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Fractional Calculation: Introduction and Rules

This article provides an introduction to fractions and the corresponding rules for fractions. First of all, it is explained what the fraction calculation is all about. Then we look at how to deal with fractions, i.e. addition, subtraction, multiplication and division of fractions. Other topics such as expanding fractions, double fractions, and more are also covered.

Fractional arithmetic is needed when it is not about "whole" things. So you often don't eat a "whole" pizza, but only part of a pizza. But how can you express something like that in mathematics? And this is exactly the question that brings us to an introduction to fractions. Everyone knows the phrase "eat half an apple" or "half a cake". Mathematically, this would be written as 1/2. This notation is called a fraction. Using the pizza as an example, this would mean that I cut the pizza into 2 pieces and eat 1 piece. If I would now say that I am eating 3/4 of the pizza, this means: I cut the pizza into 4 pieces and eat 3 pieces. The pizza is of course only a very first example in the introduction to fractions. But now it is important to first discuss the spelling.

Note: The contents of this article are also available in video. See Fractional Calculation Basics Video, Add Fractions Video, Subtract Fractions Video, Multiply Fractions Video, and Divide Fractions Video.

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Fractional Calculation Introduction: The Spelling

An important step in the introduction to fractions is of course the spelling. Here some examples:

The important thing here is:

  • The number above is the counter (in the example 1, 3, 5, 7 and 8 are the counters)
  • The number below is the denominator (2, 4, 8, 9 and 10 are denominators)
  • A fraction line is drawn in between


Fractions are often spelled differently on the Internet: 1/2 or 3/4. There is a simple reason for this: This notation is much easier for website authors or users in forums. For a better overview we use the spelling "used" from school in our articles. In the solutions to the exercises, however, we will also use the other notation for the sake of simplicity. But anyway, you should only look at the solutions to the problems after you have done the problems yourself ... Another important note: The fraction 3/4 has the same meaning as the notation used by the division: 3 : 4.

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Fractional calculation rules: addition

If I eat half a cake (1/2) and another half cake (1/2) I have eaten a whole cake in total. And with that we are in the middle of adding fractions. A few examples follow to show the rules for fractions, but first the general procedure briefly.

Add fractions:

  1. Bring fractions to a common denominator.
  2. Add the fraction.
  3. Shorten the fractions.

Step 1: bring fractions to a common denominator

Let's go to the rules of fractions: In order to add fractions, both must have the same denominator. As a reminder: The denominator was what is "below" in the fraction. And that must - as already indicated - be the same for both. The important thing is that the value of the fraction must not change. The following knowledge helps: 1 out of 2 equal pieces of a cake is the same as 2 out of 4 equal pieces of a cake. Here is a small example that shows two possible solutions.

In both cases 1/2 + 3/4 should be brought to a denominator. The two options just mentioned were used. This will work as follows.

Option A):

  • The two denominators are multiplied together. Since 2 · 4 is known to be 8, the denominator for the new fractions is 8.
  • The numerator of the first fraction is multiplied by the denominator of the second fraction. This results in 1 · 4 = 4. This 4 is the new numerator of the first fraction.
  • The numerator of the second fraction is multiplied by the denominator of the first fraction. 2 · 3 = 6. This 6 is the new numerator of the second fraction.

Tip: Read through each point of variant a) again slowly and follow it up using example a).

Variant b):

  • The 4 of the second denominator is a multiple of the 2 of the first denominator. To get from 2 to 4, multiply by 2. So multiply the numerator and denominator of the first fraction by 2. The result can be seen above.


Which variant should I use?

The variant b) is usually shorter. If you can see that one number is a multiple of the other, you can use this variant. If this does not succeed, it is best to use variant a). There is theoretically another possibility using the least common multiple (see article kgv). But that would complicate matters here unnecessarily. If you are unsure: Variant a) always works.

Step 2: add the fraction

Once the fraction has been reduced to a denominator, adding the fractions is easy. The following two examples show you how this works.

This is how the addition works:

  • The two counters are added and give the result counter.
  • The denominators are the same everywhere.

So it's very simple: just add the counters and that's it. The denominator remains.

Step 3: shorten fractions

If you did step 1 and step 2 of adding fractions correctly, you should have come up with the correct result. However, there is often the option to simplify this even further. Examples follow again.

The smaller the numbers, the easier it is. That's why the fractions are shortened. This works when the numerator and denominator are divisible by the same number without a remainder.

  • Example on the left, above: The numerator and denominator of the fraction can be divided by 3 without a remainder.
  • Example on the right, above: The numerator and denominator of the fraction can be divided by 2 without a remainder.
  • Example on the left, below: The numerator and denominator of the fraction can be divided by 2 without a remainder.
  • Example on the right, below: It is no longer possible to shorten here.


More tips for shortening:
When trimming very large numbers, it's often smart to cut multiple times. Say the numerator and denominator first with e.g. shorten with 2 or 3. And if you see "the fraction can also be shortened" then shorten it further.

Fractional calculation rules for subtracting

First, a brief overview of what needs to be done in most cases to subtract fractions:

  • Bring fractions to a common denominator
  • Subtract fractions
  • Shorten result fraction


These are the three steps that, if used correctly, will produce results. We will now address each of these points individually and explain what is behind them.

Step 1: bring fractions to a common denominator

To be able to subtract fractions, both must have the same denominator. As a reminder: The denominator was what is "below" in the fraction. And that must - as already indicated - be the same for both. The important thing is that the value of the fraction must not change. The following knowledge helps: 1 out of 2 equal pieces of a cake is the same as 2 out of 4 equal pieces of a cake. Here is a small example that shows two possible solutions.

In both cases 1/2 - 3/4 should be reduced to a common denominator. The two options just mentioned were used. It works like this.

Option A):

  • The two denominators are multiplied together. Since 2 · 4 is known to be 8, the denominator for the new fractions is 8.
  • The numerator of the first fraction is multiplied by the denominator of the second fraction. This gives 3 · 2 = 6. This 6 is the new numerator of the first fraction.
  • The numerator of the second fraction is multiplied by the denominator of the first fraction. 1 · 4 = 4. This 4 is the new numerator of the second fraction.

Tip: Read through each point of variant a) again slowly and follow it up using example a).

Variant b):

  • The 4 of the first denominator is a multiple of the 2 of the second denominator. To get from 2 to 4, multiply by 2. So multiply the numerator and denominator by 2. The result can be seen above.


Which variant should I use?

The variant b) is usually shorter. If you can see that one number is a multiple of the other, you can use this variant. If you don't succeed, it is best to use variant a). There is theoretically another possibility using the greatest common divisor. But that would complicate matters here unnecessarily. If you are unsure: Variant a) always works.

Step 2: subtract the fraction

Once you've got the fraction down to a denominator, subtracting the fractions is easy. The following two examples show you how this works.

This is how the subtraction works:

  • The two counters are subtracted and give the result counter.
  • The denominators are the same everywhere.

So it's very simple: just subtract the counters and that's it. The denominator remains. Note: If the second fraction is larger than the first, a negative result is produced, as with the subtraction. Example: 1/4 - 3/4 = -2/4.

Step 3: shorten fractions

Works in the same way as above under Adding Fractions, point 3.

Fractional calculation rules: multiplication

We come to the multiplication of fractions.

Multiplying fractions is really simple:

  • The counter is multiplied by the counter.
  • Denominator is multiplied by the denominator.


That's about it, actually. The following examples show you this again:

That was the whole secret behind multiplying the fractions: the numerator is multiplied by the numerator, the denominator is multiplied by the denominator. A small note on the side: These fractions can of course be shortened. The first fraction can be shortened to 5/6 and the second to 6/5. For anyone still struggling to trim, I recommend the section under Adding Fractions Step 3 earlier in this article.

Fractional calculation rules for division

There are still the rules of fractions for dividing fractions. The so-called reciprocal value is used for this purpose. In simple terms, this means: the second number is turned over, the numerator and denominator are swapped. For those of you who can't quite imagine it, here is a (fully calculated) example:

And this is how the division of fractions works:

  • The first break remains as it is
  • Instead of the ":" a "·" is written
  • The second fraction is reversed, the numerator and denominator swapped
  • The two "new" fractions are multiplied as you do with fractions: numerator · numerator and denominator · denominator.
  • If possible, the result will be shortened


If you still have problems shortening fractions, you will find help on this under Adding Fractions Step 3 above.

Fractional calculation rules: other topics

Further topics on fraction calculation, introduction and rules:

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