Equivalent Fractions

Date: 05/07/99 at 07:30:01
From: Sam
Subject: Math (equivalent fractions)

I am trying to understand equivalent fractions. Example:
 
  6     1
 -- =   -
 12     2

Date: 05/07/99 at 11:49:09
From: Doctor Rick
Subject: Re: Math (equivalent fractions)

Hi, Sam.

This is 6/12:

  +---+---+---+---+---+---+
  |xxx|xxx|xxx|xxx|xxx|xxx|
  |xxx|xxx|xxx|xxx|xxx|xxx|
  +---+---+---+---+---+---+
  |   |   |   |   |   |   |
  |   |   |   |   |   |   |
  +---+---+---+---+---+---+

The big rectangle is cut into 12 equal-sized boxes, and 6 are marked 
with x's. Each box is 1/12 of the whole, so 6 of them is 6/12.

This is 1/2:

  +-----------------------+
  |xxxxxxxxxxxxxxxxxxxxxxx|
  |xxxxxxxxxxxxxxxxxxxxxxx|
  +-----------------------+
  |                       |
  |                       |
  +-----------------------+

The big rectangle is cut into 2 equal-sized boxes, and 1 is marked 
with x's. Each box is 1/2 of the whole.

Both pictures show the same space marked with x's. You could imagine 
the rectangle being a chocolate cake. If you cut it in 2 equal-sized 
pieces and eat one, you have eaten the same amount as if you cut it in 
12 equal-size pieces and ate 6 of them. The two fractions are 
EQUIVALENT.

How do you find equivalent fractions? Look again at that cake cut into 
two pieces. Make 5 vertical cuts, and you have cut the cake into 12 
pieces. You multiplied the total number of pieces by 6, but you also 
multiplied the number of pieces you're going to eat by 6. 

The total number of pieces is the denominator. The number of pieces 
you chose is the numerator. You just multiplied the numerator and the
denominator by the same number, 6.

You could have made any number of vertical cuts and you'd get another
fraction equivalent to 1/2. If you multiply both the numerator and the
denominator by the same number - any whole number - you will get a
fraction equivalent to 1/2. For instance, you can multiply by 2/2 to 
get 2/4; by 3/3 to get 3/6; by 4/4 to get 4/8, and so on.

What about going the other way? Starting with 6/12, you can find some
equivalent fractions by DIVIDING the numerator and denominator by the 
same number. But you can't just pick any number, as you can when you 
multiply. The numerator and denominator must both be DIVISIBLE by the 
number.

Both 6 and 12 are divisible by 2, so one fraction equivalent to 6/12 
is 3/6 (divide 6 by 2 and divide 12 by 2).

Both 6 and 12 are divisible by 3, so another equivalent fraction is 
2/4.

Both 6 and 12 are divisible by 12, so another equivalent fraction is 
1/2.

But 1 and 2 have no common factors, so you cannot find a fraction 
equivalent to 1/2 by dividing. There is no equivalent fraction with 
smaller numbers. We say that 1/2 is IN LOWEST TERMS.

Remember I said that 4/8 is equivalent to 1/2? Two fractions that are
equivalent to the same fraction are equivalent to each other. This 
means that 4/8 and 6/12 are equivalent. But 6 is not a multiple of 4, 
and 12 is not a multiple of 8. So what's the rule?

If you want to find ALL the fractions that are equivalent to 6/12, 
first reduce it to lowest terms: find the equivalent fraction that is 
in lowest terms. That's 1/2. Then multiply the numerator and 
denominator by each number, 1, 2, 3, 4, and so on, and you will have 
a list of ALL the fractions equivalent to 6/12. The list will go on 
forever, so don't try to write them all - but you know what they are.

I've probably said more than you wanted to see. I hope it helps. If 
you have specific problems that you can't do, let us know - and tell 
us your wrong answer, that will help us figure out what you need to 
learn.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/