Question

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?

Answer 1

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.

Answer 2

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:

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.