1/4 + 1/5 = ?

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There are two fractions which we need to add together. It is not hard but requires some preparation. If the fractions would have the same denominator, adding them would be a breeze: `a / b + c / b = ( a + c ) / b .`

But this is not the case, and the denominators are 4 and 5 (i.e., they are not equal).

There is a mathematical "trick" which allows to change a fraction's denominator without changing the fraction's value. If we multiply (or divide) both numerator and denominator by the same nonzero number, the fraction will remain the same. For example, `1 / 2 = 2 / 4` and `1 / 2 = 3 / 6` and so on.

So, if we'd find such numbers to multiply (one for the first fraction and another for the second) that the denominators will become equal, we'd be able to add the fractions.

What are the numbers to multiply 4 and 5 to get the same answers? The simplest is to multiply 4 by 5 and 5 by 4, so that 4 * 5 = 5 * 4 = 20. Let's try:

`1 / 4 + 1 / 5 = ( 1 * 5 ) / ( 4 * 5 ) + ( 1 * 4 ) / ( 5 * 4 ) = 5 / 20 + 4 / 20 = ( 5 + 4 ) / 20 = 9 / 20 .`

It is the answer, and there is no way to simplify it.

Note that this method works always, but for denominators with common factors we can reduce work by multiplying by smaller numbers than the entire denominators.

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