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Add and Subtract Fractions with Different Denominators

Find the least common denominator (LCD)

Find the Least Common Denominator

In the previous section, we explained how to add and subtract fractions with a common denominator. But how can we add and subtract fractions with unlike denominators?

Let’s think about coins again. Can you add one quarter and one dime? You could say there are two coins, but that’s not very useful. To find the total value of one quarter plus one dime, you change them to the same kind of unit—cents. One quarter equals \(25\) cents and one dime equals \(10\) cents, so the sum is \(35\) cents. See .

Similarly, when we add fractions with different denominators we have to convert them to equivalent fractions with a common denominator. With the coins, when we convert to cents, the denominator is \(100.\) Since there are \(100\) cents in one dollar, \(25\) cents is \(\frac{25}{100}\) and \(10\) cents is \(\frac{10}{100}.\) So we add \(\frac{25}{100}+\frac{10}{100}\) to get \(\frac{35}{100},\) which is \(35\) cents.

You have practiced adding and subtracting fractions with common denominators. Now let’s see what you need to do with fractions that have different denominators.

First, we will use fraction tiles to model finding the common denominator of \(\frac{1}{2}\) and \(\frac{1}{3}.\)

We’ll start with one \(\frac{1}{2}\) tile and \(\frac{1}{3}\) tile. We want to find a common fraction tile that we can use to match both \(\frac{1}{2}\) and \(\frac{1}{3}\) exactly.

If we try the \(\frac{1}{4}\) pieces, \(2\) of them exactly match the \(\frac{1}{2}\) piece, but they do not exactly match the \(\frac{1}{3}\) piece.

Example

Try it.

Find the LCD for the fractions \(\frac{7}{12}\) and \(\frac{5}{18}.\)

Solution
Factor each denominator into its primes.
List the primes of 12 and the primes of 18 lining them up in columns when possible.
Bring down the columns.
Multiply the factors. The product is the LCM.LCM\(=36\)
The LCM of 12 and 18 is 36, so the LCD of \(\frac{7}{12}\) and \(\frac{5}{18}\) is 36.LCD of \(\frac{7}{12}\) and \(\frac{5}{18}\) is 36.

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

Convert Fractions to Equivalent Fractions with the LCD

Earlier, we used fraction tiles to see that the LCD of \(\frac{1}{4}\) when \(\frac{1}{6}\) is \(12.\) We saw that three \(\frac{1}{12}\) pieces exactly covered \(\frac{1}{4}\) and two \(\frac{1}{12}\) pieces exactly covered \(\frac{1}{6},\) so

\[\frac{1}{4}=\frac{3}{12}\text{ and }\frac{1}{6}=\frac{2}{12}.\]

We say that \(\frac{1}{4}\) and \(\frac{3}{12}\) are equivalent fractions and also that \(\frac{1}{6}\) and \(\frac{2}{12}\) are equivalent fractions.

We can use the Equivalent Fractions Property to algebraically change a fraction to an equivalent one. Remember, two fractions are equivalent if they have the same value. The Equivalent Fractions Property is repeated below for reference.

To add or subtract fractions with different denominators, we will first have to convert each fraction to an equivalent fraction with the LCD. Let’s see how to change \(\frac{1}{4}\) and \(\frac{1}{6}\) to equivalent fractions with denominator \(12\) without using models.

Example

Try it.

Convert \(\frac{1}{4}\) and \(\frac{1}{6}\) to equivalent fractions with denominator \(12,\) their LCD.

Solution
Find the LCD.The LCD of \(\frac{1}{4}\) and \(\frac{1}{6}\) is 12.
Find the number to multiply 4 to get 12.
Find the number to multiply 6 to get 12.
Use the Equivalent Fractions Property to convert each fraction to an equivalent fraction with the LCD, multiplying both the numerator and denominator of each fraction by the same number.
Simplify the numerators and denominators.

We do not reduce the resulting fractions. If we did, we would get back to our original fractions and lose the common denominator.

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

Add and Subtract Fractions with Different Denominators

Once we have converted two fractions to equivalent forms with common denominators, we can add or subtract them by adding or subtracting the numerators.

Example

Try it.

Add: \(\frac{1}{2}+\frac{1}{3}.\)

Solution
\(\frac{1}{2}+\frac{1}{3}\)
Find the LCD of 2, 3.
Change into equivalent fractions with the LCD 6.
Simplify the numerators and denominators.\(\frac{3}{6}+\frac{2}{6}\)
Add.\(\frac{5}{6}\)

Remember, always check to see if the answer can be simplified. Since \(5\) and \(6\) have no common factors, the fraction \(\frac{5}{6}\) cannot be reduced.

Example

Try it.

Subtract: \(\frac{1}{2}-(-\frac{1}{4}).\)

Solution
\(\frac{1}{2}-(-\frac{1}{4})\)
Find the LCD of 2 and 4.
Rewrite as equivalent fractions using the LCD 4.
Simplify the first fraction.\(\frac{2}{4}-(-\frac{1}{4})\)
Subtract.\(\frac{2\ -\ (-1)}{4}\)
Simplify.\(\frac{3}{4}\)

One of the fractions already had the least common denominator, so we only had to convert the other fraction.

Example

Try it.

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

Solution
\(\frac{7}{12}+\frac{5}{18}\)
Find the LCD of 12 and 18.
Rewrite as equivalent fractions with the LCD.
Simplify the numerators and denominators.\(\frac{21}{36}+\frac{10}{36}\)
Add.\(\frac{31}{36}\)

Because \(31\) is a prime number, it has no factors in common with \(36.\) The answer is simplified.

When we use the Equivalent Fractions Property, there is a quick way to find the number you need to multiply by to get the LCD. Write the factors of the denominators and the LCD just as you did to find the LCD. The “missing” factors of each denominator are the numbers you need.

The LCD, \(36,\) has \(2\) factors of \(2\) and \(2\) factors of \(3.\)

Twelve has two factors of \(2,\) but only one of \(3\)—so it is ‘missing‘ one \(3.\) We multiplied the numerator and denominator of \(\frac{7}{12}\) by \(3\) to get an equivalent fraction with denominator \(36.\)

Eighteen is missing one factor of \(2\)—so you multiply the numerator and denominator \(\frac{5}{18}\) by \(2\) to get an equivalent fraction with denominator \(36.\) We will apply this method as we subtract the fractions in the next example.

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

Identify and Use Fraction Operations

By now in this chapter, you have practiced multiplying, dividing, adding, and subtracting fractions. The following table summarizes these four fraction operations. Remember: You need a common denominator to add or subtract fractions, but not to multiply or divide fractions

Example

Try it.

Simplify:

  1. ⓐ \(-\frac{1}{4}+\frac{1}{6}\)
  2. ⓑ \(-\frac{1}{4}\div \frac{1}{6}\)

Solution

First we ask ourselves, “What is the operation?”

ⓐ The operation is addition.

Do the fractions have a common denominator? No.

\(-\frac{1}{4}+\frac{1}{6}\)
Find the LCD.
Rewrite each fraction as an equivalent fraction with the LCD.
Simplify the numerators and denominators.\(-\frac{3}{12}+\frac{2}{12}\)
Add the numerators and place the sum over the common denominator.\(-\frac{1}{12}\)
Check to see if the answer can be simplified. It cannot.

ⓑ The operation is division. We do not need a common denominator.

\(-\frac{1}{4}\div \frac{1}{6}\)
To divide fractions, multiply the first fraction by the reciprocal of the second.\(-\frac{1}{4}\cdot \frac{6}{1}\)
Multiply.\(-\frac{6}{4}\)
Simplify.\(-\frac{3}{2}\)

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

Use the Order of Operations to Simplify Complex Fractions

In Multiply and Divide Mixed Numbers and Complex Fractions, we saw that a complex fraction is a fraction in which the numerator or denominator contains a fraction. We simplified complex fractions by rewriting them as division problems. For example,

\[\frac{\ \frac{3}{4}\ }{\ \frac{5}{8}\ }=\frac{3}{4}\div \frac{5}{8}\]

Now we will look at complex fractions in which the numerator or denominator can be simplified. To follow the order of operations, we simplify the numerator and denominator separately first. Then we divide the numerator by the denominator.

Example

Try it.

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

Solution
\(\frac{{(\frac{1}{2})}^{2}}{4+{3}^{2}}\)
Simplify the numerator.\(\frac{\frac{1}{4}}{4+{3}^{2}}\)
Simplify the term with the exponent in the denominator.\(\frac{\frac{1}{4}}{4+9}\)
Add the terms in the denominator.\(\frac{\frac{1}{4}}{13}\)
Divide the numerator by the denominator.\(\frac{1}{4}\div 13\)
Rewrite as multiplication by the reciprocal.\(\frac{1}{4}\cdot \frac{1}{13}\)
Multiply.\(\frac{1}{52}\)
Example

Try it.

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

Solution
\(\frac{\frac{1}{2}+\frac{2}{3}}{\frac{3}{4}-\frac{1}{6}}\)
Rewrite numerator with the LCD of 6 and denominator with LCD of 12.\(\frac{\frac{3}{6}+\frac{4}{6}}{\frac{9}{12}-\frac{2}{12}}\)
Add in the numerator. Subtract in the denominator.\(\frac{\ \frac{7}{6}\ }{\ \frac{7}{12}\ }\)
Divide the numerator by the denominator.\(\frac{7}{6}\div \frac{7}{12}\)
Rewrite as multiplication by the reciprocal.\(\frac{7}{6}\cdot \frac{12}{7}\)
Rewrite, showing common factors.\(\frac{7\cdot 6\cdot 2}{6\cdot 7\cdot 1}\)
Simplify.2

Evaluate Variable Expressions with Fractions

We have evaluated expressions before, but now we can also 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

  1. ⓐ \(x=-\frac{1}{3}\)
  2. ⓑ \(x=-\frac{3}{4}.\)

Solution

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

\(x+\frac{1}{3}\)
Simplify.\(0\)

ⓑ To evaluate \(x+\frac{1}{3}\) when \(x=-\frac{3}{4},\) we substitute \(-\frac{3}{4}\) for \(x\) in the expression.

\(x+\frac{1}{3}\)
Rewrite as equivalent fractions with the LCD, 12.\(-\frac{3\cdot 3}{4\cdot 3}+\frac{1\cdot 4}{3\cdot 4}\)
Simplify the numerators and denominators.\(-\frac{9}{12}+\frac{4}{12}\)
Add.\(-\frac{5}{12}\)
Example

Try it.

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

Solution

We substitute \(-\frac{2}{3}\) for \(y\) in the expression.

\(y-\frac{5}{6}\)
Rewrite as equivalent fractions with the LCD, 6.\(-\frac{4}{6}-\frac{5}{6}\)
Subtract.\(-\frac{9}{6}\)
Simplify.\(-\frac{3}{2}\)
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. In \(2{x}^{2}y,\) the exponent applies only to \(x.\)

Simplify exponents first.
Multiply. The product will be negative.
Simplify.
Remove the common factors.
Simplify.
Example

Try it.

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

Solution

We substitute the values into the expression and simplify.

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

Key Concepts

  • Find the least common denominator (LCD) of two fractions.
    1. Factor each denominator into its primes.
    2. List the primes, matching primes in columns when possible.
    3. Bring down the columns.
    4. Multiply the factors. The product is the LCM of the denominators.
    5. The LCM of the denominators is the LCD of the fractions.
  • Equivalent Fractions Property
    • If \(a,b\), and \(c\) are whole numbers where \(b\ne 0\), \(c\ne 0\) then
      \(\frac{a}{b}=\frac{a⋅c}{b⋅c}\) and \(\frac{a⋅c}{b⋅c}=\frac{a}{b}\)
  • Convert two fractions to equivalent fractions with their LCD as the common denominator.
    1. Find the LCD.
    2. For each fraction, determine the number needed to multiply the denominator to get the LCD.
    3. Use the Equivalent Fractions Property to multiply the numerator and denominator by the number from Step 2.
    4. Simplify the numerator and denominator.
  • Add or subtract fractions with different denominators.
    1. Find the LCD.
    2. Convert each fraction to an equivalent form with the LCD as the denominator.
    3. Add or subtract the fractions.
    4. Write the result in simplified form.
  • Summary of Fraction Operations
    • Fraction multiplication: Multiply the numerators and multiply the denominators. \[\frac{a}{b}⋅\frac{c}{d}=\frac{ac}{bd}\]
    • Fraction division: Multiply the first fraction by the reciprocal of the second.\[\frac{a}{b}+\frac{c}{d}=\frac{a}{b}⋅\frac{d}{c}\]
    • Fraction addition: Add the numerators and place the sum over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.\[\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}\]
    • Fraction subtraction: Subtract the numerators and place the difference over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.\[\frac{a}{c}-\frac{b}{c}=\frac{a-b}{c}\]
  • Simplify complex fractions.
    1. Simplify the numerator.
    2. Simplify the denominator.
    3. Divide the numerator by the denominator.
    4. Simplify if possible.

Add and Subtract Fractions with Different Denominators

Find the Least Common Denominator (LCD)

In the following exercises, find the least common denominator (LCD) for each set of fractions.

Try it.

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

Try it.

\(\frac{3}{4}\) and \(\frac{2}{5}\)

Solution

20

Try it.

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

Try it.

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

Solution

48

Try it.

\(\frac{13}{30}\) and \(\frac{25}{42}\)

Try it.

\(\frac{23}{30}\) and \(\frac{5}{48}\)

Solution

240

Try it.

\(\frac{21}{35}\) and \(\frac{39}{56}\)

Try it.

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

Solution

245

Try it.

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

Try it.

\(\frac{2}{3},\frac{1}{4},\) and \(\frac{3}{5}\)

Solution

60

Convert Fractions to Equivalent Fractions with the LCD

In the following exercises, convert to equivalent fractions using the LCD.

Try it.

\(\frac{1}{3}\) and \(\frac{1}{4},\) LCD \(=12\)

Try it.

\(\frac{1}{4}\) and \(\frac{1}{5},\) LCD \(=20\)

Solution

\(\frac{5}{20},\frac{4}{20}\)

Try it.

\(\frac{5}{12}\) and \(\frac{7}{8},\) LCD \(=24\)

Try it.

\(\frac{7}{12}\) and \(\frac{5}{8},\) LCD \(=24\)

Solution

\(\frac{14}{24},\frac{15}{24}\)

Try it.

\(\frac{13}{16}\) and \(-\frac{11}{12},\) LCD \(=48\)

Try it.

\(\frac{11}{16}\) and \(-\frac{5}{12},\) LCD \(=48\)

Solution

\(\frac{33}{48},-\frac{20}{48}\)

Try it.

\(\frac{1}{3},\frac{5}{6},\) and \(\frac{3}{4},\) LCD \(=12\)

Try it.

\(\frac{1}{3},\frac{3}{4},\) and \(\frac{3}{5},\) LCD \(=60\)

Solution

\(\frac{20}{60},\frac{45}{60},\frac{36}{60}\)

Add and Subtract Fractions with Different Denominators

In the following exercises, add or subtract. Write the result in simplified form.

Try it.

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

Try it.

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

Solution

\(\frac{9}{20}\)

Try it.

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

Try it.

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

Solution

\(\frac{11}{24}\)

Try it.

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

Try it.

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

Solution

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

Try it.

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

Try it.

\(\frac{1}{2}-(-\frac{1}{6})\)

Solution

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

Try it.

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

Try it.

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

Solution

\(\frac{23}{20}\)

Try it.

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

Try it.

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

Solution

\(\frac{19}{24}\)

Try it.

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

Try it.

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

Solution

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

Try it.

\(\frac{11}{12}-\frac{3}{8}\)

Try it.

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

Solution

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

Try it.

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

Try it.

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

Solution

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

Try it.

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

Try it.

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

Solution

\(\frac{7}{60}\)

Try it.

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

Try it.

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

Solution

\(-\frac{53}{80}\)

Try it.

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

Try it.

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

Solution

\(-\frac{291}{245}\)

Try it.

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

Try it.

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

Solution

\(\frac{1}{20}\)

Try it.

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

Try it.

\(-\frac{7}{20}-(-\frac{5}{8})\)

Solution

\(\frac{11}{40}\)

Try it.

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

Try it.

\(1+\frac{5}{6}\)

Solution

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

Try it.

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

Try it.

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

Solution

\(\frac{7}{10}\)

Try it.

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

Try it.

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

Solution

\(\frac{3y+4}{6}\)

Try it.

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

Try it.

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

Solution

\(\frac{4x-5}{20}\)

Identify and Use Fraction Operations

In the following exercises, perform the indicated operations. Write your answers in simplified form.

Try it.

  1. ⓐ \(\frac{3}{4}+\frac{1}{6}\)
  2. ⓑ \(\frac{3}{4}\div \frac{1}{6}\)

Try it.

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

Solution

  1. ⓐ \(\frac{5}{6}\)
  2. ⓑ \(4\)

Try it.

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

Try it.

  1. ⓐ \(-\frac{4}{5}-\frac{1}{8}\)
  2. ⓑ \(-\frac{4}{5}\cdot \frac{1}{8}\)

Solution

  1. ⓐ \(-\frac{37}{40}\)
  2. ⓑ \(-\frac{1}{10}\)

Try it.

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

Try it.

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

Solution

  1. ⓐ \(\frac{9a}{14}\)
  2. ⓑ \(\frac{9a-14}{24}\)

Try it.

  1. ⓐ \(\frac{9}{10}\cdot (-\frac{11d}{12})\)
  2. ⓑ \(\frac{9}{10}+(-\frac{11d}{12})\)

Try it.

  1. ⓐ \(\frac{4}{15}\cdot (-\frac{5q}{9})\)
  2. ⓑ \(\frac{4}{15}+(-\frac{5q}{9})\)

Solution

  1. ⓐ \(-\frac{4q}{27}\)
  2. ⓑ \(\frac{12-25q}{45}\)

Try it.

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

Try it.

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

Solution

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

Try it.

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

Try it.

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

Solution

\(\frac{11}{24}\)

Try it.

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

Try it.

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

Solution

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

Try it.

\(\frac{3}{8}\cdot (-\frac{10}{21})\)

Try it.

\(\frac{7}{12}\cdot (-\frac{8}{35})\)

Solution

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

Try it.

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

Try it.

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

Solution

\(\frac{-33-8x}{88}\)

Try it.

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

Try it.

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

Solution

\(\frac{16}{39}\)

Use the Order of Operations to Simplify Complex Fractions

In the following exercises, simplify.

Try it.

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

Try it.

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

Solution

\(\frac{1}{81}\)

Try it.

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

Try it.

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

Solution

32

Try it.

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

Try it.

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

Solution

\(\frac{36}{25}\)

Try it.

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

Try it.

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

Solution

\(\frac{60}{7}\)

Try it.

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

Try it.

\(\frac{\frac{3}{4}+\frac{1}{2}}{\frac{5}{6}-\frac{2}{3}}\)

Solution

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

Try it.

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

Try it.

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

Solution

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

Mixed Practice

In the following exercises, simplify.

Try it.

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

Try it.

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

Solution

\(\frac{19}{30}\)

Try it.

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

Try it.

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

Solution

−9

Try it.

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

Try it.

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

Solution

\(\frac{91}{60}\)

Try it.

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

Try it.

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

Solution

\(\frac{11}{40}\)

Try it.

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

Try it.

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

Solution

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

Try it.

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

Try it.

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

Solution

1

Try it.

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

Try it.

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

Solution

\(\frac{22}{7}\)

In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary.

Try it.

\(x+\frac{1}{2}\) when

  1. ⓐ \(x=-\frac{1}{8}\)
  2. ⓑ \(x=-\frac{1}{2}\)

Try it.

\(x+\frac{2}{3}\) when

  1. ⓐ \(x=-\frac{1}{6}\)
  2. ⓑ \(x=-\frac{5}{3}\)

Solution

  1. ⓐ \(\frac{1}{2}\)
  2. ⓑ \(-1\)

Try it.

\(x+(-\frac{5}{6})\) when

  1. ⓐ \(x=\frac{1}{3}\)
  2. ⓑ \(x=-\frac{1}{6}\)

Try it.

\(x+(-\frac{11}{12})\) when

  1. ⓐ \(x=\frac{11}{12}\)
  2. ⓑ \(x=\frac{3}{4}\)

Solution

  1. ⓐ \(0\)
  2. ⓑ \(-\frac{1}{6}\)

Try it.

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

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

Try it.

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

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

Solution

  1. ⓐ \(\frac{1}{3}\)
  2. ⓑ \(-1\)

Try it.

\(\frac{7}{10}-w\) when

  1. ⓐ \(w=\frac{1}{2}\)
  2. ⓑ \(w=-\frac{1}{2}\)

Try it.

\(\frac{5}{12}-w\) when

  1. ⓐ \(w=\frac{1}{4}\)
  2. ⓑ \(w=-\frac{1}{4}\)

Solution

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

Try it.

\(4{p}^{2}q\) when \(p=-\frac{1}{2}\) and \(q=\frac{5}{9}\)

Try it.

\(5{m}^{2}n\) when \(m=-\frac{2}{5}\) and \(n=\frac{1}{3}\)

Solution

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

Try it.

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

Try it.

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

Solution

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

Try it.

\(\frac{u+v}{w}\) when \(u=-4,v=-8,w=2\)

Try it.

\(\frac{m+n}{p}\) when \(m=-6,n=-2,p=4\)

Solution

−2

Try it.

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

Try it.

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

Solution

3

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.

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

    Rivela la risposta

    \(\frac{10}{12},\frac{15}{18}\)

  2. Simplify: \(\frac{1+5\cdot 3}{{2}^{2}+4}.\)
    If you missed this problem, review .

    Rivela la risposta

    \(2\)

  3. Find the LCD for the fractions \(\frac{7}{12}\) and \(\frac{5}{18}.\)

    Rivela la risposta
    Factor each denominator into its primes.
    List the primes of 12 and the primes of 18 lining them up in columns when possible.
    Bring down the columns.
    Multiply the factors. The product is the LCM.LCM\(=36\)
    The LCM of 12 and 18 is 36, so the LCD of \(\frac{7}{12}\) and \(\frac{5}{18}\) is 36.LCD of \(\frac{7}{12}\) and \(\frac{5}{18}\) is 36.

  4. Find the least common denominator for the fractions: \(\frac{7}{12}\) and \(\frac{11}{15}.\)

    Rivela la risposta

    60

  5. Find the least common denominator for the fractions: \(\frac{13}{15}\) and \(\frac{17}{5}.\)

    Rivela la risposta

    15

  6. Find the least common denominator for the fractions \(\frac{8}{15}\) and \(\frac{11}{24}.\)

    Rivela la risposta

    To find the LCD, we find the LCM of the denominators.

    Find the LCM of \(15\) and \(24.\)

    The LCM of \(15\) and \(24\) is \(120.\) So, the LCD of \(\frac{8}{15}\) and \(\frac{11}{24}\) is \(120.\)

  7. Find the least common denominator for the fractions: \(\frac{13}{24}\) and \(\frac{17}{32}.\)

    Rivela la risposta

    96

  8. Find the least common denominator for the fractions: \(\frac{9}{28}\) and \(\frac{21}{32}.\)

    Rivela la risposta

    224

  9. Convert \(\frac{1}{4}\) and \(\frac{1}{6}\) to equivalent fractions with denominator \(12,\) their LCD.

    Rivela la risposta
    Find the LCD.The LCD of \(\frac{1}{4}\) and \(\frac{1}{6}\) is 12.
    Find the number to multiply 4 to get 12.
    Find the number to multiply 6 to get 12.
    Use the Equivalent Fractions Property to convert each fraction to an equivalent fraction with the LCD, multiplying both the numerator and denominator of each fraction by the same number.
    Simplify the numerators and denominators.

    We do not reduce the resulting fractions. If we did, we would get back to our original fractions and lose the common denominator.

  10. Change to equivalent fractions with the LCD:

    \(\frac{3}{4}\) and \(\frac{5}{6},\) LCD \(=12\)

    Rivela la risposta

    \(\frac{9}{12},\frac{10}{12}\)

  11. Change to equivalent fractions with the LCD:

    \(-\frac{7}{12}\) and \(\frac{11}{15},\) LCD \(=60\)

    Rivela la risposta

    \(-\frac{35}{60},\frac{44}{60}\)

  12. Convert \(\frac{8}{15}\) and \(\frac{11}{24}\) to equivalent fractions with denominator \(120,\) their LCD.

    Rivela la risposta
    The LCD is 120. We will start at Step 2.
    Find the number that must multiply 15 to get 120.
    Find the number that must multiply 24 to get 120.
    Use the Equivalent Fractions Property.
    Simplify the numerators and denominators.

  13. Change to equivalent fractions with the LCD:

    \(\frac{13}{24}\) and \(\frac{17}{32},\) LCD \(96\)

    Rivela la risposta

    \(\frac{52}{96},\frac{51}{96}\)

  14. Change to equivalent fractions with the LCD:

    \(\frac{9}{28}\) and \(\frac{27}{32},\) LCD \(224\)

    Rivela la risposta

    \(\frac{72}{224},\frac{189}{224}\)

  15. Add: \(\frac{1}{2}+\frac{1}{3}.\)

    Rivela la risposta
    \(\frac{1}{2}+\frac{1}{3}\)
    Find the LCD of 2, 3.
    Change into equivalent fractions with the LCD 6.
    Simplify the numerators and denominators.\(\frac{3}{6}+\frac{2}{6}\)
    Add.\(\frac{5}{6}\)

    Remember, always check to see if the answer can be simplified. Since \(5\) and \(6\) have no common factors, the fraction \(\frac{5}{6}\) cannot be reduced.

  16. Add: \(\frac{1}{4}+\frac{1}{3}.\)

    Rivela la risposta

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

  17. Add: \(\frac{1}{2}+\frac{1}{5}.\)

    Rivela la risposta

    \(\frac{7}{10}\)

  18. Subtract: \(\frac{1}{2}-(-\frac{1}{4}).\)

    Rivela la risposta
    \(\frac{1}{2}-(-\frac{1}{4})\)
    Find the LCD of 2 and 4.
    Rewrite as equivalent fractions using the LCD 4.
    Simplify the first fraction.\(\frac{2}{4}-(-\frac{1}{4})\)
    Subtract.\(\frac{2\ -\ (-1)}{4}\)
    Simplify.\(\frac{3}{4}\)

    One of the fractions already had the least common denominator, so we only had to convert the other fraction.

  19. Simplify: \(\frac{1}{2}-(-\frac{1}{8}).\)

    Rivela la risposta

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

  20. Simplify: \(\frac{1}{3}-(-\frac{1}{6}).\)

    Rivela la risposta

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

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

    Rivela la risposta
    \(\frac{7}{12}+\frac{5}{18}\)
    Find the LCD of 12 and 18.
    Rewrite as equivalent fractions with the LCD.
    Simplify the numerators and denominators.\(\frac{21}{36}+\frac{10}{36}\)
    Add.\(\frac{31}{36}\)

    Because \(31\) is a prime number, it has no factors in common with \(36.\) The answer is simplified.

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

    Rivela la risposta

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

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

    Rivela la risposta

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

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

    Rivela la risposta
    \(\frac{7}{15}-\frac{19}{24}\)
    Find the LCD.

    15 is 'missing' three factors of 2
    24 is 'missing' a factor of 5
    Rewrite as equivalent fractions with the LCD.
    Simplify each numerator and denominator.\(\frac{56}{120}-\frac{95}{120}\)
    Subtract.\(-\frac{39}{120}\)
    Rewrite showing the common factor of 3.\(-\frac{13\cdot 3}{40\cdot 3}\)
    Remove the common factor to simplify.\(-\frac{13}{40}\)

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

    Rivela la risposta

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

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

    Rivela la risposta

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

  27. Add: \(-\ \frac{11}{30}+\frac{23}{42}.\)

    Rivela la risposta
    \(-\frac{11}{30}+\frac{23}{42}\)
    Find the LCD.
    Rewrite as equivalent fractions with the LCD.
    Simplify each numerator and denominator.\(-\frac{77}{210}+\frac{115}{210}\)
    Add.\(\frac{38}{210}\)
    Rewrite showing the common factor of 2.\(\frac{19\cdot 2}{105\cdot 2}\)
    Remove the common factor to simplify.\(\frac{19}{105}\)

  28. Add: \(-\frac{13}{42}+\frac{17}{35}.\)

    Rivela la risposta

    \(\frac{37}{210}\)

  29. Add: \(-\frac{19}{24}+\frac{17}{32}.\)

    Rivela la risposta

    \(-\frac{25}{96}\)

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

    Rivela la risposta

    The fractions have different denominators.

    \(\frac{3}{5}+\frac{x}{8}\)
    Find the LCD.
    Rewrite as equivalent fractions with the LCD.
    Simplify the numerators and denominators.\(\frac{24}{40}+\frac{5x}{40}\)
    Add.\(\frac{24\ +\ 5x}{40}\)

    We cannot add \(24\) and \(5x\) since they are not like terms, so we cannot simplify the expression any further.

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

    Rivela la risposta

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

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

    Rivela la risposta

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

  33. Simplify:

    1. ⓐ \(-\frac{1}{4}+\frac{1}{6}\)
    2. ⓑ \(-\frac{1}{4}\div \frac{1}{6}\)

    Rivela la risposta

    First we ask ourselves, “What is the operation?”

    ⓐ The operation is addition.

    Do the fractions have a common denominator? No.

    \(-\frac{1}{4}+\frac{1}{6}\)
    Find the LCD.
    Rewrite each fraction as an equivalent fraction with the LCD.
    Simplify the numerators and denominators.\(-\frac{3}{12}+\frac{2}{12}\)
    Add the numerators and place the sum over the common denominator.\(-\frac{1}{12}\)
    Check to see if the answer can be simplified. It cannot.

    ⓑ The operation is division. We do not need a common denominator.

    \(-\frac{1}{4}\div \frac{1}{6}\)
    To divide fractions, multiply the first fraction by the reciprocal of the second.\(-\frac{1}{4}\cdot \frac{6}{1}\)
    Multiply.\(-\frac{6}{4}\)
    Simplify.\(-\frac{3}{2}\)

  34. Simplify each expression:

    1. ⓐ \(-\frac{3}{4}-\frac{1}{6}\)
    2. ⓑ \(-\frac{3}{4}\cdot \frac{1}{6}\)

    Rivela la risposta

    1. ⓐ \(-\frac{11}{12}\)
    2. ⓑ \(-\frac{1}{8}\)

  35. Simplify each expression:

    1. ⓐ \(\frac{5}{6}\div (-\frac{1}{4})\)
    2. ⓑ \(\frac{5}{6}-(-\frac{1}{4})\)

    Rivela la risposta

    1. ⓐ \(-\frac{10}{3}\)
    2. ⓑ \(\frac{13}{12}\)

  36. Simplify:

    1. ⓐ \(\frac{5x}{6}-\frac{3}{10}\)
    2. ⓑ \(\frac{5x}{6}\cdot \frac{3}{10}\)

    Rivela la risposta

    ⓐ The operation is subtraction. The fractions do not have a common denominator.

    \(\frac{5x}{6}-\frac{3}{10}\)
    Rewrite each fraction as an equivalent fraction with the LCD, 30.\(\frac{5x\cdot 5}{6\cdot 5}-\frac{3\cdot 3}{10\cdot 3}\)
    \(\frac{25x}{30}-\frac{9}{30}\)
    Subtract the numerators and place the difference over the common denominator.\(\frac{25x-9}{30}\)

    ⓑ The operation is multiplication; no need for a common denominator.

    \(\frac{5x}{6}\cdot \frac{3}{10}\)
    To multiply fractions, multiply the numerators and multiply the denominators.\(\frac{5x\cdot 3}{6\cdot 10}\)
    Rewrite, showing common factors.\(\frac{5\cdot x\cdot 3}{2\cdot 3\cdot 2\cdot 5}\)
    Remove common factors to simplify.\(\frac{x}{4}\)

  37. Simplify:

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

    Rivela la risposta

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

  38. Simplify:

    1. ⓐ \(\frac{(24k+25)}{30}\)
    2. ⓑ \(\frac{24k}{5}\)

    Rivela la risposta

    1. ⓐ \(\frac{(24k+25)}{30}\)
    2. ⓑ \(\frac{24k}{5}\)

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

    Rivela la risposta
    \(\frac{{(\frac{1}{2})}^{2}}{4+{3}^{2}}\)
    Simplify the numerator.\(\frac{\frac{1}{4}}{4+{3}^{2}}\)
    Simplify the term with the exponent in the denominator.\(\frac{\frac{1}{4}}{4+9}\)
    Add the terms in the denominator.\(\frac{\frac{1}{4}}{13}\)
    Divide the numerator by the denominator.\(\frac{1}{4}\div 13\)
    Rewrite as multiplication by the reciprocal.\(\frac{1}{4}\cdot \frac{1}{13}\)
    Multiply.\(\frac{1}{52}\)

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

    Rivela la risposta

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

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: Add and Subtract Fractions with Different Denominators

  1. Find the least common denominator (LCD)
  2. Convert fractions to equivalent fractions with the LCD
  3. Add and subtract fractions with different denominators
  4. Identify and use fraction operations
  5. Use the order of operations to simplify complex fractions
  6. Evaluate variable expressions with fractions
  7. Factor each denominator into its primes.
  8. List the primes, matching primes in columns when possible.

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.

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

Più in Arithmetic

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