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Add and Subtract Fractions

Add or subtract fractions with a common denominator

Add or Subtract Fractions with a Common Denominator

When we multiplied fractions, we just multiplied the numerators and multiplied the denominators right straight across. To add or subtract fractions, they must have a common denominator.

Example

Try it.

Find the sum: \(\frac{x}{3}+\frac{2}{3}.\)

Solution
\(\frac{x}{3}+\frac{2}{3}\)
Add the numerators and place the sum over the common denominator.\(\frac{x+2}{3}\)
Example

Try it.

Find the difference: \(-\ \frac{23}{24}-\ \frac{13}{24}.\)

Solution

\(-\ \frac{23}{24}-\ \frac{13}{24}\)
Subtract the numerators and place the difference over the common denominator.\(\frac{-23-13}{24}\)
Simplify.\(\frac{-36}{24}\)
Simplify. Remember, \(-\ \frac{a}{b}=\frac{\text{-}a}{b}\).\(-\ \frac{3}{2}\)

Example

Try it.

Simplify: \(-\ \frac{10}{x}-\ \frac{4}{x}.\)

Solution

\(-\ \frac{10}{x}-\ \frac{4}{x}\)
Subtract the numerators and place the difference over the common denominator.\(\frac{-14}{x}\)
Rewrite with the sign in front of the fraction.\(-\ \frac{14}{x}\)

Now we will do an example that has both addition and subtraction.

Example

Try it.

Simplify: \(\frac{3}{8}+(-\ \frac{5}{8})-\ \frac{1}{8}.\)

Solution
Add and subtract fractions—do they have a common denominator? Yes. \(\frac{3}{8}+(-\ \frac{5}{8})-\ \frac{1}{8}\)
Add and subtract the numerators and place the difference over the common denominator.\(\frac{3+(-5)-1}{8}\)
Simplify left to right.\(\frac{-2-1}{8}\)
Simplify.\(-\ \frac{3}{8}\)

Add or Subtract Fractions with Different Denominators

As we have seen, to add or subtract fractions, their denominators must be the same. The least common denominator (LCD) of two fractions is the smallest number that can be used as a common denominator of the fractions. The LCD of the two fractions is the least common multiple (LCM) of their denominators.

After we find the least common denominator of two fractions, we convert the fractions to equivalent fractions with the LCD. Putting these steps together allows us to add and subtract fractions because their denominators will be the same!

How to Add or Subtract Fractions

Try it.

Add: \(\frac{7}{12}+\frac{5}{18}.\)

Solution

When finding the equivalent fractions needed to create the common denominators, there is a quick way to find the number we need to multiply both the numerator and denominator. This method works if we found the LCD by factoring into primes.

Look at the factors of the LCD and then at each column above those factors. The “missing” factors of each denominator are the numbers we need.

In , the LCD, 36, has two factors of 2 and two factors of \(3.\)

The numerator 12 has two factors of 2 but only one of 3—so it is “missing” one 3—we multiply the numerator and denominator by 3.

The numerator 18 is missing one factor of 2—so we multiply the numerator and denominator by 2.

Example

Try it.

Subtract: \(\frac{7}{15}-\ \frac{19}{24}.\)

Solution

Do the fractions have a common denominator? No, so we need to find the LCD.

Find the LCD.\(\\)
Notice, 15 is “missing” three factors of 2 and 24 is “missing” the 5 from the factors of the LCD. So we multiply 8 in the first fraction and 5 in the second fraction to get the LCD.
Rewrite as equivalent fractions with the LCD.
Simplify.
Subtract.\(\ -\ \frac{39}{120}\)
Check to see if the answer can be simplified.\(\ -\ \frac{13⋅3}{40⋅3}\)
Both 39 and 120 have a factor of 3.
Simplify.\(\ -\ \frac{13}{40}\)

Do not simplify the equivalent fractions! If you do, you’ll get back to the original fractions and lose the common denominator!

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

Use the Order of Operations to Simplify Complex Fractions

We have seen that a complex fraction is a fraction in which the numerator or denominator contains a fraction. The fraction bar indicates division. We simplified the complex fraction \(\frac{\frac{3}{4}}{\frac{5}{8}}\) by dividing \(\frac{3}{4}\) by \(\frac{5}{8}.\)

Now we’ll look at complex fractions where the numerator or denominator contains an expression that can be simplified. So we first must completely simplify the numerator and denominator separately using the order of operations. Then we divide the numerator by the denominator.

How to Simplify Complex Fractions

Try it.

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

Solution
Example

Try it.

Simplify: \(\frac{\frac{1}{2}+\frac{2}{3}}{\frac{3}{4}-\ \frac{1}{6}}.\)

Solution

It may help to put parentheses around the numerator and the denominator.

\(\frac{(\frac{1}{2}+\frac{2}{3})}{(\frac{3}{4}-\ \frac{1}{6})}\)
Simplify the numerator (LCD = 6) and simplify the denominator (LCD = 12).\(\frac{(\frac{3}{6}+\frac{4}{6})}{(\frac{9}{12}-\ \frac{2}{12})}\)
Simplify.\(\frac{(\frac{7}{6})}{(\frac{7}{12})}\)
Divide the numerator by the denominator.\(\frac{7}{6}\div \frac{7}{12}\)
Simplify.\(\frac{7}{6}\cdot \frac{12}{7}\)
Divide out common factors.\(\frac{7\cdot 6\cdot 2}{6\cdot 7}\)
Simplify.\(2\)

Evaluate Variable Expressions with Fractions

We have evaluated expressions before, but now we can evaluate expressions with fractions. Remember, to evaluate an expression, we substitute the value of the variable into the expression and then simplify.

Example

Try it.

Evaluate \(x+\frac{1}{3}\) when ⓐ \(x=-\ \frac{1}{3}\) ⓑ \(x=-\ \frac{3}{4}.\)

Solution
  1. ⓐ To evaluate \(x+\frac{1}{3}\) when \(x=-\ \frac{1}{3},\) substitute \(-\ \frac{1}{3}\) for \(x\) in the expression.
     
    Simplify.\(\\)0


  2. ⓑ To evaluate \(x+\frac{1}{3}\) when \(x=-\ \frac{3}{4},\) we substitute \(-\ \frac{3}{4}\) for x in the expression.
    Rewrite as equivalent fractions with the LCD, 12.
    Simplify.
    Add.\(-\ \frac{5}{12}\)
Example

Try it.

Evaluate \(-\ \frac{5}{6}-y\) when \(y=-\ \frac{2}{3}.\)

Solution
Rewrite as equivalent fractions with the LCD, 6.         
Subtract.
Simplify.\(-\ \frac{1}{6}\)
Example

Try it.

Evaluate \(2{x}^{2}y\) when \(x=\frac{1}{4}\) and \(y=-\ \frac{2}{3}.\)

Solution

Substitute the values into the expression.

\(2{x}^{2}y\)
Simplify exponents first.\(2(\frac{1}{16})(-\ \frac{2}{3})\)
Multiply. Divide out the common factors. Notice we write 16 as \(2⋅2⋅4\) to make it easy to remove common factors.\(-\ \frac{2⋅1⋅2}{2⋅2⋅4⋅3}\)
Simplify.\(-\ \frac{1}{12}\)

The next example will have only variables, no constants.

Example

Try it.

Evaluate \(\frac{p+q}{r}\) when \(p=-4,q=-2,\ \text{and}\ r=8.\)

Solution

To evaluate \(\frac{p+q}{r}\) when \(p=-4,q=-2,\ \text{and}\ r=8,\) we substitute the values into the expression.

\(\frac{p+q}{r}\)
Add in the numerator first.\(\frac{-6}{8}\)
Simplify.\(-\ \frac{3}{4}\)

Key Concepts

  • Fraction Addition and Subtraction: If \(a,b,\ \text{and}\ c\) are numbers where \(c\ne 0,\) then
    \(\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}\) and \(\frac{a}{c}-\ \frac{b}{c}=\frac{a-b}{c}.\)
    To add or subtract fractions, add or subtract the numerators and place the result over the common denominator.
  • Strategy for Adding or Subtracting Fractions
    1. Do they have a common denominator?
      Yes—go to step 2.
      No—Rewrite each fraction with the LCD (Least Common Denominator). Find the LCD. Change each fraction into an equivalent fraction with the LCD as its denominator.
    2. Add or subtract the fractions.
    3. Simplify, if possible. To multiply or divide fractions, an LCD IS NOT needed. To add or subtract fractions, an LCD IS needed.
  • Simplify Complex Fractions
    1. Simplify the numerator.
    2. Simplify the denominator.
    3. Divide the numerator by the denominator. Simplify if possible.

Add and Subtract Fractions

Add and Subtract Fractions with a Common Denominator

In the following exercises, add.

Try it.

\(\frac{6}{13}+\frac{5}{13}\)

Solution

\(\frac{11}{13}\)

Try it.

\(\frac{4}{15}+\frac{7}{15}\)

Try it.

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

Solution

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

Try it.

\(\frac{8}{q}+\frac{6}{q}\)

Try it.

\(-\ \frac{3}{16}+(-\ \frac{7}{16})\)

Solution

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

Try it.

\(-\ \frac{5}{16}+(-\ \frac{9}{16})\)

Try it.

\(-\ \frac{8}{17}+\frac{15}{17}\)

Solution

\(\frac{7}{17}\)

Try it.

\(-\ \frac{9}{19}+\frac{17}{19}\)

Try it.

\(\frac{6}{13}+(-\ \frac{10}{13})+(-\ \frac{12}{13})\)

Solution

\(-\ \frac{16}{13}\)

Try it.

\(\frac{5}{12}+(-\ \frac{7}{12})+(-\ \frac{11}{12})\)

In the following exercises, subtract.

Try it.

\(\frac{11}{15}-\ \frac{7}{15}\)

Solution

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

Try it.

\(\frac{9}{13}-\ \frac{4}{13}\)

Try it.

\(\frac{11}{12}-\ \frac{5}{12}\)

Solution

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

Try it.

\(\frac{7}{12}-\ \frac{5}{12}\)

Try it.

\(\frac{19}{21}-\ \frac{4}{21}\)

Solution

\(\frac{5}{7}\)

Try it.

\(\frac{17}{21}-\ \frac{8}{21}\)

Try it.

\(\frac{5y}{8}-\ \frac{7}{8}\)

Solution

\(\frac{5y-7}{8}\)

Try it.

\(\frac{11z}{13}-\ \frac{8}{13}\)

Try it.

\(-\ \frac{23}{u}-\ \frac{15}{u}\)

Solution

\(-\ \frac{38}{u}\)

Try it.

\(-\ \frac{29}{v}-\ \frac{26}{v}\)

Try it.

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

Solution

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

Try it.

\(-\ \frac{3}{7}-(-\ \frac{5}{7})\)

Try it.

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

Solution

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

Try it.

\(-\ \frac{8}{11}-(-\ \frac{5}{11})\)

Mixed Practice

In the following exercises, simplify.

Try it.

\(-\ \frac{5}{18}\cdot \frac{9}{10}\)

Solution

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

Try it.

\(-\ \frac{3}{14}\cdot \frac{7}{12}\)

Try it.

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

Solution

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

Try it.

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

Try it.

\(-\ \frac{7}{24}+\frac{2}{24}\)

Solution

\(-\ \frac{5}{24}\)

Try it.

\(-\ \frac{5}{18}+\frac{1}{18}\)

Try it.

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

Solution

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

Try it.

\(\frac{7}{12}\div \frac{9}{28}\)

Add or Subtract Fractions with Different Denominators

In the following exercises, add or subtract.

Try it.

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

Solution

\(\frac{9}{14}\)

Try it.

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

Try it.

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

Solution

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

Try it.

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

Try it.

\(\frac{7}{12}+\frac{5}{8}\)

Solution

\(\frac{29}{24}\)

Try it.

\(\frac{5}{12}+\frac{3}{8}\)

Try it.

\(\frac{7}{12}-\ \frac{9}{16}\)

Solution

\(\frac{1}{48}\)

Try it.

\(\frac{7}{16}-\ \frac{5}{12}\)

Try it.

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

Solution

\(\frac{7}{24}\)

Try it.

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

Try it.

\(-\ \frac{11}{30}+\frac{27}{40}\)

Solution

\(\frac{37}{120}\)

Try it.

\(-\ \frac{9}{20}+\frac{17}{30}\)

Try it.

\(-\ \frac{13}{30}+\frac{25}{42}\)

Solution

\(\frac{17}{105}\)

Try it.

\(-\ \frac{23}{30}+\frac{5}{48}\)

Try it.

\(-\ \frac{39}{56}-\ \frac{22}{35}\)

Solution

\(-\ \frac{53}{40}\)

Try it.

\(-\ \frac{33}{49}-\ \frac{18}{35}\)

Try it.

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

Solution

\(\frac{1}{12}\)

Try it.

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

Try it.

\(1+\frac{7}{8}\)

Solution

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

Try it.

\(1-\ \frac{3}{10}\)

Try it.

\(\frac{x}{3}+\frac{1}{4}\)

Solution

\(\frac{4x+3}{12}\)

Try it.

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

Try it.

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

Solution

\(\frac{5y-12}{20}\)

Try it.

\(\frac{x}{5}-\ \frac{1}{4}\)

Mixed Practice

In the following exercises, simplify.

Try it.

ⓐ \(\frac{2}{3}+\frac{1}{6}\) ⓑ \(\frac{2}{3}\div \frac{1}{6}\)

Solution

ⓐ \(\frac{5}{6}\) ⓑ 4

Try it.

ⓐ \(-\ \frac{2}{5}-\ \frac{1}{8}\) ⓑ \(-\ \frac{2}{5}\cdot \frac{1}{8}\)

Try it.

ⓐ \(\frac{5n}{6}\div \frac{8}{15}\) ⓑ \(\frac{5n}{6}-\ \frac{8}{15}\)

Solution

ⓐ \(\frac{25n}{16}\) ⓑ \(\frac{25n-16}{30}\)

Try it.

ⓐ \(\frac{3a}{8}\div \frac{7}{12}\) ⓑ \(\frac{3a}{8}-\ \frac{7}{12}\)

Try it.

\(-\ \frac{3}{8}\div (-\ \frac{3}{10})\)

Solution

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

Try it.

\(-\ \frac{5}{12}\div (-\ \frac{5}{9})\)

Try it.

\(-\ \frac{3}{8}+\frac{5}{12}\)

Solution

\(\frac{1}{24}\)

Try it.

\(-\ \frac{1}{8}+\frac{7}{12}\)

Try it.

\(\frac{5}{6}-\ \frac{1}{9}\)

Solution

\(\frac{13}{18}\)

Try it.

\(\frac{5}{9}-\ \frac{1}{6}\)

Try it.

\(-\ \frac{7}{15}-\ \frac{y}{4}\)

Solution

\(\frac{-28-15y}{60}\)

Try it.

\(-\ \frac{3}{8}-\ \frac{x}{11}\)

Try it.

\(\frac{11}{12a}\cdot \frac{9a}{16}\)

Solution

\(\frac{33}{64}\)

Try it.

\(\frac{10y}{13}\cdot \frac{8}{15y}\)

Use the Order of Operations to Simplify Complex Fractions

In the following exercises, simplify.

Try it.

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

Solution

54

Try it.

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

Try it.

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

Solution

\(\frac{49}{25}\)

Try it.

\(\frac{{(\frac{3}{4})}^{2}}{{(\frac{5}{8})}^{2}}\)

Try it.

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

Solution

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

Try it.

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

Try it.

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

Solution

\(\frac{5}{21}\)

Try it.

\(\frac{\frac{3}{4}-\ \frac{3}{5}}{\frac{1}{4}+\frac{2}{5}}\)

Try it.

\(\frac{1}{2}+\frac{2}{3}\cdot \frac{5}{12}\)

Solution

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

Try it.

\(\frac{1}{3}+\frac{2}{5}\cdot \frac{3}{4}\)

Try it.

\(1-\ \frac{3}{5}\div \frac{1}{10}\)

Solution

\(-5\)

Try it.

\(1-\ \frac{5}{6}\div \frac{1}{12}\)

Try it.

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

Solution

\(\frac{19}{12}\)

Try it.

\(\frac{2}{3}+\frac{1}{4}+\frac{3}{5}\)

Try it.

\(\frac{3}{8}-\ \frac{1}{6}+\frac{3}{4}\)

Solution

\(\frac{23}{24}\)

Try it.

\(\frac{2}{5}+\frac{5}{8}-\ \frac{3}{4}\)

Try it.

\(12(\frac{9}{20}-\ \frac{4}{15})\)

Solution

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

Try it.

\(8(\frac{15}{16}-\ \frac{5}{6})\)

Try it.

\(\frac{\frac{5}{8}+\frac{1}{6}}{\frac{19}{24}}\)

Solution

1

Try it.

\(\frac{\frac{1}{6}+\frac{3}{10}}{\frac{14}{30}}\)

Try it.

\((\frac{5}{9}+\frac{1}{6})\div (\frac{2}{3}-\ \frac{1}{2})\)

Solution

\(\frac{13}{3}\)

Try it.

\((\frac{3}{4}+\frac{1}{6})\div (\frac{5}{8}-\ \frac{1}{3})\)

Evaluate Variable Expressions with Fractions

In the following exercises, evaluate.

Try it.

\(x+(-\ \frac{5}{6})\) when
ⓐ \(x=\frac{1}{3}\)
ⓑ \(x=-\ \frac{1}{6}\)

Solution

ⓐ \(-\ \frac{1}{2}\) ⓑ \(-1\)

Try it.

\(x+(-\ \frac{11}{12})\) when
ⓐ \(x=\frac{11}{12}\)ⓑ \(x=\frac{3}{4}\)

Try it.

\(x-\ \frac{2}{5}\) when ⓐ \(x=\frac{3}{5}\) ⓑ \(x=-\ \frac{3}{5}\)

Solution

ⓐ \(\frac{1}{5}\) ⓑ \(-1\)

Try it.

\(x-\ \frac{1}{3}\) when ⓐ \(x=\frac{2}{3}\) ⓑ \(x=-\ \frac{2}{3}\)

Try it.

\(\frac{7}{10}-w\) when ⓐ \(w=\frac{1}{2}\) ⓑ \(w=-\ \frac{1}{2}\)

Solution

ⓐ \(\frac{1}{5}\) ⓑ \(\frac{6}{5}\)

Try it.

\(\frac{5}{12}-w\) when ⓐ \(w=\frac{1}{4}\) ⓑ \(w=-\ \frac{1}{4}\)

Try it.

\(2{x}^{2}{y}^{3}\) when \(x=-\ \frac{2}{3}\) and \(y=-\ \frac{1}{2}\)

Solution

\(-\ \frac{1}{9}\)

Try it.

\(8{u}^{2}{v}^{3}\) when \(u=-\ \frac{3}{4}\) and \(v=-\ \frac{1}{2}\)

Try it.

\(\frac{a+b}{a-b}\) when \(a=-3,b=8\)

Solution

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

Try it.

\(\frac{r-s}{r+s}\) when \(r=10,s=-5\)

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.

  1. Find the sum: \(\frac{x}{3}+\frac{2}{3}.\)

    Sýna svarið
    \(\frac{x}{3}+\frac{2}{3}\)
    Add the numerators and place the sum over the common denominator.\(\frac{x+2}{3}\)

  2. Find the sum: \(\frac{x}{4}+\frac{3}{4}.\)

    Sýna svarið

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

  3. Find the sum: \(\frac{y}{8}+\frac{5}{8}.\)

    Sýna svarið

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

  4. Find the difference: \(-\ \frac{23}{24}-\ \frac{13}{24}.\)

    Sýna svarið

    \(-\ \frac{23}{24}-\ \frac{13}{24}\)
    Subtract the numerators and place the difference over the common denominator.\(\frac{-23-13}{24}\)
    Simplify.\(\frac{-36}{24}\)
    Simplify. Remember, \(-\ \frac{a}{b}=\frac{\text{-}a}{b}\).\(-\ \frac{3}{2}\)

  5. Find the difference: \(-\ \frac{19}{28}-\ \frac{7}{28}.\)

    Sýna svarið

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

  6. Find the difference: \(-\ \frac{27}{32}-\ \frac{1}{32}.\)

    Sýna svarið

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

  7. Simplify: \(-\ \frac{10}{x}-\ \frac{4}{x}.\)

    Sýna svarið

    \(-\ \frac{10}{x}-\ \frac{4}{x}\)
    Subtract the numerators and place the difference over the common denominator.\(\frac{-14}{x}\)
    Rewrite with the sign in front of the fraction.\(-\ \frac{14}{x}\)

  8. Find the difference: \(-\ \frac{9}{x}-\ \frac{7}{x}.\)

    Sýna svarið

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

  9. Find the difference: \(-\ \frac{17}{a}-\ \frac{5}{a}.\)

    Sýna svarið

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

  10. Simplify: \(\frac{3}{8}+(-\ \frac{5}{8})-\ \frac{1}{8}.\)

    Sýna svarið
    Add and subtract fractions—do they have a common denominator? Yes. \(\frac{3}{8}+(-\ \frac{5}{8})-\ \frac{1}{8}\)
    Add and subtract the numerators and place the difference over the common denominator.\(\frac{3+(-5)-1}{8}\)
    Simplify left to right.\(\frac{-2-1}{8}\)
    Simplify.\(-\ \frac{3}{8}\)

  11. Simplify: \(-\frac{2}{9}+(-\ \frac{4}{9})-\ \frac{7}{9}.\)

    Sýna svarið

    \(-1\)

  12. Simplify: \(\frac{5}{9}+(-\ \frac{4}{9})-\ \frac{7}{9}.\)

    Sýna svarið

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

  13. Add: \(\frac{7}{12}+\frac{5}{18}.\)

  14. Add: \(\frac{7}{12}+\frac{11}{15}.\)

    Sýna svarið

    \(\frac{79}{60}\)

  15. Add: \(\frac{13}{15}+\frac{17}{20}.\)

    Sýna svarið

    \(\frac{103}{60}\)

  16. Subtract: \(\frac{7}{15}-\ \frac{19}{24}.\)

    Sýna svarið

    Do the fractions have a common denominator? No, so we need to find the LCD.

    Find the LCD.\(\\)
    Notice, 15 is “missing” three factors of 2 and 24 is “missing” the 5 from the factors of the LCD. So we multiply 8 in the first fraction and 5 in the second fraction to get the LCD.
    Rewrite as equivalent fractions with the LCD.
    Simplify.
    Subtract.\(\ -\ \frac{39}{120}\)
    Check to see if the answer can be simplified.\(\ -\ \frac{13⋅3}{40⋅3}\)
    Both 39 and 120 have a factor of 3.
    Simplify.\(\ -\ \frac{13}{40}\)

    Do not simplify the equivalent fractions! If you do, you’ll get back to the original fractions and lose the common denominator!

  17. Subtract: \(\frac{13}{24}-\ \frac{17}{32}.\)

    Sýna svarið

    \(\frac{1}{96}\)

  18. Subtract: \(\frac{21}{32}-\ \frac{9}{28}.\)

    Sýna svarið

    \(\frac{75}{224}\)

  19. Add: \(\frac{3}{5}+\frac{x}{8}.\)

    Sýna svarið

    The fractions have different denominators.

    \(\\)
    Find the LCD.\(\\)
    Rewrite as equivalent fractions with the LCD.\(\\)
    Simplify.\(\\)
    Add.\(\\)

    Remember, we can only add like terms: 24 and 5x are not like terms.

  20. Add: \(\frac{y}{6}+\frac{7}{9}.\)

    Sýna svarið

    \(\frac{3y+14}{18}\)

  21. Add: \(\frac{x}{6}+\frac{7}{15}.\)

    Sýna svarið

    \(\frac{5x+14}{30}\)

  22. Simplify: ⓐ \(\frac{5x}{6}-\ \frac{3}{10}\) ⓑ \(\frac{5x}{6}\cdot \frac{3}{10}.\)

    Sýna svarið

    First ask, “What is the operation?” Once we identify the operation that will determine whether we need a common denominator. Remember, we need a common denominator to add or subtract, but not to multiply or divide.

    ⓐ What is the operation? The operation is subtraction.
    Do the fractions have a common denominator? No.
    Rewrite each fraction as an equivalent fraction with the LCD.
    Subtract the numerators and place the difference over the common denominators.
    Simplify, if possible.
    There are no common factors. The fraction is simplified.     		\(\begin{array}{l}\frac{5x}{6}-\ \frac{3}{10} \\ \frac{5x\cdot 5}{6\cdot 5}-\ \frac{3\cdot 3}{10\cdot 3} \\ \frac{25x}{30}-\ \frac{9}{30} \\ \frac{25x-9}{30}\end{array}\)
    ⓑ What is the operation? Multiplication.
    To multiply fractions, multiply the numerators and multiply the denominators.
    Rewrite, showing common factors. Remove common factors.
    Simplify.     		\(\begin{array}{l}\frac{5x}{6}\cdot \frac{3}{10} \\ \frac{5x\cdot 3}{6\cdot 10} \\ \frac{5x\cdot 3}{2\cdot 3\cdot 2\cdot 5} \\ \frac{x}{4}\end{array}\)

    Notice we needed an LCD to add \(\frac{5x}{6}-\ \frac{3}{10},\) but not to multiply \(\frac{5x}{6}\cdot \frac{3}{10}.\)

  23. Simplify: ⓐ \(\frac{3a}{4}-\ \frac{8}{9}\) ⓑ \(\frac{3a}{4}\cdot \frac{8}{9}.\)

    Sýna svarið

    ⓐ \(\frac{27a-32}{36}\) ⓑ \(\frac{2a}{3}\)

  24. Simplify: ⓐ \(\frac{4k}{5}-\ \frac{1}{6}\) ⓑ \(\frac{4k}{5}\cdot \frac{1}{6}.\)

    Sýna svarið

    ⓐ \(\frac{24k-5}{30}\) ⓑ \(\frac{2k}{15}\)

  25. Simplify: \(\frac{{(\frac{1}{2})}^{2}}{4+{3}^{2}}.\)

  26. Simplify: \(\frac{{(\frac{1}{3})}^{2}}{{2}^{3}+2}.\)

    Sýna svarið

    \(\frac{1}{90}\)

  27. Simplify: \(\frac{1+{4}^{2}}{{(\frac{1}{4})}^{2}}.\)

    Sýna svarið

    \(272\)

  28. Simplify: \(\frac{\frac{1}{2}+\frac{2}{3}}{\frac{3}{4}-\ \frac{1}{6}}.\)

    Sýna svarið

    It may help to put parentheses around the numerator and the denominator.

    \(\frac{(\frac{1}{2}+\frac{2}{3})}{(\frac{3}{4}-\ \frac{1}{6})}\)
    Simplify the numerator (LCD = 6) and simplify the denominator (LCD = 12).\(\frac{(\frac{3}{6}+\frac{4}{6})}{(\frac{9}{12}-\ \frac{2}{12})}\)
    Simplify.\(\frac{(\frac{7}{6})}{(\frac{7}{12})}\)
    Divide the numerator by the denominator.\(\frac{7}{6}\div \frac{7}{12}\)
    Simplify.\(\frac{7}{6}\cdot \frac{12}{7}\)
    Divide out common factors.\(\frac{7\cdot 6\cdot 2}{6\cdot 7}\)
    Simplify.\(2\)

  29. Simplify: \(\frac{\frac{1}{3}+\frac{1}{2}}{\frac{3}{4}-\ \frac{1}{3}}.\)

    Sýna svarið

    2

  30. Simplify: \(\frac{\frac{2}{3}-\ \frac{1}{2}}{\frac{1}{4}+\frac{1}{3}}.\)

    Sýna svarið

    \(\frac{2}{7}\)

  31. Evaluate \(x+\frac{1}{3}\) when ⓐ \(x=-\ \frac{1}{3}\) ⓑ \(x=-\ \frac{3}{4}.\)

    Sýna svarið
    1. ⓐ To evaluate \(x+\frac{1}{3}\) when \(x=-\ \frac{1}{3},\) substitute \(-\ \frac{1}{3}\) for \(x\) in the expression.
       
      Simplify.\(\\)0


    2. ⓑ To evaluate \(x+\frac{1}{3}\) when \(x=-\ \frac{3}{4},\) we substitute \(-\ \frac{3}{4}\) for x in the expression.
      Rewrite as equivalent fractions with the LCD, 12.
      Simplify.
      Add.\(-\ \frac{5}{12}\)

  32. Evaluate \(x+\frac{3}{4}\) when ⓐ \(x=-\ \frac{7}{4}\) ⓑ \(x=-\ \frac{5}{4}.\)

    Sýna svarið

    ⓐ \(-1\) ⓑ \(-\ \frac{1}{2}\)

  33. Evaluate \(y+\frac{1}{2}\) when ⓐ \(y=\frac{2}{3}\) ⓑ \(y=-\ \frac{3}{4}.\)

    Sýna svarið

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

  34. Evaluate \(-\ \frac{5}{6}-y\) when \(y=-\ \frac{2}{3}.\)

    Sýna svarið
    Rewrite as equivalent fractions with the LCD, 6.         
    Subtract.
    Simplify.\(-\ \frac{1}{6}\)

  35. Evaluate \(-\ \frac{1}{2}-y\) when \(y=-\ \frac{1}{4}.\)

    Sýna svarið

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

  36. Evaluate \(-\ \frac{3}{8}-y\) when \(y=-\ \frac{5}{2}.\)

    Sýna svarið

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

  37. Evaluate \(2{x}^{2}y\) when \(x=\frac{1}{4}\) and \(y=-\ \frac{2}{3}.\)

    Sýna svarið

    Substitute the values into the expression.

    \(2{x}^{2}y\)
    Simplify exponents first.\(2(\frac{1}{16})(-\ \frac{2}{3})\)
    Multiply. Divide out the common factors. Notice we write 16 as \(2⋅2⋅4\) to make it easy to remove common factors.\(-\ \frac{2⋅1⋅2}{2⋅2⋅4⋅3}\)
    Simplify.\(-\ \frac{1}{12}\)

  38. Evaluate \(3a{b}^{2}\) when \(a=-\ \frac{2}{3}\) and \(b=-\ \frac{1}{2}.\)

    Sýna svarið

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

  39. Evaluate \(4{c}^{3}d\) when \(c=-\ \frac{1}{2}\) and \(d=-\ \frac{4}{3}.\)

    Sýna svarið

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

  40. Evaluate \(\frac{p+q}{r}\) when \(p=-4,q=-2,\ \text{and}\ r=8.\)

    Sýna svarið

    To evaluate \(\frac{p+q}{r}\) when \(p=-4,q=-2,\ \text{and}\ r=8,\) we substitute the values into the expression.

    \(\frac{p+q}{r}\)
    Add in the numerator first.\(\frac{-6}{8}\)
    Simplify.\(-\ \frac{3}{4}\)

Symbols used here

\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.
i
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: Add and Subtract Fractions

  1. Add or subtract fractions with a common denominator
  2. Add or subtract fractions with different denominators
  3. Use the order of operations to simplify complex fractions
  4. Evaluate variable expressions with fractions
  5. Do they have a common denominator?
  6. Yes—go to step 2.
  7. No—rewrite each fraction with the LCD (least common denominator). Find the LCD. Change each fraction into an equivalent fraction with the LCD as its denominator.
  8. Add or subtract the fractions.

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.

Prófaðu þitt eigið

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.

Meira í Algebra

Linear equationsQuadratic equationsSystems of equationsInequalitiesFactoringExpandingSimplifying expressionsFunctions and graphsExponential and logarithmic equationsPolynomial equationsAbsolute value