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by sana.noor » Fri Oct 04, 2013 7:30 am
In the first year of a pyramid scheme, John convinced y of his friends to pay 30 dollars each to join a particular website that he created. Each of those y friends then convinced another y people to pay 15 dollars each to join the same website. If no one else joined the website that year and each person joined only once, what was the value of y?

a)The revenue for the website that year was $36,000.
b)The first y friends accounted for 1/25 of the total revenue for the website that year.

OA is D
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Source: — Data Sufficiency |
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by theCodeToGMAT » Fri Oct 04, 2013 7:41 am
Total Revenue = 30(y) + 15(y)(y)

Statement 1
30y + 15y^2 = 36000
y^2 + 2y - 2400 = 0
We will definitely get only one positive value of "y"
SUFFICIENT

Statement 2:
30y = (30y + 15y^2)/25
30 = (30 + 15y)/25
SUFFICIENT

Answer [spoiler]{D}[/spoiler]
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by Brent@GMATPrepNow » Fri Oct 04, 2013 7:50 am
sana.noor wrote:In the first year of a pyramid scheme, John convinced y of his friends to pay 30 dollars each to join a particular website that he created. Each of those y friends then convinced another y people to pay 15 dollars each to join the same website. If no one else joined the website that year and each person joined only once, what was the value of y?

a)The revenue for the website that year was $36,000.
b)The first y friends accounted for 1/25 of the total revenue for the website that year.

OA is D
Target question: What was the value of y?

Statement 1: The revenue for the website that year was $36,000.
First round of "investors": y people paying $30 each = 30y dollars
Second round of "investors": y² people paying $15 each = 15y² dollars
Total revenue = 15y² + 30y
So, 15y² + 30y = 36,000
Set equal to zero: 15y² + 30y - 36,000 = 0

IMPORTANT: This is a quadratic equation, and quadratic equations typically have 2 solutions. IF there are two valid solutions to this equation, then statement 1 is not sufficient.
Divide both side by 15 to get: y² + 2y - 2400 = 0
At this point, we should recognize that we IF we were to factor the left-side, we'd get (y + something)(y - something) = 0, which means one possible value for y is positive, and one possible value for y is negative.
Since the number of friends MUST be positive, we can be certain that the equation has ONLY ONE valid solution.
This means that IF WE WERE to solve the equation, we'd be able to target question with certainty.
So, statement 1 is SUFFICIENT

Statement 2: The first y friends accounted for 1/25 of the total revenue for the website that year
In other words, (1st round of investments) = (1/25)(total investments)
30y = (1/25)(15y² + 30y)
Multiply both sides by 25 to get: 750y = 15y² + 30y
Rearrange: 15y² - 720y = 0
Factor: 15y(y - something) = 0
At this point, we can be certain that there is only 1 valid solution for y (since y = 0 is an invalid solution).
So, IF WE WERE to solve the equation, we'd be able to target question with certainty.
Statement 2 is SUFFICIENT

Answer = D

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Brent
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by sana.noor » Fri Oct 04, 2013 8:00 am
Brent why it is in quadratic equation, i mean why it is 15y² why not 15 y.
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by Brent@GMATPrepNow » Fri Oct 04, 2013 8:06 am
sana.noor wrote:Brent why it is in quadratic equation, i mean why it is 15y² why not 15 y.
Good question. I'll answer it with an example.

Let's say John has 3 friends (i.e., y = 3)
First round of investors: 3 people paying $30 each = 30(3) dollars

Second round of investors:
Each of John's 3 friends, gets 3 people to invest.
In other words, (3)(3) people invest (aka 3² people)
Each pays $15, so the we get 15(3²) dollars


Now, let's say John has 4 friends (i.e., y = 4)
First round of investors: 4 people paying $30 each = 30(4) dollars

Second round of investors:
Each of John's 4 friends, gets 4 people to invest.
In other words, (4)(4) people invest (aka 4² people)
Each pays $15, so the we get 15(4²) dollars

And so on . . .

Does that help?

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by sana.noor » Fri Oct 04, 2013 9:49 am
Yes A LOT....Thank you.
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by [email protected] » Fri Oct 04, 2013 1:24 pm
Hi sana.noor,

Both of the other explanations provides an "algebra" approach; here's a "pattern/brute-force" approach that you might find useful in certain situations.

This DS question describes a very specific relationship that's based on Y.

If Y = 1, then the scheme gets 1(30) + 1(1)(15) = 45 dollars
If Y = 2, then the scheme gets 2(30) + 2(2)(15) = 120 dollars
If Y = 3, then the scheme gets 3(30) + 3(3)(15) = 225 dollars
etc.

You should notice that when Y increases, the total amount of money increases. If you change the Y, then the number of dollars changes. Watch how that deduction can be used in combination with the Facts....

Fact 1: the revenue = $36,000

With a total dollar amount, we know that we COULD figure out the value of Y AND there would be just one answer. Don't waste time doing work that you don't have to.
Fact 1 is SUFFICIENT.

Fact 2: the first Y friends = 1/25 of the total revenue. If you look at the data at the top of this post, you should notice that the "first Y friends" account for a smaller and smaller fraction of the total as Y increases. Knowing that the "first Y friends" would account for exactly 1/25 of the revenue, we COULD figure out the value of Y AND there would be just one answer. We don't have to do this calculation either.
Fact 2 is SUFFICIENT.

Final Answer: D

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