Is it true or false that the product of a mixed number between 4 and 5 and a fraction between 0 and 1 is always less than 4?
It is always true.
Let P = N * F
where N is the number between 4 and 5.
i.e. Let N = 4 + ε, where ε is a positive value.
and F is the fractional value
i.e. Let F = 1 – δ, where δ is a positive value.
Then,
P = N * F
P = (4 + ε)(1 - δ)
P = 4 – εδ
Since both ε and δ are positive values then so also is εδ.
Therefore P is always less than 4.