Upload
verity-sharp
View
213
Download
0
Embed Size (px)
Citation preview
1
MULTIPLYING AND DIVIDING FRACTIONS
The Number System
© 2013 Meredith S. Moody
2
Objective: You will be able to…
Convert fractions to their reciprocals and back
Divide a fraction by another fractionSolve problems by dividing fractions by
fractions
© 2013 Meredith S. Moody
3
The Fraction
When we divide a whole into equal parts, we represent those parts as fractions
© 2013 Meredith S. Moody
4
Part of a Part
Why would we need to divide a fraction by a fraction?
We may not have a whole number to divide into pieces
How much is half of a half? What about a third of a half? What about a half of a fourth?
© 2013 Meredith S. Moody
5
Example 1
Let’s say you have a cake for a partyThe day of the party, you serve half of the
cake to your guests, but have half left overThe next day, you have some friends over and
you want to serve the other half of the cake, but you want everyone to have the same number of pieces
Here, you would have to divide a fraction ( ½ of the cake) by a whole number (the number of friends you have over)
© 2013 Meredith S. Moody
6
Example 1, visual
Drawing pictures is helpful when dividing a fraction by a fraction
You start with a whole, and then divide it in half
1 whole cake ½ of the cake
© 2013 Meredith S. Moody
7
Example 1, visual, continued
When half the cake is gone, you only have the leftover half to work with
If you have 1 friend over, you have to split the half into 2 pieces (1 for you, 1 for the friend)
½ of a ½
© 2013 Meredith S. Moody
8
Example 1, visual, continued
You can look at how the smallest piece fits into the whole:
½ of a ½ is equal to ¼ of the whole
½ of a ½
© 2013 Meredith S. Moody
9
Example 1, Algebraic translation
What operation represents ‘½’ something?½ of a 1 whole is the same thing as dividing 1
whole by 2This is because the fraction bar represents
the operation of division ( ½ = 1 ÷ 2)So if we want to ‘half’ something, we divide
by 2So, ‘half’ing a half is = ½ ÷ 2½ ÷ 2 = ¼
© 2013 Meredith S. Moody
10
Reciprocals
Reciprocal fraction: The result of interchanging the numerator and denominator in a fraction
A whole number can be represented as a fraction over ‘1’ (because a number divided by ‘1’ is itself)
Therefore, the whole number 5 = 5/1 and the
reciprocal = 1/5
© 2013 Meredith S. Moody
11
Example 2, visual
Working with halves is easier than other fractions
What if I wanted to know what a third of a half is?
1 whole cake½ of the
cake
1/3 of the ½
1/6 of the whole
© 2013 Meredith S. Moody
12
Example 2, Algebraic translation
Here, I took ½ and divided by 3 to find 1/3 of the ½
½ ÷ 3 = 1/6½ ÷ 3/1 = 1/6Do you see a pattern yet?
© 2013 Meredith S. Moody
13
Example 3
What if I wanted to find something more complicated?
What if the first day of my party, I served 1/3 of the cake, so I had 2/3 left over
The next day, I had 8 friends over, and I wanted each person to have a slice, but three said ‘no thanks’ – so I only served 5/8 of the 2/3
What is the leftover 3/8 of the 2/3?So I want to know: what is 3/8 of 2/3?
© 2013 Meredith S. Moody
14
Example 3, visual
Here is the visual representation of 3/8 of 2/3
1 whole cake the cake
2/3 of
is leftwhole
of
cakethe6/243/8 of
2/3 ofthe cake
© 2013 Meredith S. Moody
15
Example 3, Algebraic translation
I divided 2/3 into 8 pieces 2/3 ÷ 8But I wasn’t interested in just 1 of those
pieces, I wanted to figure out what 3 of them were
I didn’t want 2/3 ÷ 8/1I wanted 2/3 ÷ 8/3So 2/3 ÷ 8/3 = 6/24 Do you see a pattern yet?
© 2013 Meredith S. Moody
16
Seeing the Pattern
Let’s review:½ ÷ 2 = ½ ÷ 2/1 = ¼ ½ ÷ 3 = ½ ÷ 3/1 = 1/62/3 ÷ 8/3 = 6/24What is the pattern?That’s right! The answer is the dividend (1st
fraction) multiplied by the reciprocal of the divisor (2nd fraction)!
© 2013 Meredith S. Moody
17
Using Reciprocals
We can use reciprocals and our understanding of multiplication to divide a fraction by a fraction
½ ÷ 2 = ½ ÷ 2/1 = ¼ When I divide a ½ by two, I’m finding ½ of
the ½ What operation does ‘of’ represent?That’s right! Multiplication!½ ÷ 2 = ½ ÷ 2/1 = ½ x ½ = ¼
© 2013 Meredith S. Moody
18
Check the Pattern
Does this pattern always work?Let’s check our second cake scenario, where
we wanted to find 1/3 of ½ We took ½ the cake and cut it into 3rds and
looked at one of those pieces: ½ ÷ 3 = ½ ÷ 3/1 = 1/6
1/3 of ½ = 1/3 x ½ = 1/6Does the pattern fit?Yes!
© 2013 Meredith S. Moody
19
Check the Pattern
Let’s check our last cake scenario, where we wanted to find 3/8 of 2/3
We took 2/3 of the cake and cut it into 8ths and looked at 3 of those pieces
2/3 ÷ 8/3 = 6/243/8 of 2/3 = 3/8 x 2/3 = 6/24Does the pattern fit?Yes!
© 2013 Meredith S. Moody
20
Algebraic Rule
Based on the pattern we found, we can write a rule for dividing a fraction by another fraction:
a/b ÷ c/d = a/b x d/c = ad/bc
© 2013 Meredith S. Moody
21
You try
Evaluate the following expressions. Remember to write your answer in lowest terms!
½ ÷ ⅛ 8/2 = 4
⅞ ÷ ¼ 28/8 = 7/4
© 2013 Meredith S. Moody
22
Mixed Numbers
What if I have mixed numbers as part of my problem?
Convert the mixed number to an improper fraction, then solve
© 2013 Meredith S. Moody
23
You try
6 and ¼ ÷ 3 and 5/9 25/4 ÷ 32/9 25/4 x 9/32 225/256
3 and 2/7 ÷ 2 and 5/6 21/2 ÷ 17/6 21/2 x 6/17 126/34 3 and 24/34 3 and 12/17
© 2013 Meredith S. Moody