Download Fractions - Mrs. Wallace

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Fractions Notes
Definitions:
Numerator- the top number in a fraction
Denominator- the bottom number in a fraction
Multiple- the result of multiplying by a whole number
Factor- any number that divides evenly into another number
Ex.: List the Factors of 12: 1, 2, 3, 4, 6, 12
Least Common Multiple (LCM) - The smallest multiple that two or more numbers have
in common. This is used for finding a common denominator
Ex.: Find the LCM of 3 and 5. Start by listing the multiples of each number. Then
circle the smallest number they have in common.
Multiples of 3: 3, 6, 9, 12, 15, 18, …
Multiples of 5: 5, 10, 15, 20, 25, …
The LCM is 15. This would be the number used as a common denominator
Greatest Common Factor (GCF) - the largest factor two or more numbers have in
common. This is used for reducing fractions
Ex.: Find the GCF of 12 and 30. Start by listing the factors (numbers that divide
in) of each number. Then circle the largest number they have in common.
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
The GCF is 6.
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Types of Fractions:
(Images from: www.mathisfun.com)
Proper: Numerator is smaller than the Denominator. Value is less than 1 whole.
Improper: Numerator is larger than the Denominator. Value is equal to or greater than
one whole
Mixed Numbers: Also called Mixed Fractions. It contains a whole number and a fraction.
Value is greater than one whole.
Reducing Fractions
Reducing fractions is necessary. Always reduce answers when possible.
Reducing proper fractions
1. Find the GCF of the numerator and denominator
2. Divide both the numerator and denominator by the GCF
3. This is your reduced fraction. If 1 is the only number that can go into the
numerator and denominator, it is in simplest terms. If you did not use the GCF to
reduce, you’ll have to keep reducing until the fraction is in simplest terms.
Example: Reduce the fraction 12/30
1.
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
The GCF is 6.
2.
Divide the numerator and denominator by 6
12 ÷ 6 = 2
30 ÷ 6 = 5
3.
The reduced fraction is 2/5. This is in simplest terms.
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Reducing improper fractions (changing to a mixed number)
1. Divide the numerator by the denominator. Set up a long division problem
remembering that denominator is “down (in a fraction) and out (in division)”
2. The quotient becomes the whole number in the mixed number.
3. The remainder becomes the numerator.
4. Keep the denominator from the original fraction and write it under the numerator.
5. Reduce fraction if possible.
Example: Change the Improper Fraction to a Mixed Number
Reducing Mixed Numbers (changing to improper fractions)
1. Multiply the denominator and the whole number together
2. Add the numerator to your answer from step one.
3. Put this total over the original denominator and you’re finished!
Example: Change the Mixed Number to an Improper Fraction
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Adding and Subtracting Fractions
A note about ADDING AND SUBTRACTING Fractions:
When ADDING or SUBTRACTING fractions, you must have a common denominator.
To determine what number to use as a common denominator, refer to the definition of
LCM on page one.
Adding Fractions with Like Denominators
If the fractions you are adding already have a common denominator, you may solve the
problem with the following steps:
1. Add the numerators. Write this number as the numerator in the answer
2. Keep the common denominator in your answer (just move it into your answer).
3. Reduce your answer if possible.
Example:
Adding Fractions with Different Denominators
If the fractions you are adding do not have common denominators, you must first find
one before doing the math. Follow these steps:
1. Find the lowest common denominator by listing the multiples of each
denominator and circling the lowest multiple they have in common. (See page 1
for an example of this step)
2. Raise the fractions up to the desired denominator using multiplication. Whatever
you do to the denominator you must do to the numerator.
3. Once your fractions have common denominators, add the numerators, keeping the
common denominator in your answer.
4. Reduce if possible.
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Subtracting Fractions with Like Denominators
If the fractions you are subtracting already have common denominators, use the
following steps:
1. Subtract the numerators. Write this number as the numerator in the answer
2. Keep the common denominator in your answer (just move it into your answer).
3. Reduce the answer if possible.
Example:
Subtracting Fractions with Different Denominators
If the fractions you are subtracting do not have common denominators, you must first
find one before doing the math. Follow these steps:
1. Find the lowest common denominator by listing the multiples of each
denominator and circling the lowest multiple they have in common. (See page 1
for an example of this step). This is your common denominator.
2. Raise the fractions up, if needed, to the desired denominator using multiplication.
Whatever you do to the denominator you must do to the numerator.
3. Once your fractions have common denominators, add the numerators, keeping the
common denominator in your answer.
4. Reduce if possible.
Example:
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Adding & Subtracting Mixed Numbers
Follow these steps to add or subtract mixed numbers:
1. Stack mixed numbers on top of one another, making sure fractions line up with
fractions and whole numbers line up with whole numbers. You will do the
fractions first.
2. Check for a common denominator; find the lowest common denominator if
denominators are different and raise fractions up.
3. Add/Subtract numerators and keep common denominator in answer. Reduce
fraction if possible.
4. Add/Subtract whole numbers last.
Example:
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Borrowing when Subtracting Mixed Numbers
Sometimes when subtracting mixed numbers, we have to borrow from the whole number
to complete the problem. If the numerator of the top fraction is not large enough to
subtract the numerator of the bottom fraction, we must borrow a 1 from the whole
number.
FYI: any number over itself equals one whole
5/5 = 1
13/13 = 1
2/2 = 1
Q: Why is that true?
A: Fractions also represent division (numerator
divided by denominator).
If I have 5/5, that is the same as saying, “five
divided by five”, which equals one.
Here are the steps:
1. Stack mixed numbers on top of one another, making sure fractions line up with
fractions and whole numbers line up with whole numbers. You will do the
fractions first.
2. Check for a common denominator; find the lowest common denominator before
doing any math.
3. If you are unable to subtract the bottom numerator from the top numerator, you
must borrow from the top whole number.
4. Borrow 1 from the whole number. Change the 1 into a fraction using the common
denominator and write it next to the top fraction. In other words, whatever the
common denominator is, put that number over itself and write it next to the top
fraction.
5. Add the top fraction and the new “borrowed” fraction next to it. DON’T SKIP
THIS STEP! What we borrowed must be added to what we started with.
6. Using the answer from Step 5, finish the fraction side. The top numerator is now
large enough to do the subtraction.
7. Reduce your fraction answer, if possible.
8. Lastly, subtract the whole numbers
Example:
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Multiplying Fractions
When multiplying fractions, you DO NOT NEED A COMMON DENOMINATOR!
All fractions must be written in this format:
(Source: www.professionaltutor.net)
That means Mixed Numbers must be changed to Improper Fractions (see page 3), and
whole numbers must be written over a “1”
Here are the steps to multiplying fractions:
1. Make sure all fractions are in N/D form.
a. Cross-Cancel if desired
2. Multiply the numerators straight across.
3. Multiply the denominators straight across.
4. Reduce your answer.
Example:
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Dividing Fractions
When dividing fractions, you DO NOT NEED A COMMON DENOMINATOR!
All fractions must be in N/D form (see page 8 for example).
Mixed numbers must be written as improper fractions and whole numbers must be over
the number “1”.
Here are the steps for dividing fractions:
1. Make sure all fractions are in N/D form.
2. Change division to multiplication
3. Flip the second fraction (take the reciprocal)
a. Cross-Cancel if desired
4. Multiply numerators straight across
5. Multiply denominators straight across
6. Reduce answer
Example: