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A fraction

Now imagine every number represents a unit of measurement of some sort… for example: pizzas, chocolate, etc.. The number 1 will represent 1 pizza, 2 will be 2 pizzas, and so on.

What happens if you have only one pizza, and you would like to share it with a friend?

You instinctively know that, if you want to be fair, you’ll divide that pizza into two equal parts. If our pizza represents number 1, how would we write down one half of the pizza? We would do it by using fractions.

What exactly is a fraction?
A fraction is a number which represents the number of parts of a whole.

All fractions come in the form of \frac{a}{b}. The top number in a fraction is called the numerator, and the bottom number is called denominator. The denominator shows the in how many parts is a whole divided, and the numerator shows the number of those parts in this particular fraction. The line that separates them is called the fraction line.

parts of fraction

So, when we write down one half in the form of a fraction, it looks like this: \frac{1}{2}

half of pizzaIn the context of pizza, this fraction means that you divided a pizza into two equal parts and took only one part.

What if another friend joins you and he also wants an equal part? Then you’ll divide pizza into three parts and take only one part. This means you’ll get or ‘one third’ of a pizza.

Visualizing equivalent fractions and simplification

Let’s say it’s time for dessert, and you want to share a cake with friend. Of course, the easiest way is to simply cut that cake in half and have each of you take one part. But you can’t eat that much at once, so you have to divide it into more parts to make it more manageable. If you divided it into 6 equal parts and each of you took 3 parts, both of you still got exactly one half of the cake.

two parts from one

This means that the value of one half is equal to the value of three sixths, and we can write that down as:

\frac{1}{2} = \frac{3}{6}
We can divide this cake in as many parts as we want.

\frac{1}{2} = \frac{3}{6} = \frac{4}{8} = \frac{5}{10} = ...


These fractions are called equivalent fractions because they share the same value, and this leads us to a very important property of fractions:


If we multiply both the numerator and denominator by the same positive number, the value of the fraction will not change.

This property also gives us a way to simplify a fraction whose numerator and denominator are large numbers. We can accomplish this by dividing the numerator and the denominator with their greatest common divisor.

Example 1:

Let’s simplify the fraction.

\frac{6}{12} = ?

The greatest common divisor of 6 and 12 is 6. This means that we’ll divide both the numerator and the denominator by number 6.

So, as a result we get:  \frac{6}{12} = \frac{1}{2}

When solving various problems, you may also encounter fractions that contain unknowns. The recipe for treating them is the same as is for any other fraction.


Example 2: Simplify the fraction.

\frac{12ab}{2bc} = ?

The greatest common divisor of 12ab and 2bc is 2b.

When we divide both the numerator and the denominator, we get: \frac{12ab}{2bc} = \frac{6ab}{c}.

This is it for simplifying fractions. If you would like to practice a bit more, feel free to use the worksheets we prepared for you below.

Comparing fractions

How would you compare two fractions? Compared to any other number, a fraction can be greater than, lesser than or equal to it.


1. Comparing two fractions whose numerators are the same

As you know the denominator is a number which represents the number of parts our whole is divided to. If we have two fractions with the same numerator means that we take the same amount of parts, but if a denominator is greater, that means that we take equal amount of smaller parts.

This means that if the numerator of two numbers is the same, but the denominator of the first is greater than the second, then the first fraction is smaller than the other.

Example: Compare \frac{24}{25} and \frac{24}{3}.

Numerators are the same, but the denominator of the first is larger than the second which leads us to:

\frac{24}{25} < \frac{24}{3}

2. Comparing two fractions whose denominators are the same

Let’s first think about what this means. You have two equal wholes. And you divide them into equal parts. But from the first whole you take more parts than the second. This means that if the denominators in two fractions are equal but the numerator of first is greater than the numerator of second, than the first numerator is greater.

Example: Compare \frac{22}{26} and \frac{2}{26}.

\frac{22}{26} > \frac{2}{26}

3. The general fraction comparison

If you have any two fractions and you have to compare them

For this we’ll use something called the cross multiplication. (for the missing sign we’ll use “O”)

If we have two fractions \frac{a}{b} and \frac{c}{d}. To compare them we’ll write them down as:

\frac{a}{b}\frac{c}{d} and cross multiply them.

cross multiply fractions

By doing this, we got two numbers we know how to compare.


Example: Compare two fractions.

\frac{8}{5} O \frac{4}{7}

\ 7 \cdot 8 O 5 \cdot 4

\ 56 O 20

\ 56 > 20

\frac{8}{5} > \frac{4}{7}

Here you just have to be careful where you put your sign. Remember that the numerators will always stay at the sides they are given.

If you’re wondering why can we do that:

Remember that we said that if we multiply both the numerator and denominator with the same number their value will remain the same? It may not seem like it but we did the same thing here. For every two numbers their common multiple is their product. If we have two numbers \frac{a}{b} and \frac{c}{d} we have to compare we would multiply them both with the product of their denominator \ b \cdot d. but then we would get fractions we can shorten:

\frac{abd}{b} O \frac{cbd}{d}

ad O cb.

Fractions can also be negative. With this you have to be careful. Because a minus can be anywhere in a fraction.

Of course if you have two fractions, one negative and one positive, positive one will always be greater than the negative one.

But what if you have two negative fractions?

Then you do this procedure without the minuses, and then when you add minuses you “turn your sign backwards”.

Example: Compare – \frac{2}{9} and – \frac{9}{17}.

First compare – \frac{2}{9} and \frac{9}{17}

\frac{2}{9} O \frac{9}{17}

\ 34 O 81

\ 34 > 81

\frac{9}{17} > \frac{2}{9}

\frac{9}{17} > - \frac{2}{9}

And of course two fractions can also be equal. These are, already mentioned, equivalent fractions.

Decomposing fractions

Decomposing a fraction means to break him into smaller pieces. For example, if you have one whole that is divided into five parts you can also get that by adding five pieces of fifths.

one pizza has 6 slices

This means that every fraction can be separated by his numerator, you can do this, but you have to be careful for your denominator must be the same as the given fraction, and the sum of the numerators of new fractions must be equal to the numerator of given fraction.

sum of fractions

The decomposition of a fraction is not unique. You can decompose a fraction in any way you like.

\frac{5}{6} = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6}

\frac{5}{6} = \frac{2}{6} + \frac{2}{6} + \frac{1}{6}

\frac{5}{6}= \frac{4}{6} + \frac{1}{6} and so on…

The decomposition of fraction leads us to very important operations with fractions – addition and subtraction.

These operations expect from you to have high skills in finding lowest common multiple.

To quickly solve these tasks requires a lot of work and practice so take it slow and don’t be discouraged if at first you don’t succeed.

We’ll explain how it’s done on an example:

Add two fractions \frac{1}{2} + \frac{1}{3}


First you find their least common multiple which is number 6. That number will be the denominator of our sum.

1. Step finding the denominator:

\frac{1}{2} + \frac{1}{3} = \frac{x}{6}

2. Now you divide your new denominator with the first denominator and multiply with the first numerator. 6 : 2 = 3

\frac{1}{2} + \frac{1}{3} = \frac{(3 \cdot 1 + ?)}{6}
3. You do the same with the second fraction. Divide the new denominator with the second denominator, multiply that with second numerator and add to everything we did so far. (\frac{6}{3} = 2)

\frac{1}{2} + \frac{(1)}{3} = \frac{(3 \cdot 1 + 1)}{6}

\frac{1}{2} + \frac{1}{3} = \frac{5}{6}

Note that you can check your solution by decomposing your solution:

\frac{5}{6} = \frac{3}{6} + \frac{2}{6} = \frac{1}{2} + \frac{1}{3}

Another step by step example:

\frac{2}{5} + \frac{4}{15}
Again the first thing you do is finding the denominator. The least common multiple of 5 and 15 is 15.

This means that the denominator of our sum will be number 15.

Second, the first addend in a numerator of the sum will be 6, because we divide 15 with the first denominator, number 5 which is number 3, and then multiply it with its numerator which is number 2.

The second addend in a numerator of the sum will be 4, because number 15 divided by 15 is number 1, and times 4 is 4.

This leads us to our solution:

\frac{2}{5} + \frac{4}{15} = \frac{10}{15} = \frac{2}{3}
Try to make a habit in shortening the fractions. This will really come in handy when you’re up against much more complicated expressions.

What if you get more addends? The procedure is the same, only now you have three fractions, and do the procedure three times, and of course, the denominator of that sum will be the least common multiple of all three.

Example: \frac{1}{5} + \frac{7}{30} + \frac{2}{3}

The least common multiple of 5, 30 and 3 is number 30.

\frac{30}{5} = 6; \frac{30}{30} = 1; \frac{30}{3} = 10

\frac{1}{5} + \frac{7}{30} + \frac{2}{3} = \frac{(1 \cdot 6 + 7 \cdot 1 + 2 \cdot 10)}{30} = \frac{33}{30} = \frac{11}{10}

If you have to add a whole number and a fraction, the procedure is the same, you simply think as a whole number “a” as a fraction \frac{a}{1}

What about subtracting? The process is the same, you only have to watch, as always, on a minus.

Example: Subtract \frac{5}{8} - \frac{1}{2}.

Least common multiple of 8 and 2 is number 8 and thats is the denominator of the difference.

\frac{8}{8} = 1; 1 \cdot 5 = 5, \frac{8}{2}= 4; 4 \cdot 1 = 4

 \frac{5}{8} - \frac{1}{2} = \frac{(5-4)}{8} = \frac{1}{8}

Example 2: Subtract \frac{1}{2} - \frac{5}{8}

\frac{1}{2} - \frac{5}{8} = \frac{(4-5)}{8} = - \frac{1}{8}

Reminder: if you are unsure of which number is the least common multiple of your denominators, you can put any multiple, and shorten the fraction afterwards:

\frac{1}{2} - \frac{5}{8} = \frac{(8-10)}{16} = \frac{(8-10)}{16} = - \frac{2}{16}

Multiplying fractions

Multiplying fractions is the easiest mathematical operation with fractions; you simply multiply the numerator of the first number with the numerator of the second number, and denominator of the first number with the denominator of the second number.

\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}

Example 1: Multiply.

\frac{2}{7} \cdot \frac{3}{5} = ?

\frac{2}{7} \cdot \frac{3}{5} = \frac{(2 \cdot 3)}{(7 \cdot 5)} = \frac{21}{35}

Things can even go simpler. You can shorten them before multiplication, so you get smaller numbers. You can only shorten the numerators with denominators, never numerator with numerator or denominator with denominator. But you can take any denominator and any numerator in a multiplication.

Example: Shorten the fraction before multiplication.

\frac{2}{7} \cdot \frac{14}{8} = ?

If it’s easier for you, solve this in a certain order. First you can shorten them individually:

\frac{2}{7} \cdot \frac{14}{8} = \frac{2}{7} \cdot \frac{7}{4}

From here you can see that your numerators are 2 and 7, and denominators 7 and 4. You can shorten 7 with 7, and 2 with 4. Remember never to shorten two numerators or two denominators. From here we can easily get our solution:

\frac{2}{7} \cdot \frac{14}{8} = \frac{2}{7} \cdot \frac{7}{4} = \frac{1}{1} \cdot \frac{1}{2} = \frac{1}{2}

Dividing fractions

If you have to divide two fractions, your best chance is to “turn the division into a multiplication”. When you have two fractions and have to divide them, simply use a reciprocal value of the divisor and replace the division symbol with a multiplication one.

\frac{a}{b} : \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}

Remember that division is not a commutative operation, which means that you have to be careful of which reciprocal value you take- always from the divisor.

Example: Divide two fractions.

\frac{1}{2} : \frac{5}{8} =
=\frac{1}{2} \cdot \frac{8}{5}
(we can shorten)
= \frac{4}{5}

Another thing you can do is to make a complex fraction. A complex fraction has fractions as numerators and denominators, and while writing them you have to be careful and in every point know which fraction line is your main fraction line.

As you know, every division can be written as a fraction \frac{a}{b} = a : b

dividing fractions

So how do you solve this? You multiply by the rule- external with external and internal with internal (Numerator of the first fraction with the denominator of the second, and the denominator of the first with the numerator of the second fraction).

division of reciprocal fractions

The numerator of the quotient will be the product of external numbers, and denominator the product of internal numbers.

Example: Divide two fractions using complex fraction notation.

\frac{(1)}{2} : \frac{5}{4} = ?

\frac{(1)}{2} : \frac{5}{4}=
=\frac{(1)}{2} / \frac{5}{4}
= \frac{(1 \cdot 4)}{(2 \cdot 5)}
= \frac{2}{5}

Improper fractions and mixed numbers

Let’s get back to our pizza notation. If you ate one half of a pizza and another pizza arrives, how many pizzas do you have? With the knowledge you collected so far you know how to calculate this. You have 1 + \frac{1}{2} = \frac{3}{2} pizza.

But it’s kind of silly to say you have three halves of a pizza when you can just say you have one and a half, it’s easier and more understandable.

This is why we divide fractions into proper and improper ones.

Improper fractions contains numerator that is greater than its denominator, because it contains wholes.

If you want to write down number one and one half this is how you would do it:

First you have the number of wholes which is one, and then the proper fraction.

1 \frac{1}{2} this is read as one and one half, these kind of numbers are called the mixed numbers, because they contain a whole number and a fraction.

This was easy and intuitive, but what if you have large numbers in your denominator and you would like to know how many wholes and how many parts exactly do you have.

You have to divide the numerator with a denominator, and the whole number you get is how many wholes you have. But if those two numbers are not divisible, you’ll have a remainder; this remainder will be the fraction part in your mixed number.

Example: Turn a fraction into a mixed number.

\frac{51}{4} we know that \frac{51}{4} = 12.75

This means that our improper fraction contains 12 wholes.

How do we get our proper fraction that goes with those wholes? Since you know that one whole contains 4 parts (from the denominator in your fraction) and you have 12 wholes, multiplying those two, you get 48. You have 48 parts in your wholes, but in your fraction you have 51. You simply subtract those two and get your fraction. 51 - 48 = 3. This leads us to our final solution:

\frac{51}{4} = 12 \frac{3}{4}

Next thing to learn is how to do the backwards procedure, how to turn the mixed numbers into a improper fraction?

Example: Convert 12 \frac{3}{4} back to improper fraction.

From the denominator of your proper fraction part of a number, you know that one whole contains 4 parts, and you know that you have 12 wholes and three parts. This means that you’ll simply multiply 12 with 4 and add 3.

12 \frac{3}{4}

= \frac{(12 \cdot 4 + 3)}{4}

= \frac{(48+3)}{4}

= \frac{51}{4}

Of course, now the question occurs how do you do all those mathematical operations with mixed numbers. The answer is you don’t. Always convert them into improper fraction, and then if there is a need, convert your solution back to the mixed number.


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