## Subtraction II - Fraction minus Fraction

When boiling water, it slowly evaporates. Let's do a thought experiment on this:

In a pot there is boiling ^{3}/_{4} litres of water. After half an hour, ^{1}/_{3} litres of water evaporated. How much water is left in the pot?

Finding out the task here is easy. It's just this one:

## Common mistake when subtracting Fractions

Often the mistake is made that the two numerators and the two denominators are subtracted from each other:

Unfortunately, it's not that easy. This is only works when multiplying fractions. Let's see if this makes any sense:

If the water is filled into glasses with a volume of ^{1}/_{4} litre each, exactly 3 glasses can be filled.
1 glass of it is to be taken away. **But**: this glass should contain ^{1}/_{3} litre!

And that is exactly when it comes to a problem! A glass of ^{1}/_{4} litre water could be easily removed, but ^{1}/_{4} litre is not equal to ^{1}/_{3} litre...

Even if the (wrong) result ^{2}/_{1} litre is illustrated, it doesn't fit anymore:
As a result, there should be 2 glasses each with a size of one litre? This is obviously more than the ^{3}/_{4} liters you had before subtracting.

Problems of the type *Fraction minus Fraction* can only be solved if the denominators (or the glasses) are equal.

## Formula: Subtracting Fractions having Common Denominators

*Two fractions having the same denominators or common denominators are subtracted by subtracting the numerator.
The denominator remains unchanged.*

The result often can be simplified / reduced:

^{5}/

_{8}-

^{3}/

_{8}

## Formula: Subtracting Fractions NOT having Common Denominators

*Two fractions that don't have the same denominators or common denominators cannot be subtracted directly! Instead:*

- Make the denominators the same by expanding or simplifying / reducing (→ see simplifying / reducing).
- Subtract the numerators - the denominator remains unchanged.
- Check, if you can simplify the fraction (to lowest terms).

^{3}/

_{4}-

^{1}/

_{3}

Coming back to our "hot-water" problem:

Unfortunately, the denominators are not the same, so we cannot subtract the two fractions directly.

Have a look in our toolbox and pick the right one: Common Denominator - then we can use the formula to subtract fractions that have the same denominator:

In order to get a common denominator, expand the first fraction by 3 and expand the second fraction by 4. Afterwards, the numerators can be subtracted:

There are still ** ^{5}/_{12} litres** of water in the pot.

## Online Practice: Subtracting Fractions

Here you can practice subtracting fractions online: Subtracting Fractions.

If you have any problems with the exercise, please have a look at the **hint** and the **sample solution** first.

## Congratulations! Step 4 is done!

Great! On the last pages you have learned, to calculate with fractions: **Addition** und **Subtraction**.

It's essential that you know:

- only fractions having the same denominator can be added/subtracted directly by adding/subtracting their numerators and remaining the denominator unchanged
- fractions having different denominators need to have a Common Denominator before adding/subtracting

Always check, whether fractions can be **simplified/reduced**, especially the result!
Is the result is an improper fraction (the numerator is greater than the denominator): convert into a mixed number.

To complete Step 4, please take a minute to answer the comprehension questions: