Subtraction II - Fraction minus Fraction

The Thought Experiment

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:

Example: 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:

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

Let's continue with:
Once again the most important - Test your knowledge!