MGMAT numbers

[rommysingh]

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

PS - Kindly let know how to pick smart numbers so that I can save time. Specially in testing case 1.

[GMATGuruNY]

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.

[rommysingh]

Hi GURUNY
thanks for responding. I need to know that in case 1 why did you specifically pick 6 s=as a number. I waste a lot of time trying different number. plz help

[[email protected]]

Hi rommysingh,

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

GMAT assassins aren't born, they're made,
Rich

[sandipgumtya]

Hi Mitch,
Is option II always true?The eqn is Y>2X-12 as X=10,Y=9 is valid set of no.Then how Y>X?pl explain

[[email protected]]

Hi sandipbumtya,

We're told that Norman earned MORE THAN TWICE as much as he did last week....When you choose a Y and X, you have to account for the amount of MONEY that Norman earned when selling Y bicycles and X bicycles. If the first total isn't MORE THAN TWICE the second total, then you are NOT allowed to use that pair of values.

Logically though, you don't need to do any algebra at all to properly assess the second Roman Numeral. Since we're told that Norman made MORE money this week, he had to have sold more bicycles (so Y > X).

GMAT assassins aren't born, they're made,
Rich