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Whole numerator and whole denominator
In a fraction you may always multiply or divide the whole numerator and the whole denominator with the same number. But what does that mean exactly?
Example 1
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
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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 x ≠ 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
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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 parentheses in the numerator
That is fun, we can use parentheses 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 x ≠ 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