As a bicycle salesperson, Norman earns a fixed salary of $20 per week plus $6 per bicycle for the first six bicycles he sells, $12 per bicycle for the next six bicycles he sells, and $18 per bicycle for every bicycle sold after the first 12. This week, Norman earned more than twice as much as he did last week. If he sold x bicycles last week and y bicycles this week, which of the following statements must be true?

I. y > 2x

II. y > x

III. y > 3

A)I only
B)II only
C)I and II
D)II and III
E)I, II, and III

II is obvs.
III is also pretty obvs. If we think of the least scenario: 26 (1 bike sold) if the salarly is twice as large, then it def has to be more than 3 bikes.

I: we can have 26 and thus we need to have 52 for the 2nd week salary. This is obvs > than 2x

but we can have $128 we have 6,6. Then to get double we would need 256. well x=12, that means y would be greater than 24 for I, but we can obviously see that 12*18 +128 is not going to be 256 so y does not have to be greater than 2x.


D

mbafall - let me try an equation based approach here:

The salary equation is a function of number of bicycles sold. Depending on the number of cycles sold the salary equation (for B bicycles) would be:

S = 20+6*B for B = (0,6)
S = 20+36+12*(B-6) or 12*B-16 for B = (7,12) and
S= 20 + 36 + 72 +18*(B-12) or 18*B-88 for B > 12

We need to check for three scenarios, y>3, Y>x and Y>2x.

Now the value for Salary is linearly related to the number of bicycles under all conditions, so y would be always greater than x if salary from y bicycles is to be greater than salary from x bicycles.

To test whether y needs to be greater than 3, lets see how equations behave when y is in the region (0,6)

If y is within (0,6) then salary for week 2 is 20+6*y and the equation is 20+6*y > 2*(20+6*x)
or 6*y > 12*x + 20
or y > 2*x + 3.66
If x = 0 then minimum value for y is 3.66 so y > 3

To test for y > 2x, lets just check at border points, lets say if x=12 then salary is 128 in week 1 and for salary to be 256 in week 2, the value of y would be governed by 18*y-88 (as we already know that y>x) so 18*y-88>256 or Y >344/18 or y > 19.xx or Y=20 would suffice so y need not be greater than 2x. Hence only 2 and 3 must be true.

While the equation based approach is more sound - in the current context where equation is a step function, it makes more sense to plug numbers along various ranges and see. Plugging in approach is always likely to be faster.