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Of the people who donated money to a certain local theater last year

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Bunuel
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Of the people who donated money to a certain local theater last year [#permalink] New post   Updated on: 15 Aug 2023, 00:01
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64% (02:35) correct 36% (02:48) wrong based on 288 sessions
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Of the people who donated money to a certain local theater last year, 1/4 donated $20 or less and 2/3 donated more than $20 but less than $1,000. If the average (arithmetic mean) amount donated by the people who donated more than $20 but less than $1,000 was $180, what was the average amount donated by the people who donated $1,000 or more?

(1) The average amount donated by the people who donated less than $1,000 was $132.
(2) The average amount donated by the people who donated more than $20 was $360.

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2023-08-15_11-00-18.png [ 134.86 KiB | Viewed 6778 times ]
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B

Originally posted by Bunuel on 07 Aug 2023, 05:00.
Last edited by Bunuel on 15 Aug 2023, 00:01, edited 1 time in total.
Edited the question.
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Re: Of the people who donated money to a certain local theater last year [#permalink] New post   29 Aug 2023, 09:39
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Bunuel wrote:
Of the people who donated money to a certain local theater last year, 1/4 donated $20 or less and 2/3 donated more than $20 but less than $1,000. If the average (arithmetic mean) amount donated by the people who donated more than $20 but less than $1,000 was $180, what was the average amount donated by the people who donated $1,000 or more?

(1) The average amount donated by the people who donated less than $1,000 was $132.
(2) The average amount donated by the people who donated more than $20 was $360.


Each person who donated is a member of exactly one of the following subgroups.

Subgroup 1: $20 or less
fraction of the whole = 1/4 = 3/12
average = a₁

Subgroup 2: between $20 and $1,000
fraction of the whole = 2/3 = 8/12
average = $180

Subgroup 3: $1,000 or more
fraction of the whole = 1 – 3/12 – 8/12 = 1/12
average = a₃

We need to answer the question:

a₃ = ?

Statement One Alone:

=> The average amount donated by the people who donated less than $1,000 was $132.

Using the weighted arithmetic average formula, we could determine a₁:

[(3/12)a₁ + (8/12)180]/(3/12 + 8/12) = 132

However, we can’t answer the question. Statement one is not sufficient. Eliminate answer choices A and D.

Statement Two Alone:

=> The average amount donated by the people who donated more than $20 was $360.

Using the weighted arithmetic average formula, we could determine a₃:

[(8/12)180 + (1/12)a₃]/(8/12 + 1/12) = 360

We could answer the question. Statement two is sufficient.

Answer: B
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Of the people who donated money to a certain local theater last year [#permalink] New post   15 Aug 2023, 00:59
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Bunuel wrote:
Of the people who donated money to a certain local theater last year, 1/4 donated $20 or less and 2/3 donated more than $20 but less than $1,000. If the average (arithmetic mean) amount donated by the people who donated more than $20 but less than $1,000 was $180, what was the average amount donated by the people who donated $1,000 or more?

(1) The average amount donated by the people who donated less than $1,000 was $132.
(2) The average amount donated by the people who donated more than $20 was $360.

Show Spoiler
Attachment:
2023-08-15_11-00-18.png


We have three categories of donors

1) Who donated $20 or less - We will refer to this category as Group 1
2) Who donated more than $20 but less than $1,000 - We will refer to this category as Group 2
2) Who donated more than $1,000 - We will refer to this category as Group 3

Assume that the total number of donors is \(12x\) (LCM of 3x and 4x).

Number of people in each group

  • Group 1: \(\frac{1}{4} * 12x = 3x\)
  • Group 2: \(\frac{2}{3} * 12x = 8x\)
  • Group 3: \(12x - 11x = x\)

...If the average (arithmetic mean) amount donated by the people who donated more than $20 but less than $1,000 was $180...

Amount Donated

  • Group 1: \(p\)
  • Group 2: \(180*3x\)
  • Group 3: \(q\)

Question:\( \frac{q}{x}\) ?

Statement 1

(1) The average amount donated by the people who donated less than $1,000 was $132.

\(\frac{180*3x + p }{ 11x}= 132\)

We don't have any information given in the premise or in this stem to calculate \(q\). Hence, the statement alone is not sufficient. We can eliminate A and D.

Statement 2

(2) The average amount donated by the people who donated more than $20 was $360.

\(\frac{180*3x + q }{ 17x }= 360\)

\(180*3x + q = 360 * 17x\)

\(q = 360 * 17x - 180 * 3x\)

\(q = x(360 * 17 - 180 * 3)\)

\(\frac{q}{x} = (360 * 17 - 180 * 3)\)

We have a definite answer. Hence, this statement alone is sufficient.

Option B
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Re: Of the people who donated money to a certain local theater last year [#permalink] New post   25 Jan 2024, 13:51
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gmatophobia wrote:
Bunuel wrote:
Of the people who donated money to a certain local theater last year, 1/4 donated $20 or less and 2/3 donated more than $20 but less than $1,000. If the average (arithmetic mean) amount donated by the people who donated more than $20 but less than $1,000 was $180, what was the average amount donated by the people who donated $1,000 or more?

(1) The average amount donated by the people who donated less than $1,000 was $132.
(2) The average amount donated by the people who donated more than $20 was $360.

Show Spoiler
Attachment:
2023-08-15_11-00-18.png


We have three categories of donors

1) Who donated $20 or less - We will refer to this category as Group 1
2) Who donated more than $20 but less than $1,000 - We will refer to this category as Group 2
2) Who donated more than $1,000 - We will refer to this category as Group 3

Assume that the total number of donors is \(12x\) (LCM of 3x and 4x).

Number of people in each group

  • Group 1: \(\frac{1}{4} * 12x = 3x\)
  • Group 2: \(\frac{2}{3} * 12x = 8x\)
  • Group 3: \(12x - 11x = x\)

...If the average (arithmetic mean) amount donated by the people who donated more than $20 but less than $1,000 was $180...

Amount Donated

  • Group 1: \(p\)
  • Group 2: \(180*3x\)
  • Group 3: \(q\)

Question:\( \frac{q}{x}\) ?

Statement 1

(1) The average amount donated by the people who donated less than $1,000 was $132.

\(\frac{180*3x + p }{ 11x}= 132\)

We don't have any information given in the premise or in this stem to calculate \(q\). Hence, the statement alone is not sufficient. We can eliminate A and D.

Statement 2

(2) The average amount donated by the people who donated more than $20 was $360.

\(\frac{180*3x + q }{ 17x }= 360\)

\(180*3x + q = 360 * 17x\)

\(q = 360 * 17x - 180 * 3x\)

\(q = x(360 * 17 - 180 * 3)\)

\(\frac{q}{x} = (360 * 17 - 180 * 3)\)

We have a definite answer. Hence, this statement alone is sufficient.

Option B


Why is Group 2 180*3x instead of 180*8x?
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Re: Of the people who donated money to a certain local theater last year [#permalink] New post   26 Jan 2024, 10:40
In the 2nd solution, wouldnt we divide it by 9x?





gmatophobia wrote:
Bunuel wrote:
Of the people who donated money to a certain local theater last year, 1/4 donated $20 or less and 2/3 donated more than $20 but less than $1,000. If the average (arithmetic mean) amount donated by the people who donated more than $20 but less than $1,000 was $180, what was the average amount donated by the people who donated $1,000 or more?

(1) The average amount donated by the people who donated less than $1,000 was $132.
(2) The average amount donated by the people who donated more than $20 was $360.

Show Spoiler
Attachment:
2023-08-15_11-00-18.png


We have three categories of donors

1) Who donated $20 or less - We will refer to this category as Group 1
2) Who donated more than $20 but less than $1,000 - We will refer to this category as Group 2
2) Who donated more than $1,000 - We will refer to this category as Group 3

Assume that the total number of donors is \(12x\) (LCM of 3x and 4x).

Number of people in each group

  • Group 1: \(\frac{1}{4} * 12x = 3x\)
  • Group 2: \(\frac{2}{3} * 12x = 8x\)
  • Group 3: \(12x - 11x = x\)

...If the average (arithmetic mean) amount donated by the people who donated more than $20 but less than $1,000 was $180...

Amount Donated

  • Group 1: \(p\)
  • Group 2: \(180*3x\)
  • Group 3: \(q\)

Question:\( \frac{q}{x}\) ?

Statement 1

(1) The average amount donated by the people who donated less than $1,000 was $132.

\(\frac{180*3x + p }{ 11x}= 132\)

We don't have any information given in the premise or in this stem to calculate \(q\). Hence, the statement alone is not sufficient. We can eliminate A and D.

Statement 2

(2) The average amount donated by the people who donated more than $20 was $360.

\(\frac{180*3x + q }{ 17x }= 360\)

\(180*3x + q = 360 * 17x\)

\(q = 360 * 17x - 180 * 3x\)

\(q = x(360 * 17 - 180 * 3)\)

\(\frac{q}{x} = (360 * 17 - 180 * 3)\)

We have a definite answer. Hence, this statement alone is sufficient.

Option B
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Re: Of the people who donated money to a certain local theater last year [#permalink] New post   27 Feb 2024, 07:54
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Yes. I believe it'll look like this

\(\frac{180*8x + q }{ 9x }= 360\)

\(180*8x + q = 360 * 9x\)

\(q = 360 * 9x - 180 * 9x\)

\(q = x(360 * 9 - 180 * 3)\)

\(\frac{q}{x} = 1800\)
Dhruv1212 wrote:
In the 2nd solution, wouldnt we divide it by 9x?





gmatophobia wrote:
Bunuel wrote:
Of the people who donated money to a certain local theater last year, 1/4 donated $20 or less and 2/3 donated more than $20 but less than $1,000. If the average (arithmetic mean) amount donated by the people who donated more than $20 but less than $1,000 was $180, what was the average amount donated by the people who donated $1,000 or more?

(1) The average amount donated by the people who donated less than $1,000 was $132.
(2) The average amount donated by the people who donated more than $20 was $360.

Show Spoiler
Attachment:
2023-08-15_11-00-18.png


We have three categories of donors

1) Who donated $20 or less - We will refer to this category as Group 1
2) Who donated more than $20 but less than $1,000 - We will refer to this category as Group 2
2) Who donated more than $1,000 - We will refer to this category as Group 3

Assume that the total number of donors is \(12x\) (LCM of 3x and 4x).

Number of people in each group

  • Group 1: \(\frac{1}{4} * 12x = 3x\)
  • Group 2: \(\frac{2}{3} * 12x = 8x\)
  • Group 3: \(12x - 11x = x\)

...If the average (arithmetic mean) amount donated by the people who donated more than $20 but less than $1,000 was $180...

Amount Donated

  • Group 1: \(p\)
  • Group 2: \(180*3x\)
  • Group 3: \(q\)

Question:\( \frac{q}{x}\) ?

Statement 1

(1) The average amount donated by the people who donated less than $1,000 was $132.

\(\frac{180*3x + p }{ 11x}= 132\)

We don't have any information given in the premise or in this stem to calculate \(q\). Hence, the statement alone is not sufficient. We can eliminate A and D.

Statement 2

(2) The average amount donated by the people who donated more than $20 was $360.

\(\frac{180*3x + q }{ 17x }= 360\)

\(180*3x + q = 360 * 17x\)

\(q = 360 * 17x - 180 * 3x\)

\(q = x(360 * 17 - 180 * 3)\)

\(\frac{q}{x} = (360 * 17 - 180 * 3)\)

We have a definite answer. Hence, this statement alone is sufficient.

Option B




 

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Re: Of the people who donated money to a certain local theater last year [#permalink] New post   18 Mar 2024, 19:59
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­Hi all,
I found a simpler way to solve the question than the ways shared in this post.

Imagine just drawing a BAR Graph with the 3 Groups
1) people who donated 20 or less --> 1/4 or 3/12 of the total donors
2) people who donated more than 20 and less than $1000 --> 2/3 or 8/12 of the total donors            - AVG MEAN of this donatios $180
3) people who donated more than $1000 --> 1/12 of the total donors (1 - the previous values)

Notice that for this excersie we can set a proportion of 3 : 8 : 1 respectively for the donors groups.

Statement #1
They give us just information about Group 1 and Group 2. There is no possible way to estimate how much Group 3 donated.
So discarded, but note that we could get the average of group 1 if needed.

Eliminate answers: A & D

Statement #2
They give us information about the Group 2 and Group 3! Bingo!
So $360 is the average for 9 donors -> Total donations: $3240 form Group 2 & 3.
If we estimate the total Donations for just group 2 (it would be $180 x 8 --> $1440)
So Group 3 is the difference of this values, beign $1800 for just 1 donor.

So statement 2 is correct

Answer B


 
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Re: Of the people who donated money to a certain local theater last year [#permalink]
18 Mar 2024, 19:59
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