Norman's Bicycle Sales

[atharshiraz]

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

The answer is D (which is II and III)

The explanation for I being incorrect is as follows:
"I. UNCERTAIN: It depends on how many bicycles Norman sold.

For example, if x = 4, then Norman earned $44 [= $20 + (4 × $6)] last week. In order to double his earnings, he would have to sell a minimum of 9 bicycles this week (y = 9), making $92 [= $20 + (6 × $6) + (3 × $12)]. In that case, y > 2x.

However, if x = 6 and y = 11, then Norman would have earned $56 [= $20 + (6 × $6)] last week and $116 [= $20 + (6 × $6) + (5 × $12)] this week. In that case, $116 > 2 × $56, yet y < 2x.

So, it is possible for Norman to more than double his earnings without selling twice as many bicycles."

To get to bicycle #4 and #6 to see the pattern or to disprove the pattern (y>2x) takes some time. Is there an insight or understanding that would have allowed us to show that y>2x cannot be possible? Surely we should not be expected to go through the entire calculations of 0,1,2,3,4,5,6 bikes, right?

[jnelson0612]

Actually, I apparently answered this same question this way: as-a-bicycle-salesperson-norman-earns-a-fixed-salary-of-20-t11654.html :-)

Take a look, and then if you want to discuss further let's do so. :-)