Tutorial on Fractions

1/2, 3/4, 4/10 are fractions. The numbers on top are called numerators. These are the numbers 1, 3 and 4. The numbers below are called denominators. These are the numbers 2, 4 and 10.

The denominator gives how many equal parts are there. The numerator represents how many of these are taken.

The denominator cannot be 0. For example 1/0 is not allowed. This is so as division by 0 is not defined.

However, 0/1 is allowed and is equal to 0.

If numerator and denominator are the same the fraction becomes equal to 1. For example:

2/2=1 and 9/9=1

We can also have fractions like 10/2=5 and 18/6=3.

Mixed Numbers
These numbers are made up of a whole number and a fraction. For example:

5 1/2 is a mixed number and is equal to 5 + 1/2.

A mixed number may be changed into fraction through the following steps:

Step 1
Multiply the whole number and the denominator of the fraction part.

For example in 5 1/2 we multiply 5 and 2 to get 10.

Step 2
To the result of Step 1 add the numerator of the fraction part.

In our example the numerator of the fraction part 1/2 is 1. So 10 + 1 = 11.

Step 3
Write the result of Step 2 as the numerator and the denominator of the fraction part as the denominator.

In our exampls we get 11/2.

So 11/2 = 5 1/2 which shows that we can change a fraction where the numerator is greater than the denominator into a mixed number. For doing this follow these steps:

Step 1
Divide the numerator by the denominator and obtain the remainder.

In our example 11/2, 2 goes 5 times into 11 leaving a remainder of 1.

as 2 x 5 = 10
and 11 - 10 = 1

Step 2
The remainder over the denominator gives us the fractional part. So we get.

11/2 = 5 1/2

Whole Numbers and Mixed Numbers
12/3 = 4 0/3 = 4

Here 0/3 is obviously = 0 hence 4 0/3 = 4.

However representing 4 as 12/3 is useful at times. We need to do this when we do calculations between whole numbers and fractions.

Multiplication of Fractions
Say we want to multiply 3/4 with 5/6.

We do this by multiplying 3 x 5 which gives us the result 15, which is the numerator of the result. Similarly 4 x 6 gives the result 24 which is the new denominator. We can write this multiplication as under:

3/4 x 5/6 = (3x5) / (4x6) =15/24.

Dividing Fractions
Say we want to divide 7/8 by 3/4.

For this we have the following steps:

Step 1
Invert 3/4, or in other words write it up side down with 4 as the numerator and 3 as the denominator.

Step 2
Now multiply this with the first fraction. That is:

7/8 x 4/3 = (7x4) / (8x3) = 28/24

Reducing a Fraction to Lowest Terms
Look at the two fractions given below:

5/6 and 4/8

In the first fraction 5 is a prime number. So its factors are 1 and 5. That is 5 = 1 x 5. However, 6 = 1 x 2 x 3. There are no common factors (neglecting 1). We say that 5/6 is already expressed in its lowest terms.

In the fraction 4/8, 4 = 2 x 2 and 8 = 2 x 2 x 2. Here 2 x 2 is common to both 4 and 8. Hence 4/8 can be written as;

4/8 = (2x2) / (2x2x2) = 1/2

2x2 being common to both numerator and denominator can be cancelled. This leaves us with 1/2 as the result.

By changing 4/8 to 1/2 we say that 4/8 has been reduced to its lowest terms.

The value of 4/8 and 1/2 is the same as cancelling common factors does not change the value of the fraction.

Adding Fractions
5/16 + 7/16 = (5+7)/16 = 12/16 = 3/4

Notice the denominator of 5/16 and 7/16 is the same. This has been done to aid explaining addition of fractions. For adding fractions follow the following steps:

Step 1
Write the common denominator.

Step 2
Add the numerators, which gives us 12/16 which in its lowes terms is 3/4.

If the denominators of the fractions are not the same we have to make them so before adding the fractions. For example:

5/6 + 1/4

Here 6 x 2 = 12 and 4 x 3 = 12

So we can re-write the fractions as:

(5x2) / (6x2) = 10/12 and (1x3) / (4x3) = 3/12

Then add 10/12 + 3/12 = (10+3)/12 = 13/12 = 1 1/12

Here 5/6 and 10/12 are equivalent fractions as they have the same value. 1/4 and 3/12 are also equivalent fractions for the same reason.

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

It is similar to the steps used in adding fractions as explained above.


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