## Introduction - Common Denominator

Now, let's answer the question, whether Spain or Finland is a closer holiday destination (from Germany)

The answer is ...

It depends! It depends on where you are in Germany and where you want to go.

So let's answer the other open question, whether ** ^{3}/_{7}** or

**is greater.**

^{7}/_{15}To do this, use the tool Expansion.
The purpose is to expand the two fractions so that they have the **same denominator**.

^{3}/

_{7}and

^{7}/

_{15}

## Finding the Common Denominator - "simple" procedure

This trick always works:

- Expand the first fraction by the denominator of the second fraction
- Expand the second fraction by the denominator of the first fraction

So:

Both fractions are now in the "same direction", so it is easy to see that ^{7}/_{15} is the bigger fraction.

The trick works perfectly, but has a small hook...

^{5}/

_{9}and

^{7}/

_{12}

If the "simple" procedure is also applied here, the result is:

The calculation is correct, but the numbers are greater than they should be. In this way, the calculation becomes more complicated and one can easily make a mistake.

In most cases, this is the best procedure:

## Finding the Common Denominator - trial and error

Try, if the *lower denominator "fits into" the greater denominator*.

=> If yes, the common denominator is found.

If not, try if the *lower denominator "fits into" the double greater denominator*.

=> If yes, the common denominator is found.

If not, try if the *lower denominator "fits into" the triple greater denominator*.

=> If yes, the common denominator is found.

If not, try ...

Is 9 a factor of 12? → No

Is 9 a factor of 24? → No

Is 9 a factor of 36? → Yes, because 4 × 9 = 36!

The two fractions have to be expanded to the denominator **36**:

This procedure works well if the numbers are not too big.

Else...

^{1}/

_{12}and

^{1}/

_{980}

Finally, let's find the common denominator of ^{1}/_{12} and ^{1}/_{980}.

For this we use a second tool in addition to the Expansion, the Least Common Multiple (LCM):

## Finding the Common Denominator - using LCM

- Determine the Least Common Multiple (LCM) of the denominators.
- Expand the two fractions, so that the denominator
**is equal to the LCM**.

At the page Prime Factorization we found disassemblies for the numbers 12 and 980:

2 × 2 × 3 = 12

2 × 2 × 5 × 7 × 7 = 980

Therefore, the LCM of 12 and 980 is:

2 × 2 × 3 × 5 × 7 × 7 = 2940.

Now, which is the expansion number?

The first option is to divide the LCM by the denominator:

2940 ÷ 12 = 245

2940 ÷ 980 = 3

The second option is to expand with the prime factors that are **only** contained in the other number. Sounds more complicated than it is:

12 is expanded with 5 × 7 × 7 = 245

980 is expanded with 3

## Congratulations! Step 3 is done!

In step 3 you filled your toolbox with some tools you need for calculating with fractions.

It's important to know,:

- that a fraction is
**expanded**, by multiplying the numerator and the denominator with the same number - that a fraction is
**simplified**or**reduced**, by dividing the numerator and the denominator with the same number - how to simplify a fraction to
**lowest terms** - how to find the
**common denominator**of two fractions

To complete Step 3, please take a minute to answer the comprehension questions: