The landscape company charges 30+15x
Sum/difference:
Let

This means that

Now, assume that
is rational. The sum/difference of two rational numbers is still rational (so 5-x is rational), and the division by 3 doesn't change this. So, you have that the square root of 8 equals a rational number, which is false. The mistake must have been supposing that
was rational, which proves that the sum/difference of the two given terms was irrational
Multiplication/division:
The logic is actually the same: if we multiply the two terms we get

if again we assume x to be rational, we have

But if x is rational, so is -x/15, and again we come to a contradiction: we have the square root of 8 on one side, which is irrational, and -x/15 on the other, which is rational. So, again, x must have been irrational. You can prove the same claim for the division in a totally similar fashion.
Answer:
6 pieces of rope
Step-by-step explanation:
if the total is 54 and each are 9 you can make an equation
54= 9 x a
solve for a
to do this you divide 54 by 9
54/9=6
The rate Ritz can eat apples is 8/15 apples per minute.
Set the rate (which is measured apples / minutes) equal to 16 apples / t minutes.
(8/15) = (16/t)
Cross multiply.
16*15 = 8t
t = 20
The answer is 30 minutes. You could solve the problem with basic reasoning as well. 16 is twice the number of apples as 8, so the time it takes to eat them will be double 15 minutes.