phrase into an algebraic expression: the quotient of the sum of \(p\) and \(q,\) and \(r\)
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\(\frac{p+q}{r}\)
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\(\frac{3}{8}\)
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\(\frac{6}{16},\frac{9}{24},\frac{12}{32}\) answers may vary
여기서 사용된 기호
기호를 탭하면 전체 정의, 이미지 및 각 문자의 의미를 확인할 수 있습니다.
어떻게: 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.
사람들이 묻는 질문
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.
이 페이지의 일부는 다음에서 변경되었습니다. OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0). 여기서 압축하고 다시 설명; 오류는 우리의. 소스가 비상업적 사용을 위해 라이선스되어 있기 때문에, 이 레슨은 무료이며, 아무것도 판매하지 않으며, 동일한 라이선스로 공유됩니다.
에 더 Algebra
Linear equationsQuadratic equationsSystems of equationsInequalitiesFactoringExpandingSimplifying expressionsFunctions and graphsExponential and logarithmic equationsPolynomial equationsAbsolute value