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
Edric Y.
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.

George C.
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.

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