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

Correct answer is D. OA edited.

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 I is not always true.

Answer: D.
_________________

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.

I will throw in the values of x and y since its going to be tough using algebra.

For first 6 bicycles - he gets $6 / bicycle
For next 6 bicycles - he gets $12 / bicycle
For > 12 bicycles - he gets $18 / bicycle

Constraint : This week he earned more than twice as much as he did last week.

Paraphrase I: Did he double the quantity of bicycles sold to earn more than double the revenue from last week?

I dont think so. Reasons -

Lets say last week he sold 13 bicycles.

Last week revenue = 20 + 6*6 + 6*12 + 1*18 = 146

146*2 + 1 = 292 + 1 = 293. To make this revenue he could sell (293 - 128)/18 = 165/18 i.e. 10 more than 12 bicycles

Total bicycles sold this week = 12 + 10 = 22 (which is less than twice the bicyles sold last week)

Hence I is ruled out. That leaves the options - B and D.

B Vs D. I have to verify statement III

Paraphrase III: Did he double the revenue from last week by selling minimum of 4 bicycles this week?

Lets assume the contradiction is true. He sold 3 bicycles this week and 1 bicycle last week.

Last week revenue = 20 + 6*1 = 26
This week revenue = 20 + 6*3 = 38

38 is less than twice 26. So the contradiction fails. Hence III is true.

Answer D.

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!

Answer: D

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.

The answer is choice 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?

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.

Hi Karishma,

I'm intrigued by your intuitive approach.

To backtrack a little -- word problems as a whole seem to be the biggest time suck for me. I spent 4 minutes on this problem, and although I got it right, I can't seem to figure out how to speed things up when it comes to word problems as such.

Is there a strategy you recommend to tackle word problems in general? I know that this is a vague question but any help would be appreciated. Can you recommend other word problems to do to help with practice?

Regarding what you said, to me, 2 seemed very straight forward but I still went and checked statement 3. Yes, in hindsight, all of this looks very simple after reading your explanation but I'm not as certain during the test.

Any thoughts would be appreciated.

Thanks

Hey Russ,

Familiarity creates intuition. When you see a lot of word problems, you are often able to see what is going to work and usually it is correct. Till a few years back, I use to rely on algebra (equations) to solve all word problems. Then, a mentor forced me to see the big picture, the reason behind every step and how the steps are meant for machines only - how we are quite capable of using reason and logic to solve most questions in a reasoning based test such as GMAT. Now the problem is that when you need to give a solution to someone, just saying that use intuition is not helpful. You can barely explain it in a face-to-face situation.

Also, confidence comes with practice. You will start feeling confident in your inferences from the given data once you see that you are getting most of them right on practice questions.

I will suggest you to start every word problem by trying to infer whatever you can from the given data. Try to minimize your use of equations (you can't let them go completely). Look for alternative solutions for every problem. Soon. you will start coming up with your own intuitive solutions.

i still feel option (B) is correct because it says y>3 and you guys tested the condition with y=3 and x=1
but what about when y=4 and x=1 or 2
then the earning last week add up to 26 or 32 and the earnings this week is merely 44
and either ways the earnings last week is more than half of the earnings this week

Can you guys please clarify on this approach?

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 last week, since the minimum salary is fixed to $20. So y must be more than 3.
_________________

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

What if you just do:
x=1 so y>2*1=2 so let's say y=3

20+(1*6)=26 = earnings last week
20+(3*6)=38 = earnings this week

He earned this week more than twice as much as last week so 38 must be bigger than 26*2.

38<52 so this means y>2x does not have to be true.

Is this correct or is this the wrong way?

To prove that (I) needn't hold, you need to find numbers where he earned more than twice but y was not greater than twice of x. You have done the opposite - you have taken a case where y is greater than twice of x and shown that he did not earn more than twice. This doesn't prove that (I) needn't hold.

The numbers you need to consider would be say x = 12, y = 24 (y is NOT MORE than twice of x)
Last week's earning = 20 + 6*6 + 12*6 = 128
This week's earning = 20 + 6*6 + 12*6 + 12*12 = 128 + 144 (More than twice of last week's earning)

So he could earn more than twice of last week's earning and still, y > 2x may not hold.

Oke I thought because y can be bigger than 2x and he can earn more than twice last week x=1 and y=6 for example than 56>52 but if y is 3, 4 or 5, which is bigger dan 2x, y<52 and he did not earn more than twice last week. So it can be true but also false. So that's why I thought y>2x does not stand at all time.

I now understand that I have to find y=2x and that he earned this week more than twice last week but I thought the other way around could also solve the problem.

Think logically - Say if A is true implies B must be true, does it mean that if B is true then A must be true too? Not necessary, right?
For example, you know that if it rains, the ground gets wet. Now if you see the ground wet, can you say that it must have rained? No necessary, right? Perhaps someone spilled water on the ground - we don't know.

Hi Karishma and Bunuel,

In order to prove the third statement 1.e y>3, you are assuming x=0.
Is this allowed to assume x=0.

Thanks & regards,
Sunil01

Hi Sunil,

you have to realize that the money he earned by selling y cycles is more than twice that he earns by selling x cycle..
if this is true, any increase in x would result in further increase in y..
we have to test if y>3, and we test in its lowest possible value..
y will be lowest when x=0, as y increases with increase in x..
since we find when x=0, the money earned= 20..
twice the amount will be 40 which is 20(fixed salary) + 20... so y=20/6= 3.33 so y has to be greater than 3 in all circumstances..
so MUST be TRUE..
_________________

So we need to figure out whether y must be greater than 3. So I think to myself - what is so great about y = 3 that it cannot happen while y = 4 can probably happen?
That is why I put y = 3 and see the numbers I get.

If y = 3, total earning = 20 + 3*6 = 38
I know that this week he earned more than twice of last week. So if he sold 3 bikes this week, he must have earned less than $19 last week. But last week he MUST have earned at least $20, right? That is his fixed salary. This is the reason y cannot be 3 or less. It MSUT be more than 3.
We don't assume that x is 0. We say that even if x = 0, his last week's salary cannot be $19. This means he sold at least 4 bikes this week.

Hi,
although the explanation is almost same as a post above this explanation, but since you did not understand my English, I got to start looking into my Verbal ..
Kidding... Till the time you are learning, its ok, and I am sure none is contributing for appreciation.
_________________