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Bicycles

by j_shreyans » Thu Aug 14, 2014 3:05 am
Hello Everyone ,

pls help me out from the below question.




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 > 2 x

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

[spoiler]Answer is II&III[/spoiler]

Thanks in advancs


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by GMATGuruNY » Thu Aug 14, 2014 3:46 am
j_shreyans 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
Of the 3 statements, only III offers an actual NUMBER.

Statement III: y>3
Check whether it's possible that y≤3.
If y=3, then THIS WEEK'S earnings = 20 + 3 bikes earning $6 each = 20 + 3*6 = 38.
Since THIS WEEK'S earnings must be more than twice LAST WEEK'S earnings, the amount earned last week must be LESS THAN $19.
Not possible, since the base salary each week is $20.
Since y=3 is not possible, it must be true that y>3.
Since statement III must be true, eliminate A, B and C.

Statement I: y>2x
If x=6, then LAST WEEK'S earnings = 20 + 6 bikes earning $6 each = 20 + 6*6 = 56.
Since THIS WEEK'S earnings must be more than twice LAST WEEK'S earnings, the amount earned THIS WEEK must be greater than $112.
Check whether it's possible that y≤2x.
If y=12, then THIS WEEK'S earnings = 20 + 6 bikes earning $6 each + 6 bikes earning $12 each = 20 + 6*6 + 6*12 = 128, which is greater than 112.
Since it's possible that y=2x, statement I does NOT have to be true.
Eliminate E.

The correct answer is D.
Last edited by GMATGuruNY on Thu Aug 14, 2014 3:46 pm, edited 2 times in total.
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by [email protected] » Thu Aug 14, 2014 10:14 am
Hi j_shreyans,

This Roman Numeral question can be solved with "brute force"; let's map out the possibilities and look for patterns:

Based on the given information, here's a table of how much money Norman would make (based on the number of bikes sold):

Bikes = Money
0 = 20

1 = 26
2 = 32
3 = 38
4 = 44
5 = 50
6 = 56

7 = 68
8 = 80
9 = 92
10 = 104
11 = 116
12 = 128

13 = 144
14 = 162
Etc. ($18 per additional bike)

We're told that Norman sold X bicycles last week and Y bicycles this week. We also know that he earned MORE THAN TWICE the money he earned in the prior week, so we have to use THAT fact to evaluate what the possibilities could be (within the table).

II. Y > X

Roman Numeral II is easiest, so we'll start there. Since Norman earned MORE MONEY, he had to have sold MORE bicycles. Thus Y MUST be greater than X.
Roman Numeral II is TRUE.

III. Y > 3

Here, we can look at the "top" of the table and talk through the possibilities.

If last week, Norman sold ___ bikes last week, then how many would he need to have sold this week, at the MINIMUM, to make MORE than twice the money?
0 bikes....4 or more bikes were sold
1 bike.....6 or more bikes were sold

The second number will just get bigger and bigger. This proves that Y MUST be greater than 3.
Roman Numeral III is TRUE.

I. Y > 2X

For this Roman Numeral, we can continue the work that we did in Roman Numeral II; we have to look to see whether Y is ALWAYS greater than 2X or not...

2 bikes....7 or more bikes were sold
3 bikes....8 or more bikes were sold
4 bikes....9 or more bikes were sold
At this point, notice the ratio of Y to X is getting smaller....?
5 bikes...10 or more bikes were sold.

This last example PROVES that Y isn't always greater than 2X.
Roman Numeral I is NOT ALWAYS TRUE.

Final Answer: D

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by j_shreyans » Thu Aug 14, 2014 10:23 pm
Hi Everyone ,

Well understood the solutions.

Thanks :)
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by GMATinsight » Fri Aug 15, 2014 8:46 am
j_shreyans wrote:Hello Everyone ,

pls help me out from the below question.




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 > 2 x

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

[spoiler]Answer is II&III[/spoiler]

Thanks in advancs


Shreyans
The only way forward in such questions is to Objective enough. We should try to prove that the given information can be Incorrect and we start eliminating options.

Here, We have to understand that
Fixed Salary = $20 [If he sells nothing]

Additional $6 for selling upto 6 Bicycles
i.e. $26 for 1 Bicycle
$32 for 2 Bicycle
$38 for 3 Bicycle
$44 for 4 Bicycle
$50 for 5 Bicycle
$56 for 6 Bicycle

$12 per bicycle for the next six bicycles he sells
i.e.From $56+12 = $68 to $56+(6x12) = $128 from 7th to 12th Bicycle

$18 per bicycle for every bicycle sold after the first 12
From $128+18 = $146 to any increasing in multiple of 18

This week earning > 2 x This week's earning

If he sold x bicycles last week and y bicycles this week

Let's try to prove first statement y > 2x false

If x = 6, Money earned last week = $56
and y = 11, Money earned last week = $116
Here $116 > 2 x $56
and y is NOT greater than 2x
hence proved NOT DEFINITELY TRUE

Let's try to prove Second statement y > x false

To prove this incorrect we should prove y is either less than or equal to x but then the money earned this week can't be greater than the earning last week
hence proved DEFINITELY TRUE

Let's try to prove Third statement y > 3 false

If x = 0, Money earned last week = $20
and (at the most) y = 3, Money earned last week = $38
Here $38 is NOT Greater than $20
hence proved DEFINITELY TRUE

Answer: Option D
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