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The confusion arises because we call it scratching-out, instead of dividing. In a simple fraction, there is no ambiguity, because

and you can divide the numerator and denominator by 2. Another method, that better shows what happens is


Example 2

It becomes more difficult if there are two terms in the numerator. We're going to do it wrong, and scratch-out only a part of the numerator


We have clearly not taken the whole numerator. Here is the correct approach

An alternative explanation is clearer

There is another possible explanation, which of course also yields the same answer

If you calculate the fraction otherwise you can clearly see why you must do it like that. You could have written this as


Example 3

Now we dare to tackle larger fractions. Step by step you will see

Here you look at it, and then you can see that it is true. You have to write that this solution only applies for ≠ 2, because you cannot divide by zero. The alternative explanation is even better


Example 4

Finally, an extra fraction, where you can scratch-out as you like it

A check with the alternative explanation gives

And that looks pretty clear.


Example 5

We want to write this as one fraction

Step by step we continue

We guess that the denominator must be 50a, and get

It is striking that we can scratch-out 10 in the numerator and denominator, and write


Example 6

Now it becomes more difficult

That is a surprise, however we write for the denominators

We make a single fraction

Now we calculate the numerator

and eliminate the brackets in the numerator

That is fun, we can use brackets once more in the numerator

and can divide the numerator and denominator by x – 2

Never call this scratching-out, because you really divide, and there remains 1. Moreover, dividing by 0 is not allowed, so you must indicate that the solution is only valid for ≠ 2.


Example 7

We take the fraction

You must first make the denominators equal

That is correct. You will understand that a × b is the same as b × a, because 2 × 3 is, after all, just as much as 3 × 2. In alphabetical order the answer is


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