Download Adding and Subtracting Fractions and Mixed Numbers

SOL 5.6 Computation and Estimation
NOTEPAGE FOR STUDENT
Page 1
Adding and Subtracting Fractions and Mixed Numbers
When adding and subtracting fractions, the most important part of the process is to make sure that
the denominators are the same.
Adding and subtracting fractions is easy if we have like or
common denominators.
When we have like or common denominators, we just add or subtract the numerators. The
denominators are just names of the parts we are working with.
Consider this problem:
7
8
15
+
=
10
10
10
That’s pretty easy. We add the numerators, 7 plus 8 equals 15. The denominator is tenths.
The solution is
15
or fifteen-tenths.
10
Improper Fractions
However, we are not finished with the sum in this problem. The fraction
15
is called an improper
10
fraction. An improper fraction is a fraction where the numerator is greater than the
denominator.
9
11
and
are also examples of improper fractions. We have to rewrite an
4
7
improper fraction as a mixed number.
Mixed Numbers
A mixed number is a number that has two parts: a whole number and a fraction.
The improper fraction
9
1
becomes 2 , a mixed number.
4
4
The improper fraction
11
4
becomes 1 , a mixed number. The value of a mixed number is the sum of its two parts.
7
7
©2011
SOL 5.6 Computation and Estimation
NOTEPAGE FOR STUDENT
Page 2
Adding and Subtracting Fractions and Mixed Numbers
Writing an Improper Fraction as a Mixed Number
Let’s write the improper fraction
We know that
15
as a mixed number.
10
10
15
= 1. Therefore,
represents more than one whole.
10
10
How much more? To find out we subtract:
15 10
5
=
10 10 10
15
5
can be written as 1
, a mixed number. This
10
10
5
mixed number could also be written as: 1 +
That’s because a mixed number is the sum of
10
We now know that the improper fraction
its two parts: the whole number and the fraction.
Writing an Fraction in its Simplest Form
We are not finished with this sum yet. The fraction
5
is not in its simplest form.
10
A fraction is in its simplest form when the numerator and the denominator have no common
factors other than 1. Factors are numbers that are multiplied to get a product. First, let’s look at
the factors of 5 and 10.
5
10
We can see, that other than 1, 5 is the only
– 1, 5
– 1, 2, 5, 10
Now we are ready to put
common factor of 5 and 10.
5
in simplest form.
10
To do that, we need to divide both the numerator and denominator by their greatest
common factor, which we know is 5.
This works because
5
is the same as 1, and anything divided by 1 is the same number.
5
1
5
5
÷
=
2
5
10
©2011
SOL 5.6 Computation and Estimation
NOTEPAGE FOR STUDENT
Page 3
Adding and Subtracting Fractions and Mixed Numbers
1
We can see that
is in its simplest form by checking the factors of 1 and 2.
2
1
– 1
We can see that the only common factor is 1; therefore, the
2
– 1, 2
fraction
1
is in simplest form.
2
Now, let’s put it all together!
7
10
Here is the solution to the original problem:
8
10
+
15
10
=1
5
10
=1
1
2
PRACTICE!
1.
2
3
+
8
8
2.
3 5
+
6 6
An easy way to change an improper fraction to a mixed number is to divide the numerator by
the denominator and write the remainder as a fraction. Check out the example below.
The divisor 10 becomes
the denominator again.
1
10 15
= 1
- 10
5
10
We still simplify! = 1
1
2
The remainder 5 becomes the numerator.
5
PRACTICE!
1. Change the improper fraction
12
into a mixed number by following the example above.
8
Think! Which way works best (in your opinion)?
Let’s see what the same problem would look like using a picture.
©2011
SOL 5.6 Computation and Estimation
NOTEPAGE FOR STUDENT
Page 4
Adding and Subtracting Fractions and Mixed Numbers
7
10
+
8
10
+
=
1
=
We can see from the pie pieces that adding the two pies with
a total of 1
5
.
10
1
5
1
or
10
2
7
8
and
shaded gives us
10
10
5
1
=1
10
2
PRACTICE!
Use a picture to represent the following problem:
1. 1
1
1
+
4
2
Equivalent Fractions
The fractions
5
1
and
are called equivalent fractions because they name the same amount.
10
2
We find equivalent fractions by multiplying or dividing the numerator and the denominator of a
fraction by the same non-zero number.
In our last problem, the numerator and denominator of the fraction
order to come up with
1
2
5
were both divided by 5 in
10
5
5
1
÷
=
5
10
2
When adding or subtracting fractions that have different or unlike denominators, we have to
rewrite the fractions to have a common or like denominator. The reason we do this is because
we can only add or subtract fractions that name the same kind of thing or amount.
©2011
SOL 5.6 Computation and Estimation
NOTEPAGE FOR STUDENT
Page 5
Adding and Subtracting Fractions and Mixed Numbers
Least Common Multiple (LCM) & Least Common Denominator (LCD)
In order to add two fractions with unlike denominators, we have to find a common
denominator. To find it, we have to find the LCM, or the least common multiple of the two
denominators.
When we find the LCM or least common multiple of the unlike denominators, we will have the
LCD or least common denominator.
Let’s try the example
1
2
+
to find the LCD, or least common denominator.
4
3
First, we have to find the LCM, or least common multiple for the denominators 4 and 3.
Remember, once we have found the least common multiple or LCM of the two denominators, we
can find the LCD or least common denominator.
Multiples of 4:
4, 8, 12, 16, 20, 24, etc.
Multiples of 3:
3, 6, 9, 12, 15, 18, 21, 24, etc.
We can see that 12 and 24 are common multiples, but 12 is the smallest or least common
multiple. That makes 12 the LCD or least common denominator.
Using paper and pencil, we rewrite the fractions with unlike denominators like this:
1
=
4
1 x 3
4 x 3
=
3
12
2
2 x 4
3 x 4
=
8
12
+ 3
=
We now have a common, or like denominator.
11
12
©2011
SOL 5.6 Computation and Estimation
NOTEPAGE FOR STUDENT
Page 6
Adding and Subtracting Fractions and Mixed Numbers
Here’s how that same problem can be represented using fraction strips:
1
4
+
2
3
3
12
8
12
+
We can see that
1
3
is equal or equivalent to
4
12
We can see that
2
8
is equal or equivalent to
3
12
=
When we combine the numerators 3 and 8, we get a sum of 11 or
11
12
11
12
PRACTICE!
1. Why is the addition problem below incorrect?
Show the correct way to solve the problem.
1
2
3
+
=
4
3
7
2. Solve the following example as both a paper and pencil model and an area model.
We started with
Paper & Pencil
–
3
8
1
4
3
1
of a pie. Someone ate
of the pie. How much is left?
8
4
Area Model
–
©2011
SOL 5.6 Computation and Estimation
NOTEPAGE FOR STUDENT
Page 7
Adding and Subtracting Fractions and Mixed Numbers
Adding and Subtracting Mixed Numbers
As we know, a mixed number has two parts: a whole number and a fraction. The value of a
mixed number is the sum of these two parts.
5
whole number
7
8
fraction
1. When we add or subtract mixed numbers, we add and subtract the fraction part first.
We must make sure that we have common or like denominators when we add or subtract the fraction part. That
means we have to find equivalent fractions with the least common denominator (LCD).
2. Then, we add and subtract the whole number part. Sometimes, we will have to regroup.
3. Finally, make sure the answer is in simplest form.
3
1
4
+2
7
8
Let’s look at the addition problem and follow the steps above :
1. Add the fraction parts. Before we can do this we must have like denominators. To find
like denominators we must find the LCM of the two denominators.
LCM or least common multiple:
4 – 4, 8, 12, 16
8 – 8, 16, 24
We see that 8 and 16 are common multiples, but 8 is the least common multiple.
Therefore . . .
?
?
1
x
=
4
?
8

2
2
1
x
=
4
2
8
7
?
?
x
=
8
?
8

7
1
7
x
=
8
8
1
We are now ready to add the fraction parts.
1
4
7
2
8
3
2
8
7
=+ 2
8
= 3
9
8
©2011
SOL 5.6 Computation and Estimation
NOTEPAGE FOR STUDENT
Page 8
Adding and Subtracting Fractions and Mixed Numbers
1
2
3. We are now ready to add the whole numbers.
3
= 3
4
8
7
7
2
=+ 2
8
8
5
9
8
4. Finally, we write the fraction in its simplest form.
How do we change 5
Take the fraction
9
to its simplest form?
8
9
and divide the numerator into the denominator:
8
1
The divisor 8 becomes the denominator.
18
8 9
-8
1
The remainder 1 becomes
the numerator.
We aren’t done yet!
We take the whole number 1 in 1
1
9
and add it to the whole number 5 (5 ).
8
8
1+5=6
Include the remainder
Answer = 6
1
and put it all together!
8
1
8
PRACTICE!
1. 5
2
2
+1
5
3
2. 1
2
3
+3
3
4
©2011
SOL 5.6 Computation and Estimation
NOTEPAGE FOR STUDENT
Page 9
Adding and Subtracting Fractions and Mixed Numbers
Changing a Mixed Number to an Improper Fraction
Sometimes we may want to change a mixed number into an improper fraction before we
subtract two numbers like 1
3
4
–
5
5
In this problem, our fraction parts have like denominators, but we cannot subtract a larger
number like
4
3
from a smaller number like .
5
5
In order to subtract, we need to change the mixed number 1
3
into an improper fraction.
5
Follow these steps:
Change the number 1 into the fraction
5
3
. We will use 5 as the denominator because the fraction
5
5
has a 5 as the denominator. We know
We then add
5
5
is equal to 1 ( = 1).
5
5
5
3
to the fraction part
and add:
5
5
We see that the improper fraction
5
+
5
Change the mixed numbers into improper fractions.
5
2
5
2.
1
2
3
8
5
8
3
is also 1 , the mixed number. They are equivalent!
5
5
PRACTICE!
1.
3
=
5
©2011
SOL 5.6 Computation and Estimation
NOTEPAGE FOR STUDENT
Page 10
Adding and Subtracting Fractions and Mixed Numbers
3
Using the same mixed number 1 , let’s use the algorithm or mathematical process for changing a
5
mixed number into an improper fraction:
1. Multiply the denominator by the whole number.
8
4
4
=
In this problem, we multiply 5, the denominator
5 5
5 in the fraction
whole number 1 in 1
3
, the mixed number.
5
3
5
by the
5 x 1 = 5.
2. Add the numerator.
3
5
In this problem, we added 5 + 3 ( ) = 8.
3. Use the same denominator.
In this problem, the denominator is 5. Answer:
4. Result: We turned the mixed number 1
8
5
3
8
into , the improper fraction.
5
5
Example:
Let’s try changing the mixed number 2
1
to an improper fraction following the algorithm or
4
mathematical process above.
1. Multiply the denominator 4 by the whole number 2.
4x2=8
2. Add the numerators.
8+1=9
3. Use the same denominator.
9
4
4. We turned the mixed number 2
1
9
into the improper fraction .
4
4
©2011
SOL 5.6 Computation and Estimation
NOTEPAGE FOR STUDENT
Page 11
Adding and Subtracting Fractions and Mixed Numbers
PRACTICE!
1. Change the mixed number 1
2
into an improper fraction. Be prepared to explain the process.
7
2. Change the improper fraction
8
into a mixed number. Be prepared to explain the process.
5
Adding & Subtracting Fractions in Word Problems
Example:
A family orders a pepperoni pizza. The pepperoni pizza is cut into 12 slices.
The son eats
1
1
of the pizza. The daughter eats
of the pizza. Mom eats 2 pieces.
4
2
Is there any left for dad? If so, how many pieces?
Steps:
1. Set up an estimate:
1
1
1
(daughter’s share) +
(son’s
and mom’s share)
2
2
4
1
2. Set up and solve the problem:
1
2
To add up how much of the pizza
was eaten, change unlike
denominators to common or
like denominators.
+
=
6
12
1
4
=
3
12
2
12
=
2
12
12
12
1 pizza =
–
Subtract: the whole
pizza from the part
the family ate.
11
12
1
12
11
12
There is 1 piece left out of 12.
3. Check answer for reasonableness.
Compare our estimate (1 – whole pizza) and exact answer (
The answers are definitely close.
©2011
11
or 11 pieces out of 12).
12
SOL 5.6 Computation and Estimation
NOTEPAGE FOR STUDENT
Page 12
Adding and Subtracting Fractions and Mixed Numbers
PRACTICE!
Use the problem-solving steps above to solve the following word problems:
1. 3 pizzas were delivered for the pool party. 2
1
of the pizzas were eaten.
4
How much pizza is left?
2. Sam walked 2
1
2
miles the first day. He walked 6 miles the second day.
3
2
How many total miles did Sam walk?
Remember! Addition and subtraction are inverse operations. Multiplication and division are also
inverse operations. Inverse operations are opposite operations. They undo each other.
©2011