Learning More About Fractions

 

 

Comparing Fractions

When comparing two fractions that have the same denominator you simply compare the two numerators to determine which fraction is larger. For example, to compare 4/7 and 3/7 you simply compare the numerators 4 and 3, then obviously the result of the comparison is that 4/7 is larger than 3/7. Another example is comparing 7/4 and 9/4. Can you tell which one is larger? Obviously, 9/4 is larger than 7/4.

If one of your friends offers you 3/8 of his sandwich and another friend offers you 2/5 of his same kind of sandwich, which one is larger? We need to be able to compare fractions to tell which one is larger.

In order to compare two fractions we first need to convert both fractions to have the same denominator. We can use the LCM method explained in the previous page to find the common denominator. A common denominator can also be found by simply multiplying each denominator by the other, in this case 5 x 8 = 40. This will give you a common denominator for both fractions, but it may not always be the least common denominator.

You multiply 3/8 by 5/5 to get 15/40. Then multiply 2/5 by 8/8 to get 16/40. Now you can see that 16/40 is larger than 15/40, therefore 2/5 is larger than 3/8.

Lets introduce mathematical symbols used when comparing numbers or fractions.

Symbol
Meaning
Equation
Description
<
less than
a < b
a is less than b
=
equal
a = b
a equals b
>
greater than
a > b
a is greater than b

The result of the comparison of the sandwich can be written as follows:

3/8 < 2/5
or
2/5 > 3/8

Another example is to compare 5/12 and 3/6. We only have to convert the second fraction, 3/6, to the common denominator of 12, so

3/6 = 3/6 x 2/2 = 6/12

Now we can compare 5/12 and 6/12. Obviously,

5/12 < 6/12
or
5/12 < 3/6
or
3/6 > 5/12

Is the fraction closer to 0, 1/2, or 1? - We've just learned how to compare fractions. Now we will get more practice at learning which fractions are very small, which are about one-half, and which are almost one whole unit. First we will show some examples using the integer bars. In this first example the size 10, or orange, bar represents one whole unit. The size 1, or white, bar is 1/10. As you can see in the drawing 1/10 is closer to zero. The size 9, or blue, bar is 9/10. The drawing shows that 9/10 is closer to 1.

In the next example, still using the size 10, or orange, bar to represent one unit, the yellow bar below the orange is the size 5 bar and it represents 5/10 which can be simplified to exactly 1/2. The purple bar is smaller than the yellow bar but is still closer to 1/2 than to zero. The green bar is larger than the yellow bar but is still closer to 1/2 than to 1. The fractions for the purple bar is 4/10 (or 2/5) and the green bar is 6/10 (or 3/5).

We can also find out if the fraction is closer to 0, 1/2, or 1 by comparing the numerator to the denominator.

Percentages

Percent means "part of 100". Percent is a fraction where the denominator is 100. In the integer bars we don't have a bar of size 100 but in the following example we connected 10 orange bars to make 100. We also have connected five yellow (or size 5) bars to make a train of size 25.

The fraction of the yellow train to the orange train is 25/100. We can also say that the yellow train is 25 percent of the orange train. The symbol that we use for percent is "%". So 25 percent can be written as 25%.

In this next example we use the orange bar (size 10) and the black bar (size 7). Since this fraction is 7/10 we need to find the equivalent fraction with the denominator of 100 by multipying both numbers by 10. This gives us the fraciton 70/100.

Therefore, the black bar is 70% of the orange bar.

Decimal numbers - Fractions can also be represented as decimal numbers. A decimal number has a decimal point "." and each number to the right of the decimal point has a fractional value. The first number after the decimal point has a value of tenths, the second number has a value of hundredths, etc. An example of a decimal number is 0.4 which represents four tenths or 4/10. Another example of a decimal number is 0.07 which represents seven hundredths or 7/100.

What would be the value of the decimal number 0.47? The 4 represents 4/10 and the 7 represents 7/100. We can write 4/10 as 40/100. Now we have 40/100 plus 7/100 which is 47/100 or forty seven hundredths.

This may be a new concept for you but we want to mention another way of calculating percentages. In the example above, if you divide 7 by 10 the answer would be 0.70 which is a decimal number. To write this as a percentage we need to move the decimal point two places to the right to end up with 70. Since this 70 now represents the percentage it can be written as 70%.

Let's do another example using the following fraction:

13/20

If we divide 13 by 20 we get 0.65 as our answer. Again, we move the decimal point two places to the right to get 65%.

When do we use percentages? If your teacher gives a test with 20 questions and you get 18 of them correct what would be the percentage of correct answers? This gives us a fraction of 18/20. Let's use both of the methods we have described above and show that we get the same answer.

Explanation
Fraction with denominator of 100
Explanation
Decimal Numbers
 
18 /20
    
18/20
multiply by 5/5
90/100
   divide 18 by 20
0.90
final answer
90%
  

move decimal point two places to the right
90
 
  final answer
90%

Exercises

Problem Number
Problem
Answer

Compare the two fractions in each of the following problems and indicate which is larger by using the symbols <, >, or =

1
3/5 and 5/8
 
2
3/4 and 9/11
 
3
4/9 and 9/20
 
4
1/3 and 3/8
 

For each of the following problems, find if the fraction is closer to 0, 1/2, or 1

5
3/5
 
6
1/5
 
7
5/6
 
8
4/7
 
Convert each of the following fractions into percentage
9
1/4
 
10
3/5
 
11
20/25
 
12
1/3
 

Once you have finished all of the exercises, check your answers.

Rational and Irrational Numbers

Some decimal numbers can be represented as fractions and those are called rational numbers. Other decimal numbers can not be represented as fractions and those are called irrational numbers. Here are examples of some rational numbers:

0.75 can be represented as 3/4
0.9 can be represented as 9/10
0.125 can be represented as 1/8
0.333333333333... can be represented as 1/3

Here are some examples of irrational numbers:

0.12357102... can not be represented as a fraction
the number pi, which is 3.141592654... can not be represented as a fraction
the golden ratio number, which is 1.6180339887... can not be represented as a fraction
2.236067978... can not be represented as a fraction

 


Fraction addition, subtraction, and complement
Table of Contents
Multiplying and Dividing fractions

 

 

Last Updated: Tuesday, 15-Jul-2003 23:45:56 GMT


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