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Adding, Subtracting, Multiplying, Dividing Fractions

Date: 6/7/96 at 19:29:38
From: Kenneth Andriessen
Subject: Fractions

Dear Dr. Math,

I am ten years old. I don't understand lower-term fractions. I am two 
months away from fifth grade. I was wondering if you could help me.

Date: 6/8/96 at 8:12:19
From: Doctor Anthony
Subject: Re: Fractions

I am not sure how far you have already got with fractions, but I will 
assume you are just starting.

I will use the notation 1/2, meaning 1 over 2, or 1 divided by 2 to 
represent the fraction 'a half'.  3/4 means 3 divided by 4 and is 
'three quarters'.

If I want 'three quarters' of some quantity, say a sum of money, then 
I can think of dividing the sum into four equal portions, and I will 
then take three of the portions, leaving one portion behind.  So I 
have 3/4 of the sum and there is 1/4 remaining.  The total sum of 
money is represented by the number 1 in this example, and if we add 3/
4 to 1/4 we get back to the total 1.  If the total sum had been $100 
to start with, then I simply multiply by 100 to get the actual 
portions.  So (3/4)*100 = 75,  (1/4)*100 = 25

You should note from this example that the bottom line (denominator) 
gives the number of equal portions into which the total is divided, 
and the top line (numerator) gives how many of these equal portions 
you are taking.

So 5/8 means divide the total into 8 equal portions and then take 5 of 
these portions. 

Adding Fractions:

From the example of taking 3/4 and 1/4 of something you saw that if we 
add these together we get back to the total, 1.

Thus 3/4 + 1/4 = 1

This illustrates the rule we use.  First you must have the same 
denominator, (bottom line) in each fraction, then with this as the 
common denominator, the numerator(top line) is found by simply adding 
the two top numbers, 3 and 1, in this case.

So (3+1)/4 = 4/4 = 1

The rule for subtraction is the same, except that we subtract the 
numbers on the top line.

 3/4 - 1/4 = (3-1)/4 = 2/4 = 1/2

The difficulty arises if the two denominators are not equal.

Example:  5/8 - 1/4  Now to make the two denominators equal, we first 
express 1/4 as 2/8.  (This is done by multiplying top and bottom lines 
by the SAME number, 2.  Since we multiplied by the same number we have 
not changed the value of the fraction.)

Subtraction is now 5/8 - 2/8 = (5-2)/8 = 3/8

A slightly more difficult problem arises if one denominator is not a 
multiple of the other.  Example: 5/8 + 2/3 

For this problem we make the denominator the smallest number into 
which both 8 and 3 will divide.  This is called finding the LCM 
(lowest common multiple).  For 8 and 3, it is easy to see that 24 is 
the LCM, and we could write the fractions as 

  5/8 = 15/24   we multiplied 8 by 3 to get 24, so we must also 
                multiply the top number,5, by 3 to get 15.

  2/3 = 16/24   multiply top and bottom by 8.

Now we require  15/24 + 16/24 = (15+16)/24 = 31/24
                                           = 1 +7/24

Multiplying Fractions.

This is easy.  Simply multiply the two top numbers and the two bottom 

Example: 2/3 * 5/8 = (2*5)/(3*8)
                   =  10/24     Divide top and bottom by 2.
                   =  5/12  

Dividing Fractions.

The easy rule to remember is: Turn the SECOND fraction upside down and 

Example  (2/3)/(5/8) = (2/3)*(8/5)
                     = (2*8)/(3*5)
                     = 16/15
                     = 1 +1/15

This has given a rather quick run through the basic rules. Write in 
for more help if there are any particular points you don't understand.

-Doctor Anthony,  The Math Forum
 Check out our web site!   

Associated Topics:
Elementary Fractions

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