Thursday, October 13, 2011



Multiplying Fractions by Fractions

We have studied how to find  
1
2
   of a whole number. For example 
1
2
  of 24 is  
1
2
  × 24 = 12. 
  • NOTE: The word OF translates into MULTIPLICATION.
Finding  
1
2
  of any fraction also means multiplying  
1
2
   times that fraction!
1
2
  ×  1
3
   means   
1
2
  of   1
3
 .
Half of the 1/3-piece      is   .
1
2
  ×  1
3
  = 
1
6
1
4
  ×  1
3
   means   
1
4
  of   1
3
 .
A fourth part of the 1/3-piece      is   .
1
4
  ×  1
3
  = 
1
12
1. The pictures show how much pizza is left, and you share it equally with one, two, or three other
    people. Divide the pizza. What kind of part do you get? Write a multiplication sentence.
a. Find  
1
2
  of   
1
2
  × 
1
4
  =
   
[available in the book]    
  [available in the book]
l. Find  
1
4
  of   
       ×        

  =
 
Shortcut - multiply fractions of the type 1/n
You might have noticed that in the above exercises, all of our fractions
were of the form   1
n
  (where n is a whole number),
and that we could have just multiplied the
denominators to get the new denominator.
1
4
  ×  1
5
  =  
1
20
   or   
1
2
  ×  1
6
  =  
1
12


We have now studied how to find 1/2 or 1/3 or 1/5 of some fractions. But how about finding some
other kind of fractional part? Let's again compare this to finding fractional parts of whole numbers.
Remember? To find  
3
4
  of 16, or in other words  
3
4
  × 16, you can first find  
1
4
  of 16, which is 4.
Then just take that three times, which is 12. In other words, 
3
4
  × 16 = 12.
We can use the same exact idea when finding a fractional part of another fraction.
Example. Find  
2
3
   of  
1
4
 .    First we find  
1
3
  of  
1
4
 , which is  
1
12
 .
Then,  
2
3
  of  
1
4
  is double that much, or  
2
12
  . 
2
3
  × 
 =  
Example. Find  
4
5
   of  
1
7
First we find  
1
5
  of  
1
7
 , which is  
1
35
 .  Then,  
4
5
  of  
1
7
  is four times that much, or  
4
35
  . 
Multiplying a fraction by a fraction means taking that fractional part of the fraction. 
It is just like taking a certain part of what is left over, when that leftover is already a fraction.
2. The pictures show how much pizza is left, and you get a certain part of the leftovers. How much will
    you get? Write a multiplication sentence. Color in an answer picture.
a.   
3
4
  ×    =  
    (First find 1/4 of 1/2, then multiply the result by 3.)
 
1
4
  ×  
1
2
   =  
1
8
    and   
1
8
  × 3 = 
[available in the book]
  [available in the book]
  [available in the book]


Shortcut for multiplying fractions.
Multiply the numerators to get the numerator for the answer.
Multiply the denominators to get the denominator for the answer.
Let's compare.
The roundabout way Shortcut
5
6
  × 
1
2
   = ? 
(First find 1/6 of 1/2, then multiply the result by 5.)
1
6
  ×  
1
2
   =  
1
12
    and   
1
12
  × 5 = 
5
12
  
5
6
  ×  
1
2
   =  
5 × 1
6 × 2
  = 
5
12
  
2
8
  × 
3
5
   = ? 
(First find 1/8 of 3/5, then multiply the result by 2.
And, to find 1/8 of 3/5, first find 1/8 of 1/5, and
multiply that by 3.)
1
8
  × 
1
5
   = 
1
40
    That by 3 is  
1
40
  × 3 = 
3
40
  
Then, that by 2 is  
3
40
  × 2 = 
6
40
   = 
3
20
  (simplified)
2
8
  ×  
3
5
   =  
2 × 3
8 × 5
  = 
6
40
  = 
3
20
  
With the “roundabout way” we do multiplications separately.
With the shortcut, we can just do them all in “one go”.

Study the examples on the right. Remember to always give your final
answer as a mixed number and in
a simplified form.
    
3
7
  ×  
4
9
   =   
3 × 4
7 × 9
   =  
12
63
  = 
4
21
                         
   
12
5
  ×  
9
8
   =   
12 × 9
5 × 8
   =  
108
40
   =  
27
10
   =  2
7
10
3. Multiply. Give your answers in the lowest terms (simplified) and as a mixed number, if possible.
a.  
3
9
  × 
2
9
 
 
f.   10 ×
5
7


4. Multiply. Give your answers in the lowest terms (simplified) and as a mixed number, if possible.
a.  
3
4
 ×
7
8
   
  [available in the book]  
[available in the book]  
i.  
3
5
 ×
1
10
 =
5. There was 1/4 of the pizza left. Marie ate 2/3 of that.
    What part of the original pizza did she eat?
    What part of the original pizza is left now?

6. [available in the book]


7. [available in the book]
Brownies 3 cups sweetened carob chips
8 tablespoons Extra virgin olive oil
2 eggs
1/2 cup honey
1 teaspoon vanilla
3/4 cup whole wheat flour
3/4 teaspoon baking powder
1 cup walnuts or other nuts








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