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Percentages - avg price per person

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by kobel51 » Fri Feb 14, 2014 12:06 pm
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
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by Patrick_GMATFix » Fri Feb 14, 2014 12:20 pm
At least 3 approaches

I: get the total price+gratuity per person as 207/15. This is equal to price + 15%. so 207/15 = 1.15p. Solve for p.

II: 207 is the sum-of-prices+15%. to get the sum-of-prices, set 207 = 1.15*sum. Once you have the sum, divide by 15 to get the price per person, excluding gratuity.

III: Test the answers, starting with B or D (because they're easiest). Multiply the answer by 15 to get sum of prices, then add 15% to get sum of prices plus gratuity. If you get more than 207, pick a smaller answer. The right answer will be exactly 207.

For guidelines on how to do the arithmetic quickly, watch my full solution.The illustrated solution below is taken from the GMATFix App.

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by [email protected] » Fri Feb 14, 2014 12:44 pm
Hi kobel51,

Since the question is asking us to solve for one thing (the average price per person) and the answers are numbers, we have the perfect opportunity to TEST THE ANSWERS. One of those answers, when "plugged in" to the prompt, will result in $207

If you started by testing answer D, you'd have $14(15 people) + 15% tip. $14(15) = $210 which is TOO HIGH (and we haven't even added in the tip yet. So, the answer has to be smaller than $14.

If you try answer B, you'd have $12(15 people) + 15% tip. $12(15) = $180 + 15%($180) = $180 + $27 = $207. This is a PERFECT MATCH.

Final Answer: B

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by tanvis1120 » Sat Feb 15, 2014 2:03 pm
Firstly, I'll have to deduct the gratuity from the entire bill.
Therefore, the bill comes out to be $(207-(15% of 207)), which can be approximated to $185.

Calculation:
15% of 207 Approx, 15% of 200 = 30.
Total Bill (Excluding gratuity)= 207-30 = 187 Approx 185
Average Bill per person = 185/15 which is approximately 12 (Very close, rather).
Hence The answer is B.
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by Patrick_GMATFix » Sat Feb 15, 2014 4:14 pm
Hey Tanvis1120,
You did a great job with approximations, but be careful:
the bill comes out to be $(207-(15% of 207))
If $207 is base price + 15% gratuity, then removing 15% from $207 will not get you back to the base price. Why? 15% gratuity is 15% of the base price, so when you subtract 15% of $207, you are actually subtracting a different amount.

An easy example proves the point:
Suppose base + 10% is $100. If I subtract 10% from $100 I will end up with a $90. That is not the true base however, since $90 + 10% is $99, not $100.

Since your solution relied on approximations, that's not a big deal in this case but you should be aware of this distinction.

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by Abhishek009 » Sun Feb 16, 2014 11:14 am
tanvis1120 wrote:Firstly, I'll have to deduct the gratuity from the entire bill.
Therefore, the bill comes out to be $(207-(15% of 207)), which can be approximated to $185.

Calculation:
15% of 207 Approx, 15% of 200 = 30.
Total Bill (Excluding gratuity)= 207-30 = 187 Approx 185
Average Bill per person = 185/15 which is approximately 12 (Very close, rather).
Hence The answer is B.
Let the actual price for lunch be 100 , price after Gratuity becomes 115

When Price Including Gratuity is 115 , actual price (Ex gratuity) is 100

When Price Including Gratuity is 1 , actual price (Ex gratuity) is 100/115

When Price Including Gratuity is $207 , actual price (Ex gratuity) is (100/115 )*207 => $ 180


Now we know $ 207 is the price of 15 persons , so price of lunch per person is $ 180 / 15 => 12

Hence answer is (B)..
Abhishek
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by parveen110 » Sun Feb 16, 2014 1:09 pm
Patrick_GMATFix wrote:Hey Tanvis1120,
You did a great job with approximations, but be careful:
the bill comes out to be $(207-(15% of 207))
If $207 is base price + 15% gratuity, then removing 15% from $207 will not get you back to the base price. Why? 15% gratuity is 15% of the base price, so when you subtract 15% of $207, you are actually subtracting a different amount.

An easy example proves the point:
Suppose base + 10% is $100. If I subtract 10% from $100 I will end up with a $90. That is not the true base however, since $90 + 10% is $99, not $100.

Since your solution relied on approximations, that's not a big deal in this case but you should be aware of this distinction.

-Patrick
Hi Guys, there is one more faster approach if your calculations are good enough.

We can solve this problem with product-constancy principle of percentages.

If 207$ include 15%(i.e.3/20) gratuity, we can go backwards from 207$ to 207$ less 15%.

x ---(+3/20)---->207$
180 <--(-3/23)---- 207$ i.e. 207(1-20/23)=180$

Now, 180/15=12$.

Hope, i've communicated the math involved here properly.
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