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Adding Fractions
Adapted from MathisFun.com
There are 3 Simple Steps to add fractions:
Step 1: Make sure the bottom numbers (the denominators) are the same Step 2: Add the top numbers (the numerators). Put the answer over the denominator. Step 3: Simplify the fraction (if needed).
Example 1:
Step 1. The bottom numbers (the denominators) are already the same. Go straight to step 2.
Step 2. Add the top numbers and put the answer over the same denominator:
Step 3. Simplify the fraction:
In picture form it looks like this:
(^1) / 4 +^
4 =^
4 =^
2
(If you are unsure of the last step see Equivalent Fractions.)
Example 2:
Step 1 : The bottom numbers are different. See how the slices are different sizes?
(^1) / 3 +^
6 =^?
We need to make them the same before we can continue, because we can't add them like that.
The number "6" is twice as big as "3", so to make the bottom numbers the same we can multiply the top and bottom of the first fraction by 2 , like this:
× 2
× 2
Important: you multiply both top and bottom by the same amount, to keep the value of the fraction the same
Now the fractions have the same bottom number ("6"), and our question looks like this:
6 +^
6 =^
6 =^
2
A Rhyme To Help You Remember
♫ "If adding or subtracting is your aim, The bottom numbers must be the same! ♫ "Changing bottom use multiply or divide, But the same to top must be applied, ♫ "And don't forget to simplify, Before its time to say good bye"
Example 3:
Again, the bottom numbers are different (the slices are different sizes)!
3 +^
5 =^?
But let us try dividing them into smaller sizes that will each be the same :
15 +^
15
The first fraction: by multiplying the top and bottom by 5 we ended up with 5 / 15 :
× 5
× 5
The second fraction: by multiplying the top and bottom by 3 we ended up with 3 / 15 :
× 3
× 3
The bottom numbers are now the same, so we can go ahead and add the top numbers:
(^5) / 15 +^
15 =^
15
Example 1:
Step 1. The bottom numbers are already the same. Go straight to step 2.
Step 2. Subtract the top numbers and put the answer over the same denominator:
3
Step 3. Simplify the fraction:
2
Example 2:
Step 1. The bottom numbers are different. See how the slices are different sizes? We need to make them the same before we can continue, because we can't subtract them like this:
2 -^
6 =^?
To make the bottom numbers the same, multiply the top and bottom of the first fraction (^1 / 2 ) by 3 like this:
× 3
× 3
And now our question looks like this:
(^3) / 6 -^
6
The bottom numbers (the denominators) are the same, so we can go to step 2.
Step 2. Subtract the top numbers and put the answer over the same denominator:
In picture form it looks like this:
(^3) / 6 -^
6 =^
6
Convert to Improper Fractions:
2 3 / 4 = 11 / 4
3 1 / 2 = 7 / 2
Common denominator of 4:
(^11) / 4 stays as^
4 (^7) / 2 becomes^
4 (by multiplying top and bottom by 2)
Now Add:
(^11) / 4 +^
4 =^
4
Convert back to Mixed Fractions:
(^25) / 4 = 6^
4
When you get more experience you can do it faster like this:
Example: What is 3 5 / 8 + 1 3 / 4
Convert them to improper fractions:
3 5 / 8 = 29 / 8 1 3 / 4 = 7 / 4
Make same denominator: 7 / 4 becomes 14 / 8 (by multiplying top and bottom by 2)
And add:
(^29) / 8 +^
8 =^
8 = 5^
8
Subtracting Mixed Fractions
Just follow the same method, but subtract instead of add:
Example: What is 15 3 / 4 - 8 5 / 6?
Convert to Improper Fractions:
Common denominator of 12:
(^63) / 4 becomes^
12 (^53) / 6 becomes^
12
Now Subtract:
(^189) / 12 -^
12 =^
12
Convert back to Mixed Fractions:
(^83) / 12 = 6^
12
Step 1. Multiply the top numbers:
×
1 × 9
Step 2. Multiply the bottom numbers:
×
1 × 9
3 16 3 × 16 48
Step 3. Simplify the fraction:
9
A Rhyme To Help You Remember
♫ "Multiplying fractions: no big problem, Top times top over bottom times bottom." ♫
Multiplying Mixed Fractions
("Mixed Fractions" are also called "Mixed Numbers")
To multiply Mixed Fractions:
- convert to Improper Fractions
- Multiply the Fractions
- convert the result back to Mixed Fractions
Example: What is 1
/ 8 × 3?
Think of Pizzas.
1 3 / 8 is 1 pizza and 3 eighths of another pizza.
First, convert the mixed fraction (1 3 / 8 ) to an an improper fraction (^11 / 8 ):
Cut the whole pizza into eighths and how many eighths do you have in total?
1 lot of 8, plus the 3 eighths = 8+3 = 11 eighths.
Now multiply that by 3:
1 3 / 8 × 3 = 11 / 8 × 3 / 1 = 33 / 8
You have 33 eighths.
And, lastly, convert to a mixed fraction (only because the original fraction was in that form):
Convert both to improper fractions
1 1 / 2 × 2 1 / 5 = 3 / 2 × 11 / 5
Multiply the fractions (multiply the top numbers multiply bottom numbers):
(^3) / 2 ×^
5 =^
(3 × 11)/
(2 × 5) =^
10
Convert to a mixed number
(^33) / 10 =^^3
10
If you are clever you can do it all in one line like this:
1 1 / 2 × 2 1 / 5 = 3 / 2 × 11 / 5 = 33 / 10 = 3 3 / 10
One More Example: What is 3
/ 4 x 3
/ 3?
Convert both to improper fractions
3 1 / 4 × 3 1 / 3 = 13 / 4 × 10 / 3
Multiply
(^13) / 4 ×^
3 =^
12
Convert to a mixed number (and simplify):
(^130) / 12 =^^10
12 =^^10
6
Once again, here it is in one line:
3 1 / 4 × 3 1 / 3 = 13 / 4 × 10 / 3 = 130 / 12 = 10 10 / 12 = 10 5 / 6
This One Has Negatives: What is -
/ 9 × -
/ 7?
Convert Mixed to Improper Fractions:
1 5 / 9 = 9 / 9 + 5 / 9 = 14 / 9 2 1 / 7 = 14 / 7 + 1 / 7 = 15 / 7
Then multiply the Improper Fractions (Note: negative times negative gives us a positive) :
-14/ 9 ×^
7 =^
-14×-15 /
9×7 =^
63
I then decided to simplify next, first by 7 (because I noticed that 21 and 63 are both multiples of 7), then again by 3 (but I could have done it in one step by dividing by 21):
(^210) / 63 =^
9 =^
3
Finally convert to a Mixed Fraction (because that was the style of the question):
(^10) / 3 =^
3 =^
3 +^
3 = 3^
3
÷
How many
in
Now look at the pizzas below ... how many "1/6th slices" fit into a "1/2 slice"?
How many in? Answer: 3
So now you can see that
÷
= 3 really does makes sense! 2 6
Example 2
÷
Step 1. Turn the second fraction upside-down (the reciprocal ):
1 becomes
Step 2. Multiply the first fraction by that reciprocal :
1 ×
1 × 4
8 1 8 × 1 8
Step 3. Simplify the fraction:
4
And that is all you have to do.
A Rhyme To Help You Remember
♫ "Dividing fractions, as easy as pie, Flip the second fraction, then multiply." ♫
But maybe you want to know why we do it this way...
Why Turn the Fraction Upside Down?
Well ... what Does a Fraction Do?
A fraction says to:
multiply by the top number divide by the bottom number
Example: 3 / 4 means to cut into 4 pieces, and then take 3 of those.
So you:
divide by 4 multiply by 3
But when you DIVIDE by a fraction, you are asked to do the opposite of multiply ...
So you:
divide by the top number multiply by the bottom number
So dividing by 5 / 2 is the same as multiplying by 2 / 5
Because:
