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A radio station is giving away a total of $124,000 in August. If they

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A radio station is giving away a total of $124,000 in August. If they [#permalink]

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A radio station is giving away a total of $124,000 in August. If they increase the amount given away each day by $100, how much should they give away the first day?

A. $2,300
B. $2,500
C. $2,700
D. $2,900
E. $3,500
[Reveal] Spoiler: OA

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A radio station is giving away a total of $124,000 in August. If they [#permalink]

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Bunuel wrote:
A radio station is giving away a total of $124,000 in August. If they increase the amount given away each day by $100, how much should they give away the first day?

A. $2,300
B. $2,500
C. $2,700
D. $2,900
E. $3,500


Let first day amount given be \(a\), each day increment \(d=100\) and number of days in Aug \(= 31\). Hence using the Sum formula for AP series we get

\(Sum = \frac{n}{2}[2a+(n-1)*d]\)

\(124000=\frac{31}{2}[2a+(31-1)*100]\)

or \(8000=2a+3000=> a= 2500\)

Option B

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Re: A radio station is giving away a total of $124,000 in August. If they [#permalink]

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New post 09 Oct 2017, 03:31
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Bunuel wrote:
A radio station is giving away a total of $124,000 in August. If they increase the amount given away each day by $100, how much should they give away the first day?

A. $2,300
B. $2,500
C. $2,700
D. $2,900
E. $3,500


Another Method:

Sum = n * Average

124,000 = 31 * Average

Average = 4000 = (First + Last)/2

First + First + 30*100 = 8000

First term = 2500

Answer (B)
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A radio station is giving away a total of $124,000 in August. If they [#permalink]

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New post 09 Oct 2017, 07:59
Bunuel wrote:
A radio station is giving away a total of $124,000 in August. If they increase the amount given away each day by $100, how much should they give away the first day?

A. $2,300
B. $2,500
C. $2,700
D. $2,900
E. $3,500

To find the first term, I used the general formula:

Sum = (average)(# of terms)

. . .because the average contains the first term.

VeritasPrepKarishma 's version is similar but shorter (kudos).

First we need average, i.e., the middle term's value. One way to find that is to find the formula for the sequence.

\(A_1 = A_1\)
\(A_2 = A_1 + 100\)
\(A_3 = A_1 + 100 + 100\)
\(A_4 = A_1 + 100 + 100 + 100\)

Each term equals \(A_1\) plus $100 multiplied by one fewer than the term's subscript (because $100 is NOT added to the first term).

So \(A_{n} = A_1 + (n - 1)100\)
(arithmetic series, common difference)

In an arithmetic series, median = mean (average)

August has 31 days. Median is the 16th term. (Median =\(\frac{(n+1)}{2}\))

\(A_{16} = A_1 + 15(100)\)

Sum = (average = \(A_{16}\))(# of terms)
124,000 = (\(A_1\) + 1500)(31)
4,000 = \(A_1\) + 1500
2,500 = \(A_1\)

Answer B

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Re: A radio station is giving away a total of $124,000 in August. If they [#permalink]

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New post 09 Oct 2017, 10:25
Let the amount given away in first day be x.
Amount given in subsequent day=x+100 and so on.
Amount given in 31st day (August has 31 days) = x+3000
Sum of all amounts = x+(x+100)+(x+200)+.....(x+3000)=124000
Sum of 31 amounts = 31x + 30/2*(100+3000)=31x+15*3100=31x+46500
31x+46500=124000
x=2500
Answer B.
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Re: A radio station is giving away a total of $124,000 in August. If they [#permalink]

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New post 09 Oct 2017, 10:28
Bunuel wrote:
A radio station is giving away a total of $124,000 in August. If they increase the amount given away each day by $100, how much should they give away the first day?

A. $2,300
B. $2,500
C. $2,700
D. $2,900
E. $3,500


Let the amount given away on first day = a
Increase per day ,d = 100
then amount given away on last day = a + 30d = a+ 30*100 = a+ 3000
Total amount given away in August = 1,24,000
=> 31/2 ( 2a + 3000) = 1,24,000
=> 2a+ 3000 = 8000
=> a = 2500

Answer B
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A radio station is giving away a total of $124,000 in August. If they [#permalink]

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New post 09 Oct 2017, 15:46
My approach to the problem was as follows: Since each day is increased by a consistent amount, you can view the problem as one dealing with an evenly spaced set. We know that for an evenly spaced set the mean= median. In this case the mean/ median= $4,000 (124,000/ 31). From there we know that the mean day is the 16th. So the the 1st day will be $1,500 less (15x100), or $2,500. Therefore, the answer is B

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Re: A radio station is giving away a total of $124,000 in August. If they [#permalink]

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New post 12 Oct 2017, 17:55
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Bunuel wrote:
A radio station is giving away a total of $124,000 in August. If they increase the amount given away each day by $100, how much should they give away the first day?

A. $2,300
B. $2,500
C. $2,700
D. $2,900
E. $3,500


We can let x = the amount given away on the first day; thus, x + 100 = the amount given away on the second day, x + 200 = the amount given away on the third day, ... , x + 3,000 = the amount given away on the 30th day. We can create the following equation:

x + (x + 100) + (x + 200) + … + (x + 3,000) = 124,000

However, since the terms on the left-hand side constitute an arithmetic progression, we can use the formula average x quantity = sum.

First, we can determine the average by using the formula for the average of an evenly spaced set: average = (first + last)/2.

average = [x + (x + 3,000)]/2

average = (2x + 3000)/2 = x + 1500

Let’s summarize the information that we have: average = (x + 1,500), quantity = 31, and sum = 124,000. Thus, we have:

(x + 1,500)(31) = 124,000

x + 1,500 = 4,000

x = 2,500

Answer: B
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Re: A radio station is giving away a total of $124,000 in August. If they [#permalink]

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New post 15 Nov 2017, 10:46
Bunuel wrote:
A radio station is giving away a total of $124,000 in August. If they increase the amount given away each day by $100, how much should they give away the first day?

A. $2,300
B. $2,500
C. $2,700
D. $2,900
E. $3,500


Can someone tell me where I went wrong? Why is it that when I use 30 days for part 1 below instead of 31 days, I get the right answer?

1) 31 days in August therefore the sum of the incremental increases is: (first day + last day) / 2 .....(100+(31*100))/2 = $49,600
2) Test answers... $2500*31=$77,500
3) $77,500+$49,600=$127,100


Thanks!!

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Re: A radio station is giving away a total of $124,000 in August. If they [#permalink]

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New post 15 Nov 2017, 11:49
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YYZ wrote:
Bunuel wrote:
A radio station is giving away a total of $124,000 in August. If they increase the amount given away each day by $100, how much should they give away the first day?

A. $2,300
B. $2,500
C. $2,700
D. $2,900
E. $3,500


Can someone tell me where I went wrong? Why is it that when I use 30 days for part 1 below instead of 31 days, I get the right answer?

1) 31 days in August therefore the sum of the incremental increases is: (first day + last day) / 2 .....(100+(31*100))/2 = $49,600
2) Test answers... $2500*31=$77,500
3) $77,500+$49,600=$127,100


Thanks!!


Hi YYZ

the highlighted part is incorrect. 31 days are in totality. the 31st day i.e the last day will come after 30 days from the first day. hence you need to use 30 here

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Re: A radio station is giving away a total of $124,000 in August. If they   [#permalink] 15 Nov 2017, 11:49
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