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Twenty people had dinner together at a restaurant. The cost of the mea

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Twenty people had dinner together at a restaurant. The cost of the mea  [#permalink]

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New post 01 Apr 2018, 10:34
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Twenty people had dinner together at a restaurant. The cost of the meal plus an automatic gratuity of 18% for a large party was $531.00, before sales tax was added. What was the average price of the meal per person NOT INCLUDING the gratuity or sales tax?

(A) $21.77
(B) $22.50
(C) $26.55
(D) $30.50
(E) $36.55

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Twenty people had dinner together at a restaurant. The cost of the mea  [#permalink]

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New post 01 Apr 2018, 11:07
Bunuel wrote:
Twenty people had dinner together at a restaurant. The cost of the meal plus an automatic gratuity of 18% for a large party was $531.00, before sales tax was added. What was the average price of the meal per person NOT INCLUDING the gratuity or sales tax?

(A) $21.77
(B) $22.50
(C) $26.55
(D) $30.50
(E) $36.55


Given:
20 people,
Total cost + 18 % Gratuity = 531

Need to find per person cost excluding the gratuity...
Method 1: Normal Calculations

average total cost will be less than average of total Cost + 18% Gratuity
=> average of total cost + Gratuity = 531/20
=> \(total cost < 531/20\)
=> total cost < 26.55
eliminate option C, D and E

Gratuity is little less than 20%
so for ease of calculation lets take gratuity as 20%

=> Average Total cost * 1.20 (20% gratuity) = Average (total cost + Gratuity)
=> \(Average total cost = 26.55/1.20\)
Approx = 22
as we increased the denominator from 1.18 to 1.2 the answer should be greater than 22 hence Option B

Method 2: Use the options

Pick the mid value -> 26.55
if that is the average cost than overall cost = 26.55 * 20 = 513.00,
Now 18% gratuity will result in an amount that is more than 531 hence all options equal to or greater than 26.55 are disqualified

move on to the next number 22.50
22.50 * 20 = 450
increase by 18% = 450 *1.18 = 531
this is what we were looking for hence answer is B

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Twenty people had dinner together at a restaurant. The cost of the mea  [#permalink]

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New post 01 Apr 2018, 15:59
Bunuel wrote:
Twenty people had dinner together at a restaurant. The cost of the meal plus an automatic gratuity of 18% for a large party was $531.00, before sales tax was added. What was the average price of the meal per person NOT INCLUDING the gratuity or sales tax?

(A) $21.77
(B) $22.50
(C) $26.55
(D) $30.50
(E) $36.55

Answer choices
Easier, I think. Work from the answer choices, use fractions, and start with round numbers.

D) $30.50 each? Round to $30
Total before gratuity is already too great.
\(\frac{$30}{person}* 20(persons) = $600\) - REJECT

Per person cost must be lower. Answers A, B, and C remain. Choose the round number, B.

Either B is too low and the answer is C; B is too high and the answer is A; or B is the answer.

B) $22.50 each?
\($22.50 = 22\frac{1}{2}=\frac{$45}{2}\)

Total before gratuity: \(\frac{$45}{2}*20=$450\)

Total with gratuity? Find 18% of $450, add it to $450

\((\frac{18}{100}*$450)=(\frac{9}{50}*$450)=$81\)

\(($81 + $450) = $531\)
MATCH

Answer B

Algebra
Let \(x\) = total cost of dinner without gratuity

$531 includes a tip of 18% on original total, \(x\)

\($531 = 1.18x\)
\(x = \frac{$531}{1.18}\)

\(1.18=1\frac{18}{100}=1\frac{9}{50}=\frac{59}{50}\)

\(x =\frac{$531}{(\frac{59}{50})}=$531*\frac{50}{59}\)

\(x=($531*\frac{50}{59})=($9*50)=$450\)


Per person cost? \((\frac{$450}{20})=(\frac{$45}{2})=$22.50\)

Answer B
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Re: Twenty people had dinner together at a restaurant. The cost of the mea  [#permalink]

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New post 01 Apr 2018, 22:32
Bunuel wrote:
Twenty people had dinner together at a restaurant. The cost of the meal plus an automatic gratuity of 18% for a large party was $531.00, before sales tax was added. What was the average price of the meal per person NOT INCLUDING the gratuity or sales tax?

(A) $21.77
(B) $22.50
(C) $26.55
(D) $30.50
(E) $36.55


Cost + Gratuity @18% = 531

i.e. C+ (18/100)*C = 531

i.e. 1.18 C = 531

i.e. C = 531/1.18 = 450

The cost is shared by 20 people hence Cost per person = C/20 = 450/20 = 22.5

Answer: option B
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Re: Twenty people had dinner together at a restaurant. The cost of the mea &nbs [#permalink] 01 Apr 2018, 22:32
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