If you multiply two decimals less than 1, can you predict whether the product will be less than or greater than either of the factors?

2 Answers
Nov 26, 2015

It will always be less than both.

Explanation:

Multiplying by something smaller than 1 will always make the result smaller, so taking something like
0.8*0.5 means that we are taking half of 0.8
OR 4/5 of 0.5 (same thing)
either way it makes it smaller, which means we will get something smaller than both.

Nov 26, 2015

If 0 < x < 1 and 0 < y < 1 then 0 < xy < x and 0 < xy < y.

If either or both x <= 0 and y <= 0 then it's more complicated.

Explanation:

If 0 < x < 1 and 0 < y < 1 then:

xy = (1 - (1 - x)) y = y - (1-x)y < y

xy = (1 - (1 - y)) x = x - (1-y)x < x

since all of x, y, (1-x) and (1-y) are positive.

If we allow zero and/or negative values then there are quite a few cases as follows:

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Looking at all of these cases, you will see that the inequality of xy with x or y is the same as x or y's inequality with 0. That is, if x < 0 then x < xy, if y = 0 then y = xy, etc.