Problem Solving

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


OA coming after few folks reply

Since a Fixed cost of 20$ dollars plus twice the cost af a x cycle in first week will always be involved the product of y cycle in second week and cost of each should always be greater.

also y>x always.
answer is E.

please tell what i left out if am wrong

Answer D.

If Norman sold 0 bicycles, then he earned 20$. The week after, he earns at least 40$, it's significate that y>3 (Y=3 => salary = 38$).

OA is D.

The key here is to consider the conditions at the boundary of the
price range(s)

Take y = 7 for example which just crosses the border of 6$ per bicycle
to 12$ per bicycle. Total price = 20 + 6*6 + 1*12 = 68

Now, take x=4 to disprove (y > 2x). Price = 20 + 6*4 = 44. Clearly
y not > 2x.

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

Statement II: Obviously true since the more bicycles Norman sells, the more commission (variable component) he earns. So y>x
Statement III: y>3. If y <= 3; then total amount earned by Norman would be $20 (fixed component) + Maximum 3*6 = $18 (variable component). So total amount earned by Norman would be 38. However, Norman is guaranteed a fixed return of $20 (even if he sells 0 bicycles) and $38 is not greater than twice of $20. Hence Statement III must be true

Statement I: y>2x; If y = 6 then amount earned = (6*6) +20 = 36+20 = 56
If x = 2 amount earned = 2*6 + 20 = 32; Here y>2x but earning of Norman in the second week is not more than twice the amount earned in the first week. Hence I may or may not be true

Answer should be D