Download Unit 3- Multiplying Fractions

Multiplying & Dividing Fractions and Mixed Numbers
Example: If there was ½ of a whole pie left, and you wanted to take 2/3 of
the half to eat, what fraction of the entire pie would you take to eat?
Visually, we are taking 2/3 of the half left, and as you can see below, this
amounts to 2/6 of the whole pie. We divide our half pieces into thirds and then
take two of those pieces.
This example illustrates the fraction multiplication 1/2 x 2/3. We end up
multiplying the denominators to make smaller piece sizes (6ths) but we also end
up multiplying the numerators to make more pieces.(2 times as many) Note that
no common denominator is needed!
To multiply fractions, multiply numerators together and
also multiply denominators together.
Example: Multiply
Here, we multiply 3 x 2 to get 6 (pieces), and multiply 4 x 5 to get 20 (each piece
is now one 20th in size). So our result is 6/20 which may be reduced to 3/10.
Example: Multiply
Simplifying Before You Multiply Fractions
In the previous two examples, we got results of 3/10 and 3/8 after reducing.
Could we have reduced before doing any multiplying? The answer is “Yes!”
In the first example, we could divide out a factor of 2 from both the 4 and 2 and
then multiply the resulting fractions of 3/2 x 1/5, as shown below.
In the second example where 36/96 reduced to 3/8, we could have divided out
factors of 3, factors of 2, and again factors of 2, leaving only 1/1 x 1/1 x 3/8
which multiplies to 3/8.
Why does this work? We can reduce the fraction 36/96 by dividing out
numerator and denominator by 2 to get 18/48, then divide out by 2 again to get
9/24, and then divide out by 3 to get 3/8. Whether we divide out factors before
multiplying or after, we are reducing the end result in the same way.
To simplify fraction multiplication by canceling first, you may
divide common factors from any numerator and denominator.
Then multiply what is left.
CONTINUED ON NEXT PAGE . . .
Example – Multiply
factors.
by first canceling common
As you can see below, we are able to divide out by factors of 10, 2, 2, and 2 to
end up with 1/1 x 1/5 x 1/1 which multiplies to (1x1x1)/(1x5x1) = 1/5. Note that if
we did not cancel, we would have an answer or 240/1200 that would then require
quite a bit of effort to reduce!
In this example, we may divide out by a common factor of y and then divide out
a common factor of 2 to get a final simplified answer of ½
What is so important about canceling factors before multiplying?
In Algebra II, we will be multiplying fractions like the ones below, and you will
have to factor the expressions and then cancel factors before multiplying!
Multiplying Mixed Numbers
To multiply mixed numbers
1. Convert the mixed numbers into fractions.
2. Multiply the fractions. Then convert back to a mixed
number, if desired.
Example – Multiply 1 ½ by 3 ¾
We convert each mixed number into a fraction. 1 ½ = 3/2 and 3 ¾ = 15/4. Now
multiply 3/2 times 15/4. The result is 5 5/8. Note we could not cancel this time.
Example – Multiply 3/8 x 1600
In this case, 1600 is our “mixed number” and we write it as 1600/1. This results in
the fraction multiplication shown below. Notice how much easier this problem is if
you divide out common factors of 8 before multiplying!
Dividing Fractions and Mixed Numbers
Division of a number is defined as the multiplication by the reciprocal of the
number. For example 8  2 = 8  ½ = 4. We can apply this definition to dividing
fractions as well where the reciprocal consists of the fraction being “turned
upside down” or as we say “inverted”.
To divide by a fraction, multiply by the reciprocal of the
fraction you are dividing by.
Example: Divide 4  3/8
CONTINUED ON NEXT PAGE . . .
A Fraction Represents Division
We may always interpret a fraction as the division of the
numerator by the denominator.
10/2 = 10  2 = 5
Remember that a fraction always may be interpreted as a division!
Multiplying & Dividing Fractions With Negatives
When multiplying or dividing fractions or mixed numbers that are negative,
you multiply or divide as if they were positive fractions or mixed numbers.
Then apply the rules for multiplying and dividing negatives: If there is one
negative and one positive, the answer is negative. If both numbers
multiplied or divided are negative, the answer is positive.
Here, we invert the second fraction and multiply, as if the two negative signs
were not even there. Since we have a negative times a negative, the answer is
positive, as shown below. Hence the negative signs are not even included in the
last step.
Example: Write the fraction – (½) in 2 different ways.
In the first fraction, we are taking the negative of the whole fraction, in the second
fraction, we have a fraction equal to –1 divided by 2 which results in a negative
and in the last fraction we have 1 divided by –2 which results in a negative. All
three results are equal to negative one-half, or –0.50 as a decimal.
The fraction of fraction implies that the two fractions are divided. Note that we
can drop the two negative signs of the second fraction since a negative divided
by a negative is positive. But there is still one negative sign left, so the final
answer is negative.
Here, “a” is a variable. We are multiplying the fraction a/3 by itself to get
Since we do not know the value of “a” we leave a2 as it is.
First, convert both mixed numbers into improper fractions. Then invert and
multiply to get
Note that absolutely nothing could cancel before multiplying. This means that our
final answer can not be reduced any further.