Multiplying and Dividing Fractions
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Multiplying and Dividing Fractions |
Multiplying Fractions
When we multiply integers, the answer is always a larger value than the two numbers being multipled. Something different about multiplying proper fractions is that the answer will always be smaller than both of the fractions being multiplied. You could say that when you multiply two fractions you are getting a portion of a portion of a whole. For example, let's multiply 1/8 x 1/4. If you start with a whole pizza and take 1/8 of the pizza, you have one slice. Now if you take 1/4 of that slice of pizza you end up with a very small portion of the whole pizza. One fourth of one eigth of the pizza is one thirtysecondth of the whole pizza. Mathematically, this can be written as:
1/8 x 1/4 = 1/32
When you multiply fractions you multiply the numerators to get the numerator for the answer and you multiply the denominators to get the denominator for the answer.
1 x 1 = 1
8 x 4 = 32
therefore 1/8 x 1/4 = 1/32Let's look at an example using the integer bars. In this next example we multiply 1/2 x 1/4. We start with a size 8 bar as one unit, then split it in two halves (two size 4 bars). Next the size 4 bar is split into four quarters (four size 1 bars). The answer is one of the size 1 bars or 1/8 of the unit bar. Here is the picture that shows the process:
Mathematically, this is written as:
1/2 x 1/4 = 1/8
Our next example multiplies a proper fraction and an improper fraction:
1/2 x 4/3
We start with a size 6 bar because it is the multiple of the two denominators, 2 x 3 = 6. First we divide it into two halves because the first fraction is 1/2. Then we divide one half into thirds and take four of them because the second fraction is 4/3. The result is 4/6 which can be simplified to 2/3.
Mathematically, this is written as:
1/2 x 4/3 = (1 x 4) / (2 x 3) = 4/6 = 2/3
Now let's multiply a fraction and an integer. We will solve the equation 2/3 x 9. First we convert the value of 9 to the equivalent fraction 9/1. We then multiply the numerators to get 18 and the denominators to get 3. The fraction 18/3 is then simplified to 6.
2/3 x 9 = 2/3 x 9/1 =(2 x 9) / (3 x 1) = 18/3 = 6
Therefore, 2/3 of 9 is 6.
For our last example we will multiply four fractions. The process is the same as for multiplying only two fractions. We multiply all of the numerators to get the numerator for the answer and multiply all of the denominators to get the denominator for the answer. Another way to solve an equation which multiplies four fractions is to multiply two fractions then multiply the remaining two fractions, then multiply the two results. Let's solve the following equation using both of these methods. First we will multiply all the fractions together:
2/3 x 9/2 x 3/8 x 4
(2 x 9 x 3 x 4) / (3 x 2 x 8 x 1)
216 / 48 = 108/24 = 54/12 = 27/6 = 9/2 = 4 1/2Now we will solve the same equation by multiplying 2/3 x 9/2 then 3/8 x 4 then multiplying those two results to get the final answer:
2/3 x 9/2 x 3/8 x 4
(2/3 x 9/2) x (3/8 x 4)
18/6 x 12/8which can be simplified to:
3/1 x 3/2 = 9/2 = 4 1/2
The answer is 4 1/2 with either method that is used.
Dividing Fractions
When dividing with integers we learned that dividing 8 by 4 (8 ÷ 4) we need to find how many size 4 bars will fit in one size 8 bar. We use the same method when dividing fractions. For example, let's divide 1/2 ÷ 1/4. How many 1/4's can you fit in 1/2? Or how many quarters can you have in one half? Let's use integer bars to solve this equation.
The purple bar represents one unit, the red bar represents 1/2 unit, and each white bar represents 1/4 unit. We need to find out how many 1/4 (white) bars will fit into one 1/2 (red) bar. Using the picture above, can you figure it out? The picture shows that 2 white (1/4) bars fit in one red (1/2) bar, so the answer is 2.
1/2 ÷ 1/4 = 2
To solve this problem mathematically we use the concept that division is the opposite of multiplication. Dividing by 1/4 is the same as multiplying by the inverse of 1/4, which is 4/1. Let's see what our answer will be if we solve this same problem mathematically:
1/2 ÷ 1/4 = 1/2 x 4/1 = (1 x 4) / (2 x 1) = 4/2 = 2
Voilà! We got the same answer!
Let's try another example:
4/3 ÷ 1/3
How many 1/3's are in 4/3? How will this look using the integer bars?
The green bar represents one unit and each white bar represents 1/3 unit. On top of the green bar we have a train of 4 white bars which represents 4/3. Can you tell how many 1/3's are in 4/3? I see 4 white bars, therefore there are 4 white (1/3) bars in the 4/3 train. Now let's solve this same problem mathematically:
4/3 ÷ 1/3 = 4/3 x 3/1 = (4 x 3) / (3 x 1) = 12/3 = 4
Voilà! The answer is 4 using either method.
One more example:
4/10 ÷ 6/5
Let's draw the integer bars to solve this problem.
The orange bar represents one unit, the purple bar represents 4/10 unit, and each red bar represents 1/5 unit. Since we need 6/5 we have a red train of 6 red bars. How many red trains (6/5) will fit into the purple (4/10) bar? The red train is too big to fit in the purple bar. We can only fit 2 of the six red bars or 2/6 of the red train in the purple bar. We simplify 2/6 to 1/3 and the answer is 1/3. Let's use the mathematical method again to see if we get the same answer:
4/10 ÷ 6/5 = 4/10 x 5/6 = (4 x 5) / (10 x 6) = 20/60 = 1/3
Voilà!
Exercises
Fraction Multiplication Problems 3/4 x 4/3 5/7 x 2/5 5/8 x 4/10 8/5 x 1/10 x 5/2 2 1/4 x 3 1/3 2/5 x 1 3/4 x 5/7 5/6 x 6/7 x 2 1/3 Fraction Division Problems 3/4 ÷ 2/8 9/2 ÷ 3/4 7/3 ÷ 7/12 4/9 ÷ 8/3 5 ÷ 1/2 2 3/4 ÷ 1 3/8 8 1/4 ÷ 8/11 After you have completed all the problems you can check your answers.
Comparing Fraction and Percentages
Table of Contents
Learning basic concepts of geometry
Last Updated: Wednesday, 16-Jul-2003 01:15:06 GMT
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