adding and subtracting fractions and mixed...
TRANSCRIPT
Adding and Subtracting Fractions and Mixed Numbers
SOL 5.6 Computation and Estimation
NOTEPAGE FOR STUDENT
Page 5
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:
+ =
15
10
That’s pretty easy. We add the numerators, 7 plus 8 equals 15. The denominator is tenths.
The solution is
15
10
or fifteen-tenths.
Improper Fractions
However, we are not finished with the sum in this problem. The fraction
15
10
is called an improper fraction. An improper fraction is a fraction where the numerator is greater than the denominator.
4
9
and
7
11
are also examples of improper fractions. We have to rewrite an 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
4
9
becomes 2
4
1
, a mixed number. The improper fraction
7
11
becomes 1
7
4
, a mixed number. The value of a mixed number is the sum of its two parts.
Writing an Improper Fraction as a Mixed Number
Let’s write the improper fraction
15
10
as a mixed number.
We know that
10
10
= 1. Therefore,
15
10
represents more than one whole.
7
10
How much more? To find out we subtract:
We now know that the improper fraction
15
10
can be written as 1
5
10
, a mixed number. This mixed number could also be written as: 1 +
5
10
That’s because a mixed number is the sum of 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
10
is not in its simplest form.
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.
15
10
10
10
5 – 1, 5
10 – 1, 2, 5, 10
Now we are ready to put
5
10
in simplest form.
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
5
is the same as 1, and anything divided by 1 is the same number.
5
10
÷
5
5
=
1
2
We can see that
1
2
is in its simplest form by checking the factors of 1 and 2.
5
10
1
1
2
– 1
2 – 1, 2
Now, let’s put it all together!
Here is the solution to the original problem:
7
10
+
8
10
15
10
= 1
5
10
= 1
1
2
PRACTICE!
1.
8
2
+
8
3
2.
6
3
+
6
5
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.
1
15
10
= 1
5
10
We still simplify! = 1
2
1
8
1
5
10
5
5
PRACTICE!
1. Change the improper fraction
8
12
into a mixed number by following the example above.
Think! Which way works best (in your opinion)?
Let’s see what the same problem would look like using a picture.
1
2
3
5
3
5
3
5
8
5
3
5
8
5
4
1
7
10
+
8
10
= 1
5
10
or
1
2
+
=
We can see from the pie pieces that adding the two pies with
7
10
and
8
10
shaded gives us a total of 1
5
10
. 1
5
10
= 1
1
2
PRACTICE!
Use a picture to represent the following problem:
1
2
Equivalent Fractions
The fractions
5
10
and
1
2
are called equivalent fractions because they name the same amount.
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
5
10
were both divided by 5 in order to come up with
1
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.
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
4
1
+
3
2
to find the LCD, or least common denominator.
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
4
EMBED Equation.3
x
x
EMBED Equation.3
3
3
=
12
3
3
2
=
3
2
x
x
EMBED Equation.3
4
4
=
12
8
12
11
Here’s how that same problem can be represented using fraction strips:
4
1
3
2
12
3
12
8
12
11
We can see that
1
4
is equal or equivalent to
3
12
We can see that
2
3
is equal or equivalent to
8
12
When we combine the numerators 3 and 8, we get a sum of 11 or
11
12
PRACTICE!
1. Why is the addition problem below incorrect? Show the correct way to solve the problem.
1
4
+
2
3
=
3
7
2. Solve the following example as both a paper and pencil model and an area model.
We started with
3
8
of a pie. Someone ate
1
4
of the pie. How much is left?
Paper & Pencil Area Model
8
3
4
1
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
7
8
Let’s look at the addition problem and follow the steps above : 3
1
4
+ 2
7
8
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
4
x
?
?
=
8
?
(
1
4
x
2
2
=
8
2
8
7
x
?
?
=
8
?
(
8
7
x
1
1
=
8
7
We are now ready to add the fraction parts.
3
1
4
= 3
2
8
2
7
8
=+ 2
7
8
9
8
3. We are now ready to add the whole numbers. 3
1
4
= 3
2
8
2
7
8
=+ 2
7
8
5
9
8
4. Finally, we write the fraction in its simplest form.
How do we change 5
9
8
to its simplest form?
Take the fraction
9
8
and divide the numerator into the denominator:
1
9
8
We aren’t done yet!
We take the whole number 1 in 1
1
8
and add it to the whole number 5 (5
9
8
).
1 + 5 = 6
Include the remainder
1
8
and put it all together!
Answer = 6
1
8
PRACTICE!
1. 5
5
2
+1
3
2
2. 1
3
2
+ 3
4
3
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
5
–
4
5
In this problem, our fraction parts have like denominators, but we cannot subtract a larger number like
4
5
from a smaller number like
3
5
.
In order to subtract, we need to change the mixed number 1
3
5
into an improper fraction.
Follow these steps:
Change the number 1 into the fraction
5
5
. We will use 5 as the denominator because the fraction
3
5
has a 5 as the denominator. We know
5
5
is equal to 1 (
5
5
= 1).
We then add
5
5
to the fraction part
3
5
and add:
5
5
+
3
5
=
8
5
We see that the improper fraction
8
5
is also 1
3
5
, the mixed number. They are equivalent!
PRACTICE!
Change the mixed numbers into improper fractions.
1. 5
5
2
2. 1
3
2
Using the same mixed number 1
3
5
, let’s use the algorithm or mathematical process for changing a mixed number into an improper fraction:
8
5
-
4
5
=
4
5
Example:
Let’s try changing the mixed number 2
4
1
to an improper fraction following the algorithm or mathematical process above.
1. Multiply the denominator 4 by the whole number 2.
4 x 2 = 8
2. Add the numerators.
8 + 1 = 9
3. Use the same denominator.
4
9
4. We turned the mixed number 2
4
1
into the improper fraction
4
9
.
PRACTICE!
1. Change the mixed number 1
7
2
into an improper fraction. Be prepared to explain the process.
2. Change the improper fraction
5
8
into a mixed number. Be prepared to explain the process.
Adding & Subtracting Fractions in Word Problems
Example:
A family orders a pepperoni pizza. The pepperoni pizza is cut into 12 slices.
The son eats
of the pizza. The daughter eats
of the pizza. Mom eats 2 pieces.
Is there any left for dad? If so, how many pieces?
Steps:
1. Set up an estimate:
(daughter’s share) +
(son’s
and mom’s share) 1
2. Set up and solve the problem:
=
1 pizza =
=
–
+
=
12
11
3. Check answer for reasonableness.
Compare our estimate (1 – whole pizza) and exact answer (
or 11 pieces out of 12).
The answers are definitely close.
PRACTICE!
Use the problem-solving steps above to solve the following word problems:
1. 3 pizzas were delivered for the pool party. 2
of the pizzas were eaten.
How much pizza is left?
2. Sam walked 2
2
1
miles the first day. He walked 6
2
3
miles the second day.
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.
�EMBED Equation.DSMT4��� - �EMBED Equation.DSMT4��� = �EMBED Equation.DSMT4���
We can see, that other than 1, 5 is the only common factor of 5 and 10.
We can see that the only common factor is 1; therefore, the fraction �EMBED Equation.DSMT4��� is in simplest form.
�EMBED Excel.Chart.8���
�EMBED Excel.Chart.8���
�EMBED Excel.Chart.8���
�EMBED Excel.Chart.8���
+
� EMBED Equation.3 ���
There is 1 piece left out of 12.
=
+
+
–
–
fraction
whole number
5
�EMBED Equation.DSMT4��� ÷ �EMBED Equation.DSMT4��� = �EMBED Equation.DSMT4���
We now have a common, or like denominator.
-8
1
The divisor 8 becomes the denominator.
The remainder 1 becomes
the numerator.
Multiply the denominator by the whole number.
In this problem, we multiply 5, the denominator in the fraction �EMBED Equation.DSMT4��� by the whole number 1 in 1�EMBED Equation.DSMT4��� , the mixed number. 5 x 1 = 5.
Add the numerator.
In this problem, we added 5 + 3 (�EMBED Equation.DSMT4���) = 8.
Use the same denominator.
In this problem, the denominator is 5. Answer: �EMBED Equation.DSMT4���
Result: We turned the mixed number 1�EMBED Equation.DSMT4��� into �EMBED Equation.DSMT4���, the improper 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.
The remainder 5 becomes the numerator.
- 10
1. 1� EMBED Equation.3 ��� + �EMBED Equation.DSMT4���
The divisor 10 becomes
the denominator again.
To add up how much of the pizza was eaten, change unlike denominators to common or like denominators.
Subtract: the whole pizza from the part the family ate.
©2011
Chart1
1
1
1
1
1
1
1
1
1
1
Sheet1
1
1
1
1
1
1
1
1
1
1
Sheet2
Sheet3
Chart1
1
1
1
1
1
1
1
1
1
1
Sheet1
1
1
1
1
1
1
1
1
1
1
Sheet2
Sheet3
Chart1
1
1
1
1
1
1
1
1
1
1
Sheet1
1
1
1
1
1
1
1
1
1
1
Sheet2
Sheet3
Chart1
1
1
1
1
1
1
1
1
1
1
Sheet1
1
1
1
1
1
1
1
1
1
1
Sheet2
Sheet3