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1.6 Add and subtract fractions

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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 ≠ 0 , then

a c + b c = a + b c and a c − b c = 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: x 3 + 2 3 .

Solution

x 3 + 2 3 Add the numerators and place the sum over the common denominator. x + 2 3

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Find the sum: x 4 + 3 4 .

x + 3 4

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Find the sum: y 8 + 5 8 .

y + 5 8

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Find the difference: − 23 24 − 13 24 .

Solution

− 23 24 − 13 24 Subtract the numerators and place the difference over the common denominator. −23 − 13 24 Simplify. −36 24 Simplify. Remember, − a b = − a b . − 3 2

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Find the difference: − 19 28 − 7 28 .

− 26 28

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Find the difference: − 27 32 − 1 32 .

− 7 8

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Simplify: − 10 x − 4 x .

Solution

− 10 x − 4 x Subtract the numerators and place the difference over the common denominator. −14 x Rewrite with the sign in front of the fraction. − 14 x

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Find the difference: − 9 x − 7 x .

− 16 x

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Find the difference: − 17 a − 5 a .

− 22 a

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

Simplify: 3 8 + ( − 5 8 ) − 1 8 .

Solution

Add and subtract fractions—do they have a common denominator? Yes. 3 8 + ( − 5 8 ) − 1 8 Add and subtract the numerators and place the result over the common denominator. 3 + ( −5 ) − 1 8 Simplify left to right. −2 − 1 8 Simplify. − 3 8

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Simplify: 2 5 + ( − 4 9 ) − 7 9 .

−1

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Simplify: 5 9 + ( − 4 9 ) − 7 9 .

− 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: 7 12 + 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.”
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Add: 7 12 + 11 15 .

79 60

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Add: 13 15 + 17 20 .

103 60

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Add or subtract 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.

Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
Eunice Reply
what is a capacitor?
Raymond Reply
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
Maria Reply
please solve
Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
Saheed Reply
50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
Ramon Reply
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
Someone
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Haryormhidey Reply
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Esrael Reply
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Felix Reply
physics, biology and chemistry this is my Field
ALIYU
field is a region of space under the influence of some physical properties
Collete
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WISDOM Reply
determine the slope giving that 3y+ 2x-14=0
WISDOM
Another formula for Acceleration
Belty Reply
a=v/t. a=f/m a
IHUMA
innocent
Adah
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
Nassze Reply
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Esrael
Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
JALLAH Reply
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Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
What is specific heat capacity
Destiny Reply
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
ROKEEB
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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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