Which of the following two fractions is larger?89 or 1112?

It’s not easy to say, because fractions with different denominators are not easy to compare. How you can still solve this task, you will find out in this article!

## Why you need fractions of the same name – overview

You often have to compare two fractions and, for example, determine which fraction has the greater or smaller value. Or you have to add or subtract fractions. However, you can only cope with these tasks if the fractions have the same denominator, i.e. if these fractions **same name **are.

Two fractions are called **same name **called if they have the same denominator.

fractions **make same name** means that several fractions are reduced to the same denominator.

Reminder: the numerator is the number above the fraction bar and the denominator is the number below the fraction bar.

However, fractions of the same name are not only required to be able to compare two fractions, as for the introductory example. Fractions of the same name are also needed when subtracting and adding two fractions. When multiplying and dividing fractions, however, you are not dependent on fractions of the same name.

So let’s go straight to how to make fractions equal!

## Make fractions equal – rules

Fractions can be made equivalent in different ways.

For example, you can combine the two denominators of fractions **multiply**. But it quickly happens that you get a very large number. It usually makes more sense to close the fractions **shorten **or to **extend**. Another possibility is that you **least common multiple** determine the denominator.

### Find common denominator by multiplying

Two fractions can be made the same by multiplying the denominators of two fractions together. By multiplying the two denominators, you get a number that represents a suitable denominator for both fractions.

Don’t forget to expand the counter as well. To do this, you need to multiply the numerator of one fraction by the denominator of the other fraction.

#### Task 1

You gave the two fractions 13 and 42.

To find a common denominator for these two fractions, simply multiply the two denominators of the fractions together in the next step

3 * 2 = 6

Now you have found a common denominator between the two fractions.

Now add the denominator of the other fraction to the numerators.

1 x 26 = 264 x 36 = 126

You have now made two fractions of the same name by multiplying the denominators together.

### Find common denominator via least common multiple

That **least common multiple** – short **kgV** – of two or more whole numbers is the smallest natural number that is shared by both of these numbers.

There are three ways to find the least common multiple of two numbers. Once you can find the LCM using a series of numbers, or you can do a prime factorization. You can also calculate it using the GCD if you know it.

It is best to look at the exact procedure for these methods in the article «least common multiple»!

At this point we look at an example in which we calculate the LCM using the prime factor analysis, and thus make two fractions of the same name.

#### exercise 2

Given are the two fractions 46 and 710.

First, do a prime factorization for the two denominators.

6 = 3 * 210 = 2 * 5

You can see that the numbers 3 and 5 appear once each in the two prime factorizations. The 2 occurs in both prime factorizations, but is only multiplied once.

Now multiply all the numbers together. In this case it is the 2, the 3 and the 5.

2 * 3 * 5 = 30

So the LCM you are looking for is 30. Now expand the two fractions so that both get the denominator 30.

4 * 56 * 5 = 20307 * 310 * 3 = 2130

If you had made the fractions the same by multiplying both denominators together, the denominator would now be 60. With the kgV you get the smallest possible common denominator.

### Find common denominator by expanding

You expand a fraction by multiplying the numerator and denominator by the same number.

If you want to add a number c to a fraction, you multiply a and b by c.

The numbers a, b and c are so-called whole numbers ℤ. These are negative and positive integers!

For some problems both fractions have to be expanded. However, sometimes it is sufficient to expand just one fraction. Therefore, always check first whether one of the two denominators is a multiple of the other fraction. If that is the case, all you have to do is expand the fraction with the smaller denominator. This saves you unnecessary calculation work.

#### task 3

You should bring the two fractions 38 and 524 to the same denominator.

Since 24 is a multiple of 8, all you have to do is expand the 38 so that there is a 24 in the denominator.

3 * 38 * 3 = 924

Now you have the two fractions 924 and 524

#### task 4

Given are two unlike fractions 34 and 15. You can find a common denominator for these two fractions by multiplying the denominators of the two fractions together, i.e. expanding them.

3 * 54 * 5 = 15201 * 45 * 4 = 420

### Find common denominator by abbreviating

You can find a common denominator not only by expanding, but also by shortening. A fraction is reduced by dividing the numerator and denominator of the fraction by the same number.

If you want to reduce a fraction ab by a number c, you have to divide a and b by c.

The numbers a, b and c are so-called whole numbers ℤ. These are negative and positive integers!

Even if you want to make two fractions of the same name by reducing them, you may often only have to reduce one of the two fractions, because even when reducing, one denominator may be a multiple of the other denominator. If that’s the case, all you have to do is reduce the fraction that has the larger number in the denominator.

#### task 5

The two fractions 1230 and 415 can be reduced to a common denominator by reducing the first fraction.

1230 = 12 : 230 : 2 = 615

Now you have obtained the two fractions 615 and 415 by reducing.

#### task 6

You should reduce the two fractions 1442 and 2030 to a common denominator.

First you should think about which number divides both 42 and 30. For example, you can write down a series of numbers in which you write down all the numbers by which the two denominators can be divided.

42 → 1, 2, 3, 6, 7, 14, 2130 → 1, 2, 3, 5, 6, 10, 15

In the rows of numbers, look for all the numbers that appear in both rows of numbers. Now choose one of these numbers that appears in both rows of numbers. This number is the denominator that you can bring both fractions to. In this example, that is 6. Now think about the number you have to use to shorten the two denominators so that they get the denominator 6.

14 : 742 : 7 = 2620 : 530 : 5 = 46

## Compare equal fractions

As you saw in the introductory example, two fractions can only be compared well if both fractions have the same denominator. Therefore, before you want to compare two fractions, you should always make the fractions the same.

Once you have made both fractions of the same name, all you have to do is see which fraction has the larger or smaller numerator. The fraction with the larger numerator is larger than that with the smaller numerator.

#### task 7

The two fractions 89 and 1112 from the introductory example can be given the same name in different ways.

8 * 49 * 4 = 323611 * 312 * 3 = 3336

After we have made the two fractions of the same name by expanding, we can now see which of the two fractions is larger. All we have to do is see which of the two fractions has the larger numerator.

3236 < 3336

We see that 1112 is the larger fraction.

Let’s look at one more example in a moment.

#### task 8

Given are two unlike fractions 13 and 14. You can find a common denominator for these two fractions by multiplying the denominators of the two fractions together, i.e. expanding them.

This gives you the two fractions 1 43 4 = 412 and 1 34 3 = 312

Now you can easily compare these two fractions by looking at which of the two fractions has the larger numerator.

It follows: 412 > 312

Want to know more about comparing fractions? Then you should definitely take a look at the article Comparing and arranging fractions!

## Adding and subtracting fractions

As already mentioned, adding and subtracting fractions also requires that the fractions have the same name.

When both fractions have the same denominator, addition and subtraction are easy: you can just add or subtract the numerators of the fractions. The denominator remains unchanged.

Let’s look at this directly with an example.

#### task 9

Add the two fractions 811 and 23.

#### solution

1. In the first step, we expand the two fractions to make them the same. Expanding is particularly suitable for this, in order to bring the two fractions to the same denominator.

8 * 311 * 3 = 2433

2 x 113 x 11 = 2233

2. Now that we’ve brought the two fractions to the same denominator, we can easily add them together. For this we only have to add the two numerators together, since the denominator no longer changes.

2433 + 2233 = 4633

## equal fractions – examples and tasks

Now that you know all the rules for making fractions of the same name, you can test your new knowledge directly with a few tasks.

## equal fractions – The most important

- Like fractions are needed for comparing, adding, and subtracting fractions.
- Fractions can be made the same by expanding and truncating.
- Fractions can only be compared if they have the same name.
- A common denominator can be found by expanding or contracting.