As a bicycle salesperson, Norman earns a fixed salary of $20 per week plus $6 per bicycle for the first 6 bicycles he

[BTGmoderatorDC]

As a bicycle salesperson, Norman earns a fixed salary of $20 per week plus $6 per bicycle for the first 6 bicycles he sells, $12 per bicycle for the next 6 bicycles he sells, and $18 per bicycle for every bicycle sold after first 12. This week, he 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, III



OA D

Source: Manhattan Prep

[Scott@TargetTestPrep]

As a bicycle salesperson, Norman earns a fixed salary of $20 per week plus $6 per bicycle for the first 6 bicycles he sells, $12 per bicycle for the next 6 bicycles he sells, and $18 per bicycle for every bicycle sold after first 12. This week, he 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, III



OA D

Source: Manhattan Prep

Statements II and III are easy, so let’s start with them.

Statement II: Since he sold y bicycles this week and x bicycles last week, and it is given that he earned more than twice this week as last week, then y must be greater than x. II is true.

Statement III: Let’s consider the case that y = 3. Then this week Norman would have earned 20 + (3 * 6) = $38. This cannot be true because we are given that he earned more than twice what he earned last week, but his minimum for last week would be the fixed $20 salary. Since $38 is not more than twice $20, we see that y must be greater than 3. III is true.

Statement I: The possible earnings are 20, 26, 32, 38, 44, 50, 56, 68, 80, 92, 104, 116, 128, 146, 164, etc. which respectively correspond to selling 0 bikes, 1 bike, 2 bikes etc.

Notice that an earning of 104 (which corresponds to the sale of 10 bikes) is more than twice of an earning of 50 (which is from the sale of 5 bikes). Thus, if we take x = 5 and y = 10, we see that the condition “the earnings this week are more than twice the earnings of last week” is satisfied, however, the condition y > 2x is not satisfied. Therefore, we see that I is not always true.

Answer: D

[swerve]

As a bicycle salesperson, Norman earns a fixed salary of $20 per week plus $6 per bicycle for the first 6 bicycles he sells, $12 per bicycle for the next 6 bicycles he sells, and $18 per bicycle for every bicycle sold after first 12. This week, he 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, III



OA D

Source: Manhattan Prep

Test the inequalities:

I. \(y>2x\)
Let \(x = 1\) bicycle; Earnings: 26 dollars
Let \(y = 3\) bicycle; Earnings: 38 dollars

Is 38 more than twice of 26? NO! \(\Large{\color{red}\chi}\)

II. \(y>x\)
Surely, there must be more bicycles sold in the second week. Always true! YES! \(\Large{\color{green}\checkmark}\)

III. \(y>3\)
Testing I, we found that when \(y = 3\) and \(x = 1\), we still couldn't achieve the condition that the second week's earning is more than twice the first. Therefore, \(y\) must be greater than 3. YES! \(\Large{\color{green}\checkmark}\)

Therefore, the correct answer is D