How to simplify fractions?

Learn this magic trick!

Do you remember our party with the wizard?

After the party, there were ten pieces of cake left. The last trick of our magician was to reassemble 8 pieces of it into a whole cake - like a puzzle.

Now, you should learn a similar trick, too!

Suppose you have four small pieces of cake:

4/16 cake

However, your friends and you have now decided on cookies so that the cake should be put back into the fridge. And if it hadn't been cut yet, it would hold up even better. So what do we do?

Now you should be able to do magic - turn 4 small pieces into 1 giant piece!

Based on the whole cake: How big would that giant piece be?

When expanding, you divided a large piece into two small pieces. Let's try it the other way round: turn 2 into 1!

4 small pieces become 2 large pieces of cake.

How big are these pieces? Twice as big as the small pieces, i. e. when dividing the whole cake you would only get 8 pieces instead of 16 pieces:

   

4/16 cake    =    2/8 cake

As it worked out so well, we repeat the trick again: The 2 pieces become 1 piece with the size of a quarter of the whole cake:

       

4/16 cake    =    2/8 cake    =    1/4 cake

Both the numerator and the denominator have been divided by the number 2.

Formula: Simplifying/Reducing fractions

Symplify or reduce a fraction by dividing the numerator and the denominator by the same number. The value of the fraction does not change!

Another example:

Simplify a fraction - Example

Both the numerator and the denominator are divided by 7. Calculate: 6 divided by 3 and 42 divided by 21 both give the same result: 2.

Lowest Terms Fractions

The purpose of symplifying is usually to bring the fraction into a state in which the fraction cannot be simplified further. Then you can say that it's the lowest terms fraction. This is the case if there is no common divider (greater than 1) of the numerator and the denominator.

If you want to reduce a fraction to its lowest terms fraction in one single step, you have to reduce the numerator and the denominator with its Greatest Common Divisor (GCD). There are two methods for its calculation: prime factorization and the Euclidean algorithm. You can choose one of these two options.

When simplifying a fraction to its lowest terms, take a close look at it first. In most cases it is not so difficult to find any number to simplify the fraction. Maybe you can repeat this one or two times further. It is not necessary to find the GCD immediately, usually it is easier to simplify in more than one operation.

In what cases can a fraction not be simplified?

If one of these conditions is met, it is certainly not possible to simplify a fraction further:

1) The numerator or the denominator is equal to 1

The number 1 has only one divider: itself. Therefore the Greatest Common Divisor (GCD) of 1 and any other number must always be 1. However, shortening a fraction with 1 is pointless, since the numerator and denominator do not change when dividing by 1.

2) The numerator or the denominator is equal to a prime number

If you recognize a prime number in the numerator or denominator, you only have to check that both are not a multiple of each other. Either you can see it directly or you have to calculate it quickly.

This is one reason why you should know at least the first prime numbers (2, 3, 5, 7, 11, 13, 17, 19) by heart.

3) The difference between numerator and denominator is 1

If the difference between numerator and denominator is only 1, then you can safely say that the fraction cannot be shortened any further.

In what cases can a fraction further be simplified?

A fraction can be simpified, if there is a common divisor for the numerator and the denominator.

In some cases you can say without much calculation whether a number A (= numerator and denominator) can be divided by a number B (possible number for reduction):

Divisibility by 2

A number can be divided by 2, if the last digit is even: 0, 2, 4, 6, 8.

Divisibility by 3

A number can be divided by 3, if its cross sum is divisible by 3.

Divisibility by 4

A number can be divided by 4, if its last two digits are divisible by 4.

Divisibility by 5

A number can be divided by 5, if its last digit is 0 or 5.

Divisibility by 6

A number can be divided by 6, if its last digit is even AND its cross sum is divisible by 3.

Divisibility by 8

A number can be divided by 8, if its last three digits are divisible by 8.

Divisibility by 9

A number can be divided by 9, if its cross sum is divisible by 9.

Divisibility by 10

A number can be divided by 10, if its last digit is 0.

Divisibility by 25

A number can be divided by 25, if its last two digits are divisible by 25, i.e. 00, 25, 50 or 75.

Divisibility by 100

A number can be divided by 100, if its last two digits are equal to 0.

Divisibility by 1000

A number can be divided by 1000, if its last three digits are equal to 0.

Simplifying fractions by trial and error

If you are experienced with the simplification of fractions, just have a close look on the fraction. Often you already have a number in suspicion with which the fraction could be reduced. Just try it out with this number.

Sometimes, it is a good trick to try the difference between numerator and denominator. Often a fraction can be simplified by this number.

Online Exercises: Simplifying/Reducing fractions

Here you can practise simplifying (or reducing) fractions online. The first 3 exercises are just "warm up" ;-)

Number 4 and 5 are very important exercises for the calculation with fractions! As a prerequisite for these two I recommend the exercise Is A divisible by B?.

  1. Simplify the fraction with the specified reduction number.
  2. Determine the reduction number and the denominator.
  3. Determine the reduction number and the numerator.
  4. If possible without calculating: Tell whether the fraction is a lowest terms fraction.
  5. Simplify the fraction to a lowest terms fraction.

If you have any problems with the exercises, please have a look at the hint and the sample solution first.

Video: Simplifying Fractions Rap

Finally, a nice music video about: Simplifying Fractions:


Let's continue with: "prime factorization"