Reduce fractions: explanation, rules & tasks

In this article, you will learn how to reduce fractions. This is a skill that is used very often in fractions. Surely you have stumbled across the hint «shorten completely» while doing your homework. You can learn how to do this here.

What does reduce fractions mean?

Figure 1: Graphic representation of shortening

I am sure you will agree that the same area is marked in pink in all four rectangles. But as you can see from the fractional numbers next to the respective area, these fractional numbers are not equal. When the rectangle is top left one half marked, they are at the top right rectangle two quarters.

Of the fraction does not change if you multiply the numerator and denominator of a fraction by the same natural number or divide them by the same natural number. This is called «dividing by the same number». shortenthe «multiply by the same number» is that Extend. Expanding and truncating are therefore reverse operations.

Figure 2: Expanding and truncating as inverse operations

These findings are recorded mathematically in a definition:

It applies to all fractions:

pq=p q a for all numbers a∈ℕ.

In doing so

  • the transformation from p q a to pqshorten called and
  • the conversion of pq to p q aExtend called.

So you can reduce the fractions and from the introductory example to the fraction. How this works exactly and which fractions you can reduce at all, we will show you in the further course of the article. But first, you can learn why reducing fractions makes sense.

Reduce fractions – why do you need that?

An important reason why you should reduce fractions is that when Calculating with fractions is often clearer when the numerator and denominator are smaller numbers. It’s also often easier if you can calculate with smaller numbers. This mainly applies to multiplying and dividing.

Task 1

Multiply 360216 * 199398

solution

If you were to start multiplying without abbreviating, you would have to multiply the two numerators – i.e. 360 and 199 – together, as well as the two denominators – 216 and 398. This is tiresome work, and most people can no longer do it with mental arithmetic.

But if you reduce the fractions first, then this calculation becomes much easier. Because 360216 is nothing but 53, and 199398 is the same as 12. So these two fractions are much easier to multiply, right?

In the next section you will learn why these fractions can be reduced in this way. Therefore, there is no detailed calculation at this point.

Adding and subtracting, on the other hand, requires fractions to have the same denominator, so you often need to expand rather than contract fractions.

In addition, the result of calculations is often reduced when it is in the form of a fraction. This is because the result then clearer is. If you look back at the fractions from the last example, surely you can imagine a result of 12 being more than if the result was 199398, right?

Shortening fractions – procedures and tricks

Now that you’ve seen what it means to reduce a fraction and why you should be able to do it, in this section you’ll learn exactly how to reduce a fraction.

The general rule is:

To reduce a fraction, you need to find the numerator and denominator of the fraction by the same natural number (≠0) divide. This keeps the value of the fraction unchanged.

You are probably already familiar with natural numbers. If you can’t remember exactly, just read the corresponding chapter on sets of numbers again.

That doesn’t sound all that complicated, does it? You simply divide the numerator and denominator by the same number. This number will then cut number called.

Also important is the sentence in the definition that the value of the fraction remains the same when shortened. This means, for example, that a fraction is still in the same place on the number line after it has been reduced. You also agree that half a pie is equal to two quarters of a pie, right?

But shortening a fraction is not always possible. It is only possible if there is a natural number dividing both the numerator and the denominator. If there is no such number, we call it a numerator and a denominator coprime.

If a fraction cannot be reduced at all or not further, that is what it is called completely abridged.

In the Special Fractions section, you’ll be shown some fractions that can’t be reduced.

In the following sections, however, you will first be shown various ways of reducing a fraction that can be reduced.

Reduce fractions with a common divisor

Fractions can be reduced by any divisor of the numerator and denominator. This often has the advantage that you can quickly find a reduction number. For example, it is often easy to see when a fraction with the reduction number 2 or 3 can be reduced.

However, this type of truncation has the disadvantage that it takes several steps to fully truncate. The larger the reduction numbers, the fewer steps it takes to fully reduce the fraction.

exercise 2

The fraction should be completely reduced.

solution

Since the numerator and denominator are both divisible by 2, the fraction can be reduced to 2, for example:

1824=18:224:2=912

and are therefore equivalent and the same rational number.

If you don’t know exactly what the concept of rational numbers is all about, just read the corresponding sub-chapter on sets of numbers.

But that is still no fully reduced fraction, because numerator and denominator can be further reduced: Both numbers are divisible by 3:

912=9:312:3=34

Now we’re done, because the numerator 3 and the denominator 4 have no other common divisor.

When reducing fractions, it helps to keep the divisibility rules in mind. Do you remember when a number is divisible by 4 or by 9? You can look at our article on divisibility rules in the chapter on number theory in the area of ​​algebra again!

Nevertheless, we repeat the most important divisibility rules at this point:

A number is divisible by……if..2…it’s even.3…its digit sum is divisible by 3.5…it ends in 5 or 0. 9…their digit sum is divisible by 9.

Reduce fractions with the greatest common divisor (gcd)

In the example above, you would also have in one step can be truncated completely if truncated directly with 6:

We take the same fraction again, this time dividing it by the number 6 at the top and bottom.

1824=18:624:6=34

The 6 is the greatest common divisor of 18 and 24, since the fraction is completely reduced afterwards.

You see, both ways lead to the same result. So if you are not sure whether you can reduce with a higher number, you can also reduce with a smaller number first and do several calculation steps.

Now that the term «greatest common divisor» has been used, it should be defined at this point.

Of the Groasted Gcommon Thurry – short gcd – of two integers a and b is the largest natural number dividing both a and b.

The greatest common divisor is written mathematically by gcd(a,b).

If you want to know exactly how to find the gcd of two numbers, you can read about it in the article greatest common divisor.

Reduce fractions using prime factorization

A third way to reduce fractions is to use the prime factorization.

Every natural number greater than 1 can be written as a unique product of prime numbers. This product will too prime factorization called the number.

«Unique» is the prime factorization up to order. This means that the prime numbers that occur are unique, but the order in which they are written is not unique.

Example: The prime factorization of the number 12 is 12=22 3 or 12=2 3 2. The prime factors are the same but written in a different order.

So any number can be written as a product of several prime numbers, provided it is greater than 1 and not itself a prime number.

To reduce a fraction using prime factorization, first break down the numerator and denominator of the fraction into prime factors. Then you can see directly which factors are the same at the top and bottom. You can then highlight the same factors. But be careful, the same number of factors must be crossed out in the numerator and denominator of the fraction!

task 3

The fraction 3260 is to be reduced completely using the prime factorization.

solution

To do this, the numerator and denominator are first broken down into prime factors:

32=2*2*2*2*260=2*2*3*5

Only the factor 2 occurs in the numerator, but it is the same five times. The factor 2 also occurs in the denominator, twice, and the factors 3 and 5 once each.

Only factors that are in the numerator can now be reduced and occur in the denominator, so in this case only the 2. Since they are in the numerator and denominator just as often have to be deleted can only be done in each case two two be shortened, although the numerator would be five twos!

So by deleting it we get:

3260=2 2 2 2 22 2 3 5=2 2 2 2 22 2 3 5=2 2 23 5=815

Not sure how prime factorization works anymore? No problem. For more information on this topic, you can read the separate chapter on .

Notation of the truncation number

Sometimes when trimming, the trimming number below des = noted.

1045=529

The numerator and denominator of the fraction are each divided by the number 5.

Especially at the beginning of the fraction calculation, noting down the reduction number can be helpful in order to keep track. For example, if your result is wrong, you can easily check where your mistake is. That’s why some math teachers ask for it at the beginning. However, it’s not mathematically wrong if you don’t do this, and especially if you’re already good at short-cutting, you can omit it.

Reduce special fractions

The aim of reducing fractions is to reduce the fraction as much as possible. This is the case when you have two numbers in the numerator and denominator coprime are. Sometimes you have to check this a bit cumbersome (for which it is again important to know the rules of divisibility! You see, this is the key to reducing fractions!). In two cases, however, it is immediately apparent:

There is 1 in the numerator or denominator

If you have a 1 in the numerator or denominator, you can’t reduce further because the 1 can only be divided by itself. So the gcd with any other number is always 1. So you could just reduce with 1, but that doesn’t change the fraction anymore.

The difference between the numerator and denominator is 1

If the difference between the numerator and denominator is 1, you can’t reduce the fraction any further. This is because two consecutive numbers cannot be divided by the same natural number (greater than 1).

Reduce fractions with variables – rule

Fractions with variables are also called fractional terms.

A term of the form ST, where…