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As a bicycle salesperson, Norman earns a fixed salary of $20

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As a bicycle salesperson, Norman earns a fixed salary of $20 per week [#permalink]

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New post 15 Jan 2008, 11:20
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Question Stats:

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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

OPEN DISCUSSION OF THIS QUESTION IS HERE: http://gmatclub.com/forum/as-a-bicycle- ... 11038.html
[Reveal] Spoiler: OA

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Re: As a bicycle salesperson, Norman earns a fixed salary of $20 per week [#permalink]

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New post 15 Jan 2008, 11:37
Right off the bat it's obvious that II works. Unfortunately that only eliminates option A. Let's try statement 1.

1. y > 2x. This doesn't necessarily have to be true. It would be true if each bicycle made him the same amount of commission, but that's not true. There are different tiers of commission that make it possible to sell fewer than 2x bikes and still make more than twice as much money.

Example:

Selling 6 bikes: $20 + 6(6) = $56
Selling 11 bikes: $20+6(6)+5(12) = $116 which is more than twice as much as $56, but you don't have to sell 2x bikes.

2. y > x

no way you can sell fewer bikes than last week and earn twice the money

3. y>3

This is true because even if he sold 0 bikes last week. He still made $20 base rate. If y = 3 then he made $20+3(6) = $38 which doesn't even double his base rate of $20. y must be > 3 for this to work.

Answer D

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Re: As a bicycle salesperson, Norman earns a fixed salary of $20 per week [#permalink]

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New post 15 Jan 2008, 13:04
JCLEONES wrote:
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

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As a bicycle salesperson, Norman earns a fixed salary of $20 [#permalink]

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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

OPEN DISCUSSION OF THIS QUESTION IS HERE: as-a-bicycle-salesperson-norman-earns-a-fixed-salary-of-111038.html

Last edited by Bunuel on 26 Sep 2013, 08:40, edited 1 time in total.
Renamed the topic, edited the question and added the OA.

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Re: Bicycles [#permalink]

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New post 02 Nov 2009, 10:28
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This is kind of like doing your taxes, lets summarize the data in terms of bike sales and salary -

0 Bikes - $20
6 Bikes - $56
12 Bikes - $128
More than 12 bikes - 128 + 18*(x-12) where x is the total number of bikes sold (when x>12).

X = cycles sold last week
XS = Salary last week.
Y = cycles sole this week
YS = Salary this week.

given YS>2XS

lets test all statements with X=0;
XS = 20
YS >40
True when Y = 4
satisfies all the statements but we can only be sure of iii as Y will only increase from here. So iii must be true. Eliminate A,B,C

lets test all statements with X=1;
XS = 26
YS>52
True when Y=6
Still satisfies all the statements - Nothing achieved

lets test all statements with X=6;
XS = 56
YS>112
True when Y = 11
This proves statement i wrong, hence eliminate E so we are left with

Answer D
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Re: As a bicycle salesperson, Norman earns a fixed salary of $20 per week [#permalink]

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New post 26 Feb 2011, 07:00
Can some one help me formulate an equation for this:

Id like to take the guess work out

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Re: As a bicycle salesperson, Norman earns a fixed salary of $20 per week [#permalink]

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New post 26 Feb 2011, 11:49
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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.

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Re: As a bicycle salesperson, Norman earns a fixed salary of $20 per week [#permalink]

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New post 26 Feb 2011, 12:24
beyondgmatscore wrote:
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.



Well done! i was trying to form one equation for all three cases. Clearly three different equations are required based on the range!

Thanks

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Re: Bicycles [#permalink]

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New post 26 Sep 2013, 08:38
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Re: As a bicycle salesperson, Norman earns a fixed salary of $20 [#permalink]

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New post 26 Sep 2013, 08:41
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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.

OPEN DISCUSSION OF THIS QUESTION IS HERE: as-a-bicycle-salesperson-norman-earns-a-fixed-salary-of-111038.html
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Re: As a bicycle salesperson, Norman earns a fixed salary of $20   [#permalink] 26 Sep 2013, 08:41
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