Having trouble understanding this % problem (OFG13 #135)

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DFritschy: Junior | Next Rank: 30 Posts; Posts: 28; Joined: Thu Jun 21, 2012 4:25 pm

by DFritschy » Sun Jul 22, 2012 6:04 am

OFG13, Problem solving #135

Q: The price of lunch for 15 people was $207.00, including a 15 % gratuity for service. What was the average price per person, EXCLUDING the gratuity?

A: Let c be the total price of lunch for everyone excluding the gratuity. Since $207.00 is given as the total price including the 15% gratuity, the total price for the group lunch excluding the gratuity can be expressed as $207 = 1.15c, or $207/1.15 = $180 = c. c / $180.00 = the price each paid on average.

MY QUESTION: I've been spending too much time on %'s I think....can someone please explain how you get $207/1.15 = $180 out of the following two equations that I was considering solving this problem with? Thank you.

Equation 1: % increase = (amount of change) / (original amount)
Equation 2: % of = portion / whole

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Source: Beat The GMAT — Problem Solving | Sun Jul 22, 2012 6:04 am

kartikshah: Master | Next Rank: 500 Posts; Posts: 156; Joined: Sun Jul 15, 2012 12:03 pmFollowed by: 1 member; Location: New York, USA; Thanked: 34 times

by kartikshah » Sun Jul 22, 2012 6:27 am

I did this problem in two steps and in under a minute I guess. This does not answer your question directly but I'd still like to share my working:

First calculate average price per person including gratuity:
$207 for 15 people
So, 207/15 for 1 person = $(69/5)

100% = $ (69/5)
85% = $ ?
Percentages cancel out:

= (85*69)/(100*5)
Reduce 85 with 5; Multiply 17*69 and a 100 in the denominator should make division very easy= $11.73 !
But on the actual exam, I would start approximating after finding (69/5) ~ 13
So my price per person excluding gratuity HAS TO BE LESSER.
10% of 13 = 1.3 and 15% is little over this; So my answer should be $10-12

Only if options are TOO CLOSE, I would calculate as indicated above.

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eagleeye: Legendary Member; Posts: 520; Joined: Sat Apr 28, 2012 9:12 pmFollowed by: 49 members; Thanked: 339 times; GMAT Score:770

by eagleeye » Sun Jul 22, 2012 6:39 am

DFritschy wrote:OFG13, Problem solving #135

Q: The price of lunch for 15 people was $207.00, including a 15 % gratuity for service. What was the average price per person, EXCLUDING the gratuity?

A: Let c be the total price of lunch for everyone excluding the gratuity. Since $207.00 is given as the total price including the 15% gratuity, the total price for the group lunch excluding the gratuity can be expressed as $207 = 1.15c, or $207/1.15 = $180 = c. c / $180.00 = the price each paid on average.

MY QUESTION: I've been spending too much time on %'s I think....can someone please explain how you get $207/1.15 = $180 out of the following two equations that I was considering solving this problem with? Thank you.

Equation 1: % increase = (amount of change) / (original amount)
Equation 2: % of = portion / whole

For the way you were going to do it:

eqn 1: 15% = (207-c)/c
=> 15/100 = (207-c)/c
=> 3/20 = (207-c)/c
=> 3c = 207*20 - 20c
=> 23c = 207*20
=> c = 207/23*20 = 9*20 = 180.

Eqn 2: (1/15) = portion paid by each / 180
=> portion paid by each = 180/15 = 12.

I would, however, suggest you set up the question like this:
Let x be the price each pays, without gratuity.
Then with gratuity each pays = 1.15x
Total 15 people pay = 15*1.15x = 207 (given)
=> x = 207/(15*(1.15)) = 207*20/(23*15) = 23*9*20/(15*23) = 3*4 = 12.

Cheers!

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Mon Jul 23, 2012 8:59 am

The price of lunch for 15 people was $207.00,
including a 15 percent gratuity for service. What was
the average price per person, EXCLUDING the
gratuity?

(A) $11.73
(B) $12.00
(C) $13.80
(D) $14.00
(E) $15.87

We can plug in the answers, which represent the cost per person.
Given that the total cost is an INTEGER, the correct answer is almost certainly an integer itself.
Answer choice D is too great, since 15 people paying $14 each = 15*14 = 210 -- beyond the total cost of 207.

Answer choice B: $12 per person without gratuity
Total price for 15 people = 15*12 = 180.
Price + gratuity = 180 + .15(180) = 180+27 = 207.
Success!

The correct answer is B.

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kartikshah: Master | Next Rank: 500 Posts; Posts: 156; Joined: Sun Jul 15, 2012 12:03 pmFollowed by: 1 member; Location: New York, USA; Thanked: 34 times

by kartikshah » Mon Jul 23, 2012 9:10 am

Hi Mitch!

I worked backwards 100% --> 85%
I got the answer as 11.73$
Is it because I calculated 15% of 100 instead of 85?
Thanks for clarifying.

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Tue Jul 24, 2012 3:48 am

kartikshah wrote:Hi Mitch!

I worked backwards 100% --> 85%
I got the answer as 11.73$
Is it because I calculated 15% of 100 instead of 85?
Thanks for clarifying.

The gratuity is 15% of the meal cost WITHOUT gratuity, not 15% of the TOTAL PRICE that was paid.
Thus, we can't simply subtract 15% from the total price.
We could work backwards as follows:
Since the total price is 115/100 of the meal cost, the meal cost is 100/115 of the total price:
c = (100/115)207 = (20/23)207 = 180.
Since there are 15 people, the per person cost = 180/15 = 12.

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armand_h: Junior | Next Rank: 30 Posts; Posts: 15; Joined: Wed Jul 25, 2012 9:02 am; Thanked: 1 times

by armand_h » Thu Jul 26, 2012 6:48 am

let's set P the price per person excluding gratuity
then
15*P*115/100=207
1.5*11.5*P=207

which means P should be a multiple of 4
the only solution that is multiple of 4 is 12
Correct answer B

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theCEO: Master | Next Rank: 500 Posts; Posts: 363; Joined: Sun Oct 17, 2010 3:24 pmFollowed by: 3 members; Thanked: 115 times

by theCEO » Thu Jul 26, 2012 4:01 pm

armand_h wrote:let's set P the price per person excluding gratuity
then
15*P*115/100=207
1.5*11.5*P=207

which means P should be a multiple of 4
the only solution that is multiple of 4 is 12
Correct answer B

Where do you see the choices?

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armand_h: Junior | Next Rank: 30 Posts; Posts: 15; Joined: Wed Jul 25, 2012 9:02 am; Thanked: 1 times

by armand_h » Mon Jul 30, 2012 12:03 am

theCEO wrote:
armand_h wrote:let's set P the price per person excluding gratuity
then
15*P*115/100=207
1.5*11.5*P=207

which means P should be a multiple of 4
the only solution that is multiple of 4 is 12
Correct answer B
Where do you see the choices?

The choices were posted by GMATGuruNY, post Mon Jul 23, 2012 5:59 pm

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