1.6 Add and subtract fractions

Elementary algebra Page 1 / 4

By the end of this section, you will be able to:

Add or subtract fractions with a common denominator
Add or subtract fractions with different denominators
Use the order of operations to simplify complex fractions
Evaluate variable expressions with fractions

A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, Fractions .

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.

Fraction addition and subtraction

If $a, b, and c$ are numbers where $c \neq 0,$ 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.

Doing the Manipulative Mathematics activities “Model Fraction Addition” and “Model Fraction Subtraction” will help you develop a better understanding of adding and subtracting fractions.

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

Solution

$\begin{matrix} \frac{x}{3} + \frac{2}{3} \\ \begin{matrix} Add the numerators and place the sum over \\ the common denominator. \end{matrix} & \frac{x + 2}{3} \end{matrix}$

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Find the sum: $\frac{x}{4} + \frac{3}{4} .$

$\frac{x + 3}{4}$

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Find the sum: $\frac{y}{8} + \frac{5}{8} .$

$\frac{y + 5}{8}$

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

Solution

$\begin{matrix} - \frac{23}{24} - \frac{13}{24} \\ \begin{matrix} Subtract the numerators and place the \\ difference over the common denominator. \end{matrix} & \frac{−23 - 13}{24} \\ Simplify. & \frac{−36}{24} \\ Simplify. Remember, - \frac{a}{b} = \frac{− a}{b} . & - \frac{3}{2} \end{matrix}$

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Find the difference: $- \frac{19}{28} - \frac{7}{28} .$

$- \frac{26}{28}$

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Find the difference: $- \frac{27}{32} - \frac{1}{32} .$

$- \frac{7}{8}$

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Simplify: $- \frac{10}{x} - \frac{4}{x} .$

Solution

$\begin{matrix} - \frac{10}{x} - \frac{4}{x} \\ \begin{matrix} Subtract the numerators and place the \\ difference over the common denominator. \end{matrix} & \frac{−14}{x} \\ \begin{matrix} Rewrite with the sign in front of the \\ fraction. \end{matrix} & - \frac{14}{x} \end{matrix}$

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Find the difference: $- \frac{9}{x} - \frac{7}{x} .$

$- \frac{16}{x}$

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Find the difference: $- \frac{17}{a} - \frac{5}{a} .$

$- \frac{22}{a}$

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

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

Solution

$\begin{matrix} \begin{matrix} Add and subtract fractions—do they have a \\ common denominator? Yes. \end{matrix} & \frac{3}{8} + (- \frac{5}{8}) - \frac{1}{8} \\ \begin{matrix} Add and subtract the numerators and place \\ the result over the common denominator. \end{matrix} & \frac{3 + (−5) - 1}{8} \\ Simplify left to right. & \frac{−2 - 1}{8} \\ Simplify. & - \frac{3}{8} \end{matrix}$

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

$−1$

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

$- \frac{2}{3}$

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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.

Least common denominator

The least common denominator (LCD) of two fractions is the least common multiple (LCM) of their denominators.

Doing the Manipulative Mathematics activity “Finding the Least Common Denominator” will help you develop a better understanding of the LCD.

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

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

$In this figure, we have a table with directions on the left, hints or explanations in the middle, and mathematical statements on the right. On the first line, we have “Step 1. Do they have a common denominator? No – rewrite each fraction with the LCD (least common denominator).” To the right of this, we have the statement “No. Find the LCD 12, 18.” To the right of this, we have 12 equals 2 times 2 times 3 and 18 equals 2 times 3 times 3. The LCD is hence 2 times 2 times 3 times 3, which equals 36. As another hint, we have “Change into equivalent fractions with the LCD,. Do not simplify the equivalent fractions! If you do, you’ll get back to the original fractions and lose the common denominator!” To the right of this, we have 7/12 plus 5/18, which becomes the quantity (7 times 3) over the quantity (12 times 3) plus the quantity (5 times 2) over the quantity (18 times 2), which becomes 21/36 plus 10/36.$ $The next step reads “Step 2. Add or subtract the fractions.” The hint reads “Add.” And we have 31/36.$ The final step reads “Step 3. Simplify, if possible.” The explanation reads “Because 31 is a prime number, it has no factors in common with 36. The answer is simplified.”

The final step reads “Step 3. Simplify, if possible.” The explanation reads “Because 31 is a prime number, it has no factors in common with 36. The answer is simplified.”

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Add: $\frac{7}{12} + \frac{11}{15} .$

$\frac{79}{60}$

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

$\frac{103}{60}$

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Add or subtract fractions.

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.
Add or subtract the fractions.
Simplify, if possible.

Questions & Answers

how did you get 1640

Noor Reply

If auger is pair are the roots of equation x2+5x-3=0

Peter Reply

Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.

MATTHEW Reply

420

Sharon

from theory: distance [miles] = speed [mph] × time [hours] info #1 speed_Dennis × 1.5 = speed_Wayne × 2 => speed_Wayne = 0.75 × speed_Dennis (i) info #2 speed_Dennis = speed_Wayne + 7 [mph] (ii) use (i) in (ii) => [...] speed_Dennis = 28 mph speed_Wayne = 21 mph

George

Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5. Substituting the first equation into the second: W * 2 = (W + 7) * 1.5 W * 2 = W * 1.5 + 7 * 1.5 0.5 * W = 7 * 1.5 W = 7 * 3 or 21 W is 21 D = W + 7 D = 21 + 7 D = 28

Salma

Devon is 32 32 years older than his son, Milan. The sum of both their ages is 54 54. Using the variables d d and m m to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.

Aaron Reply

find product (-6m+6) ( 3m²+4m-3)

SIMRAN Reply

-42m²+60m-18

Salma

what is the solution

bill

how did you arrive at this answer?

bill

-24m+3+3mÁ^2

Susan

i really want to learn

Amira

I only got 42 the rest i don't know how to solve it. Please i need help from anyone to help me improve my solving mathematics please

Amira

Hw did u arrive to this answer.

Aphelele

Bajemah

-6m(3mA²+4m-3)+6(3mA²+4m-3) =-18m²A²-24m²+18m+18mA²+24m-18 Rearrange like items -18m²A²-24m²+42m+18A²-18

Salma

complete the table of valuesfor each given equatio then graph. 1.x+2y=3

Jovelyn Reply

x=3-2y

Salma

y=x+3/2

Salma

Enock

given that (7x-5):(2+4x)=8:7find the value of x

Nandala

3x-12y=18

Kelvin

please why isn't that the 0is in ten thousand place

Grace Reply

please why is it that the 0is in the place of ten thousand

Grace

Send the example to me here and let me see

Stephen

A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg

Marry Reply

how far

Abubakar

cool u

Enock

state in which quadrant or on which axis each of the following angles given measure. in standard position would lie 89°

Abegail Reply

hello

BenJay

Method

I am eliacin, I need your help in maths

Rood

how can I help

Sir

hmm can we speak here?

Amoon

however, may I ask you some questions about Algarba?

Amoon

Enock

what the last part of the problem mean?

Roger

The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.

cameron Reply

Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.

mahnoor Reply

I'm guessing, but it's somewhere around $4335.00 I think

Lewis

12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67. Check: Sales = 3542 Commission 12%=425.04 Pay = 500 + 425.04 = 925.04. 925.04 > 925.00

Munster

difference between rational and irrational numbers

Arundhati Reply

When traveling to Great Britain, Bethany exchanged $602 US dollars into £515 British pounds. How many pounds did she receive for each US dollar?

Jakoiya Reply

how to reduced echelon form

Solomon Reply

Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?

Zack Reply

d=r×t the equation would be 8/r+24/r+4=3 worked out

Sheirtina

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Source: OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2

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