Download Steps to adding and subtracting fractions

NOTES & GUIDE TO WORKING WITH
FRACTIONS!
Name _________________________________
ESTIMATING FRACTIONS:
Before you begin adding & subtracting fractions, you need to know how to estimate
the sums and differences so you know if you have a reasonable answer.
Consider the following problem:
4 3/8 + 2 5/6
4 3/8 is rounded to 4. Why? Because 3/8 is less than half of one whole. When
something is cut into 8ths, half would be 4/8. So, 3/8 is not at least half,
so we don’t round up.
2 5/6 is rounded to 3. Why? Because 5/6 is more than half of one whole. When
something is cut into 6ths, half would be 3/6. So, since 5/6 is more than
half, we round up to the next whole number. In this case, 2 5/6 is
rounded to 3.
We don’t have to actually add the fractions to estimate what the answer will be close
to! It is important to do this so that you know if your answer makes sense!
Try estimating this problem!
7 3/9 +
5 4/5
_______ + _______ = ______________
The same works for subtraction!
6¾
-
1 5/8
_______ + _______ = ______________
Steps to adding and subtracting fractions
Regular Fractions
1.
Look at the denominators. Are they the same? You cannot add or subtract
fractions that don’t have the same denominator.
⅜ + ¾ = ________
(can’t do it this way!)
2.
Change the denominators by finding the LCM (lowest common multiple) for the
two denominators.
Factors of 8 & 4:
⅜ + ¾ = ________
8, 16, 24, 32
4, 8, 12, 16, 20, 24, 28, 32
3.
Change both denominators to the LCM.
⅜= ?/8
4.
¾ = ?/8
Next, determine what the original denominator was multiplied by to get to the
new denominator. Multiply the original numerator by that number to get the
new denominator. (Rule: What ever you do to the denominator, you need to
do to the numerator. This is the only way to keep the new fraction equal to
the original fraction.)
⅜ = ?/8
(Eight times 1 equals 8, so multiply the numerator by 1. One times 3 is 3, so this fraction doesn’t change.)
¾ = ?/8
(Four times 2 equals 8, so multiply the numerator by 2. Two times 3 is 6, so the new numerator is 6.)
⅜=⅜
¾ = 6/8
5.
Now just add the new fractions. 3/8 + 6/8 = 9/8.
6.
Change the fraction to a mixed number, if necessary. 9/8 = 1 ⅛
7.
How did we get that?
9 ÷ 8 = 1 R1
Eight goes into 9 one time. There is a remainder of 1 – which is interpreted as 1/8
.
Now, the short cut method!
8.
Simply multiply the denominators. The product is the new common
denominator.
(8 x 4 = 32) (Your new denominator will probably be a lot larger than necessary.)
9.
Now cross multiply the numerator with the denominator of the opposite
fraction.
⅜ + ¾ = ________
(3 x 4)
32
12
32
10.
+
24
32
=
+
24
32
= 36
32
Is the numerator larger than the denominator? Turn into a mixed number by
dividing the numerator by the denominator.
36
32
12.
(8 x 3)
32
When the denominators are the same, just add or subtract as instructed!
12
32
11.
+
=
___
32√ 36
=
1r4 =
Reduce the fraction part to lowest terms!
Find the GCF to divide by!
1
4
32
4
32
÷
4
4
=
1
8
(Here you can divide both 4 and 32 by 4!)
13.
So, the answer is 1 ⅛! (The same as if you did it with the LCM!)
This method requires a bit more work, though some think of it as a short cut. You will
be working with larger numbers when you cross multiply, and if the LCM is less than
the product of the denominators, you will need to reduce your answer to lowest terms.
One way to make it easier is to put your fractions in lowest terms before you begin the
cross-multiplication! (Always check to see if the fraction is in lowest terms!)
Adding Mixed Fractions
1.
Now you’ve got whole numbers and fractions to deal with. Oh, brother! First,
let’s look at the denominators. Are they the same or different? Let’s try it when
they are the same.
8 2/3 + 9 2/3 = ___________
2.
Okay, easy! Add the whole numbers . . .
3.
Now add the fractions . . . 2/3 + 2/3 = 4/3
4.
The answer of course, is 17 and 4/3, right? WRONG! Look at 4/3. A few bells
should go off in your head! It is an improper fraction and needs to be turned
back to a mixed number.
4/3 = 1 1/3
8 + 9 = 17
BUT WAIT! That was only the fraction part! So you need
to add the whole number in too!
17 + 1 1/3 = 18 1/3
This one is in lowest terms, so you don’t have to worry about reducing it.
Moving on . . .
Adding Mixed Fractions with Uncommon Denominators
1.
When you have a mixed number, you have to convert to improper fractions
before you find common denominators.
5 2/5 + 2 3/8 = _________
2.
Find the LCM of the denominators . . .
Factors of 5 – 5, 10, 15, 20, 25, 30, 35, 40 (We can stop at 40 because 8 x 5 = 40)
Factors of 8 – 8, 16, 24, 32, 40
3.
The LCM is 40, so the new denominators should be 40.
4.
Find the new numerators:
2/5 = ?/40 (5 x 8 = 40, so multiply 2 x 8.) 2/5 = 16/40
3/8 = ?/40 (8 x 5 = 40, so multiply the 3 by 5.) 3/8 = 15/40
5.
Add the fractional parts:
16/40 + 15/40 = 31/40
6.
Next, add the whole numbers: 5 + 2 = 7
7.
So the sum of the two fractions = 7 31/40.
8.
Is this in lowest terms? Yes, so you don’t have to reduce.
(31 is a prime number!)
So What About Subtraction?
Let’s do the same problem, but instead of addition, we’ll change it to subtraction.
1.
When you have a mixed number, you have to convert to improper fractions
before you find common denominators.
5 2/5 - 2 3/8 = _________
(This part is the same as in the previous problem!)
2. Find the LCM of the denominators . . .
Factors of 5 – 5, 10, 15, 20, 25, 30, 35, 40 (We can stop at 40 because 8 x 5 = 40)
Factors of 8 – 8, 16, 24, 32, 40
3. The LCM is 40, so the new denominators should be 40.
4. Find the new numerators:
2/5 = ?/40 (5 x 8 = 40, so multiply 2 x 8.) 2/5 = 16/40
3/8 = ?/40 (8 x 5 = 40, so multiply the 3 by 5.) 3/8 = 15/40
5. First, subtract the fractions: 16/40 – 15/40 = 1/40.
6. Next, subtract the whole numbers: 5 – 2 = 3.
7. The answer is 3 1/40. This is in lowest terms because the numerator is 1
and 1 is a prime number and you can’t go any lower!
So what does this mean? It means that the method for adding or subtracting fractions
with mixed numbers is the same, until you get to the actual adding or subtracting
parts!
Okay, got it? Try this one!
1.
First, make sure you understand how to do the problems above before you try
this one!
2.
Let’s look at the following problem:
6 2/3 + 4 3/8 = ___________
3.
Convert only the fraction part by finding the LCM. Then add.
2/3
(2 x 8)
24
4.
+
+
3/8
=
(3 x 3)
24
?/24
=
+
16
24
?/24
+
9
24
= 25
24
Change 25/24 to a mixed number.
24 goes into 25 only 1 time, with a remainder of 1, so the fraction is
1 and 1/24
5.
Add the whole numbers from the fraction . . . 6 + 4 = 10
6.
Add the fraction part to the whole numbers . . . 10 + 1 and 1/24 = 11 and 1/24
What if the first fraction is smaller than the second?
Okay, that’s harder! Are you sure you want to go there?
1.
Consider the following problem:
3 1/3 - 1 2/4
If you consider only the fraction part, then you’ll get something like this:
1/3
- 2/4
=
???
First, you can’t subtract until the denominators are the same. Find the LCM and
convert the denominators.
Factors of 3 _______________________
Factors of 4 _______________________
Least Common Multiple? ______
Convert each of the fractions to equivalent fractions with the denominator 12.
1/3 = ?/12
2/4 = ?/12
NOW, try to subtract!
4/12
–
6/12
Of course, you can’t subtract 6 from 4, so you have to borrow from the whole
number.
If we borrow from the 3, we cross out and make it a 2. Then the one whole that
we borrowed gets added to the fraction. Since we changed to 12 as a denominator,
then 1 whole is equal to 12/12. Add 12/12 to the 4/12.
12/12 +
4/12 = 16/12
Now we have enough to subtract.
16/12 - 6/12 = 10/12
OKAY, WAIT! STOP! You’re not done yet! Don’t forget about the whole numbers!
You began with 3, but borrowed 1, so you have 2 left. 2 – 1 = 1.
Add that to your fraction and then reduce to lowest terms. 1 10/12 = 1 5/6.
2.
Now try this one:
4 1/8 - 2 1/6 = __________.
Cross multiply the fractions:
1/8 - 1/6 =
6/48 - 8/48
Nope, can’t do it – have to borrow!
The whole number 4 becomes 3, and the fraction 6/48 becomes 54/48. (48 + 6)
54/48 - 8/48 =
46/48
Reduce to lowest terms:
23/34
Don’t forget the whole numbers!
We borrowed from the 4, so now the problem is 3 – 2 = 1.
So the entire answer is 1 and 23/24.
Hopefully this has helped you with the steps and not confused you further!
Please, ask questions in class, and don’t forget to estimate answers before you start so
that you know if you’re in the right ‘ballpark’ when you get your answer.
REVIEW: Finding the LCM (Lowest Common Multiple)
To find the LCM, list the multiples of the denominators you are adding or subtracting.
5/6 + 4/8
Start with the larger denominator (in this case, 8)
Factors of 8 = 8, 16, 24, 32, 40, 48
*You can stop after 6 factors, because 6 is the other denominator!
Factors of 6 = 6, 12, 18, 24, 30, 36, 42, 48
*You can stop after 8 factors, because 8 is the other
denominator.
The lowest common denominator is 48.
To finish this problem, you would need to convert each fraction to a fraction that is
equal to the original. Remember the rule: What ever you do to the denominator, you
must do to the numerator.
Find the LCM of these fractions!
3/9 + 2/6
Factors of 9: _______________________________________
Factors of 6: _______________________________________
LCM = ______
Convert 3/9 to ______/_______*(new denominator)
Convert 2/6 to ______/_______*(new denominator)
Now you can add the two easily!
REVIEW:
Finding the GCF (Greatest Common Factor)
First, why do we need the GCF?
When you have added two or more fractions together, or if you subtract one
fraction from another, you may have had to find equivalent fractions. This makes the
denominators larger, but in order to show the fraction in lowest terms, you have to
figure out what the GCF is so that you can divide by that number and successfully
reduce the fraction in one try!
Let’s try an easy one!
4/16
Factors of 4 = {1, 4; 2, 2}
Factors of 16 = {1, 16; 2, 8; 4, 4)
Common factors are {1, 2, 4)
Greatest Common Factor is 4
Now, divide both the numerator and the denominator by the GCF (4).
4÷4=1
16 ÷ 4 = 4
so the fraction in lowest terms is 1/4
Try this one!
6/15
Factors of 6 = __________________________________
Factors of 15 = _________________________________
Greatest common factor = __________
Divide the numerator and denominator by the GCF.
The fraction in lowest terms is _______________________.