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As a bicycle salesperson, Norman earns a fixed salary of $20 [#permalink]
17 Mar 2011, 05:39

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Difficulty:

45% (medium)

Question Stats:

48% (02:22) correct
51% (01:24) wrong based on 110 sessions

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

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

II and III are obviously always true:

II. y>x --> since this week, Norman earned more than he did last week and the total salary is in direct relationship with the # of bicycle sold, then y (# of bicycle sold this week) must be more than x (# of bicycle sold last week);

III. y>3 --> if Norman sold 3 bicycles this week then this week he earned 20+3*6=$38, which cannot be more than twice as much as he earned the last week, since the minimum salary is fixed to $20. So y must be more than 3;

I. y>2x --> if y=12 and x= 6 then this week Norman earned 20+6*6+6*12=$128, and the last week he earned 20+6*6=$56. $128 is more than twice as much as $56, so the condition in the stem holds but y=2x, which means that III is not always true.

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

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

I think II and III are pretty straight forward and I am assuming you have no problem deciding about those.

Let me add here what I thought about I. One way is that you can try to find a case where he earns twice as much but doesn't sell twice as many bikes. Another is a more intuitive approach. You know that initially, he has to sell more bikes to make some money (he earns only $6 from first 6 bikes and $12 from next 6 bikes. First $20 is too small an amount). Later on, he gets $18 per bike which means he makes money at a much faster rate. Hence, later on, he can double the amount he made previously very quickly and by selling far fewer bikes. Hence it is not essential that he needs to sell twice as many bikes to make twice as much money. Hence y may not be greater than 2x.
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Re: As a bicycle salesperson, Norman earns a fixed salary of $20 [#permalink]
01 Dec 2012, 22:00

Test the inequalities: I. y>2x Let x = 1 bicycle; Earnings: 26 dollars Let y = 3 bicylce; Earnings: 38 dollars Is 38 more than twice of 26? NO! II. y > x Surely, there must be more bicycles sold in the second week. Always true! YES! 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!

Re: As a bicycle salesperson, Norman earns a fixed salary of $20 [#permalink]
29 Mar 2013, 00:19

AnkitK wrote:

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

Given:

1. The number of bicycles sold last week = x 2. The number of bicycles sold this week = y 3. Let earnings of last week and this week be s1 and s2 resp. s2> 2s1

Question:

1. Is y > 2x 2. Is y > x 3. Is y > 3

Basically the question asks us to relate number of bicycles sold in each of 2 weeks based on the relation between the earnings in those 2 weeks.

1. Earnings in the current week can be higher than that of the last week only when the number of bicycles sold is higher in the current week. i.e., only when y>x 2. If the number of bicycles sold during the current week <4, then the earnings in the current week cannot be more than double that of the previous week. 3. Now let us assume y=2x. Since we are assuming twice the bicycles are sold this week over that of the previous week , if we take x=18, then y=36. 4. Let us calculate s1 and s2. s1= earnings from the first 12 bicycles + earnings from the next 6 bicycles = 128+ 108= 236 s2= earnings from the 12 bicycles+ earnings from the next 24 bicycles= 128+ 432= 560 5. s2>2s1 even when y=2x

We see from (1) above statement II is true, from (2) above statement III is true, from (5) above statement I need not be true.

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

II and III are obviously always true:

II. y>x --> since this week, Norman earned more than he did last week and the total salary is in direct relationship with the # of bicycle sold, then y (# of bicycle sold this week) must be more than x (# of bicycle sold last week);

III. y>3 --> if Norman sold 3 bicycles this week then this week he earned 20+3*6=$38, which cannot be more than twice as much as he earned the last week, since the minimum salary is fixed to $20. So y must be more than 3;

I. y>2x --> if y=12 and x= 6 then this week Norman earned 20+6*6+6*12=$128, and the last week he earned 20+6*6=$56. $128 is more than twice as much as $56, so the condition in the stem holds but y=2x, which means that III is not always true.

Answer: D.

Bunuel, nice approach +1

On I though, I'm having some issues picking the correct numbers, how can I decide which numbers to use to prove this case not necessarily true?

I'm having some issues picking the correct numbers, how can I decide which numbers to use to prove this case not necessarily true?

Cheers! J

There are no correct/incorrect numbers. You can just try to understand the logic using numbers.

6 bikes - $6 each i.e. total $36 next 6 bikes - $12 each i.e. total $72 So 12 bikes for a total sum of $108

But for every subsequent bike, he gets $18 so the next $108 he will be able to make by selling just 6 bikes. So even if he earns twice as much as before, he doesn't need to sell twice as many bikes.
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