(Fractions are easy to learn if you know how to add and multiply WELL.)
Adding fractions with the same denominators is easy. Let’s add the fractions 1/6 and 4/6 together.
1
6 
+

4
6 
=

5
6 
If the denominators are the same then just add the numerators (1 + 4) together and write the sum (5) as a numerator. When adding and subtracting fractions, the denominators stay the same as those being added and subtracted. Denominators don’t get added!
If the denominators are not the same, find the least common multiple of the two denominators then write equivalent fractions and add the numerators as shown in the example below.
Let’s add the fractions 1/4 and 2/7.
1
4 
+

2
7 
=

The first thing is to find common denominators. This is the same as finding the least common multiple.
So we have to find a number shared by both denominators. In this case it’s going to be 28 (4 prime factors to 2 Ã— 2 and 7 primes to 1 Ã— 7….then multiply the primes 2 Ã— 2 Ã— 7 = 28).
Now we have to find the factor of 4 Ã— ____ = 28 and we know it’s 7. And we need to know 7 Ã— _____= 28 and we know it’s 4.
Now we need to make equivalent fractions as shown below.
1
4 
x

7
7 
=

7
28 
+

2
7 
x

4
4 
=

8
28 
Now that you have equivalent fractions (in bold) with common denominators you can carry out the addition as shown in the first example above or as shown here:
7
28 
+

8
28 
=

15
28 
We take the 7 and add it straight across to the 8 and write the sum 15 as the numerator. Then, just carryacross the denominator 28 and write that below the 15. Easy! Now there’s usually an easier way to add fractions…. and below is one of them.
Multiply the denominators together then multiply each of the numerators by the other fractions denominator.
1
4 
x

7
7 
=

7
28 
+

2
2 
x

4
4 
=

8
28 
Take another look at the above problem…see how we can multiply the denominator of the second fraction (7) to the numerator and denominator of the first fraction (1/4)? And see how we can multiply the denominator of the first fraction (4) to both the numerator and denominator of the second fraction ?
1
4 
x

7
7 
+

2
7 
The example above shows how the second fraction’s denominator gets multiplied to the first fraction’s numerator and denominator.
1
4 
x

7
7 
+

2
7 
x

4
4 
And the example above shows how the first fraction’s denominator gets multiplied to the second fraction’s numerator and denominator.
And finally, the easiest way of all….cross multiply the numerator of the first fraction with the denominator of the second fraction and write the answer as a numerator. Then multiply the first fraction’s denominator with the second fraction’s numerator. Add the two numerators and multiply the two denominators together and write that as the denominator with the sum of the numerators as it’s numerator as shown below.
If you study the example above, you’ll see that this is a pretty good short cut.
Now here’s your chance to add fractions with and without common denominators.
Subtracting fractions is very similar to adding fractions. To subtract fractions with common denominators, simply subtract the numerators then use the same denominator again. In the example below, we’ll subtract 2/3 – 1/3 for a result of 1/3.
2
3 


1
3 
=

1
3 
Notice that the denominators stay the same. The numerators are subtracted much like whole numbers.
If the denominators are not the same, find the least common multiple of the two denominators then write equivalent fractions and subtract the numerators as shown in the example below.
Let’s subtract the fraction 2/7 from 3/4.
3
4 


2
7 
=

The first thing is to find common denominators. This is the same as finding the least common multiple.
So we have to find a number shared by both denominators. In this case it’s going to be 28 (4 prime factors to 2 Ã— 2 and 7 primes to 1 Ã— 7….then multiply the primes 2 Ã— 2 Ã— 7 = 28).
Now we have to find the factor of 4 Ã— ____ = 28 and we know it’s 7. And we need to know 7 Ã— _____= 28 and we know it’s 4.
Now we need to make equivalent fractions as shown below.
3
4 
x

7
7 
=

21
28 


2
7 
x

4
4 
=

8
28 
Now that you have equivalent fractions (in bold) with common denominators you can carry out the addition as shown in the first example above or as shown here:
21
28 


8
28 
=

13
28 
We subtract the 8 and from the 21 and write the result 13 as the numerator. Then, just move the denominator 28 to the right and write that below the 13. Easy! As with addition there’s an easier way to subtract fractions….and below is one of them.
Multiply the denominators together then multiply each of the numerators by the other fractions denominator.
3
4 
x

7
7 
=

21
28 


2
7 
x

4
4 
=

8
28 
Take another look at the above problem…see how we can multiply the denominator of the second fraction (7) to the numerator and denominator of the first fraction (3/4)? And see how we can multiply the denominator of the first fraction (4) to both the numerator and denominator of the second fraction?
3
4 
x

7
7 


2
7 
The example above shows how the second fraction’s denominator gets multiplied to the first fraction’s numerator and denominator.
3
4 
x

7
7 


2
7 
x

4
4 
And the example above shows how the first fraction’s denominator gets multiplied to the second fraction’s numerator and denominator.
And finally, the easiest way of all….cross multiply the numerator of the first fraction with the denominator of the second fraction and write the answer as a numerator. Then multiply the first fraction’s denominator with the second fraction’s numerator. Add the two numerators and multiply the two denominators together and write that as the denominator with the sum of the numerators as it’s numerator as shown below.
If you study the example above, you’ll see that this is a pretty good short cut.
If you’ve been reading this section on fractions you should have a good idea on how to multiply fractions from the adding and subtracting fractions section above. But if you’ve just skipped to this point, here’s how to multiply two fractions.
First, multiply the numerators and write those as a numerator.
Second, multiply the denominators and write the product as a denominator. At this point we usually reduce the fractions to lowest terms to complete the process.
In the following example, we’ll multiply the fractions 2/3 and 4/5 for a result of 8/15 which cannot be reduced since the numerator and denominator do not share any similar factors or prime numbers.
As you can see from the above example, we multiplied the numerators 2 and 4 and wrote the product, 8 as a numerator. Then, we multiplied the denominators 3 and 5 for a product of 15 and wrote the product as a denominator. That’s it!!
Now it’s your turn to try multiplying fractions. Use the problems below to see if you understand multiplying fractions.
Dividing fractions is almost as easy as multiplying fractions. For example, to divide the fractions 2/5 and 3/4 simply invert the last fraction (turn it upsidedown) then multiply straight across as we did above in the multiplying fractions section.
So, to multiply 2/5 and 3/4 we would turn the 3/4 upsidedown and make it 4/3. Then, we’d multiply 2 and 4 for a product of 8 and write that as a numerator. Next, we’d multiply the 5 and the 3 for a product of 15 and write that as a denominator. The example below shows this process again.
Now it’s your turn to try dividing fractions. Use the problems below to see if you understand dividing fractions. Use the boxes to invert the second fraction then multiply straight across.