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Algebra

This topic has 7 expert replies and 0 member replies
marat_isr Newbie | Next Rank: 10 Posts Default Avatar
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Posted:
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Algebra

Post Tue Feb 28, 2017 8:38 am
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) $11.73
b) $12.00
c) $13.80
d) $14.00
E) $15.87

Why is it wrong to get the price before the gratuity by simply multiplying 207*0.85 ?

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Post Wed Mar 01, 2017 3:24 pm
marat_isr wrote:
Why is it wrong to get the price before the gratuity by simply multiplying 207*0.85 ?
Because percentages aren't quite the same going up as going down: you're working with different bases.

For example, suppose I have a $100 stock. If it goes up 20% today, it's $120. If it then goes down 20% tomorrow, it's $96. So $100 + 20% - 20% doesn't get me back to $100!

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Post Thu Mar 02, 2017 5:18 pm
marat_isr wrote:
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) $11.73
b) $12.00
c) $13.80
d) $14.00
E) $15.87
An efficient way to solve this problem is to translate what is provided in the question into a mathematical equation. We know that the total bill for 15 people was 207 dollars, including the 15 percent tip. We use the value 1.15 in the equation below to account for the meal itself plus the gratuity, and x as the individual cost per person for the meal before the gratuity. Thus, we can say:

(1.15)(15)(x) = 207

(115)(15)(x) = 20700

Dividing both sides by 15, we have:

115x = 1380

x = 1380/115

x = 12

Answer: B

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Post Tue Feb 28, 2017 9:09 am
marat_isr wrote:
Why is it wrong to get the price before the gratuity by simply multiplying 207*0.85 ?
Your reasoning is that, since $207 is 15% GREATER than the pre-tip total, then the pre-tip total must be 15% LESS than $207 (i.e., pre-tip total = 85% of $207)

This is a common misconception that goes something like this: If A is k percent greater than B, then B must be k percent less than A.

The truth of the matter is as follows: If A is k percent greater than B, then B is NOT k percent less than A.

I wrote an article about this misconception: http://www.gmatprepnow.com/articles/common-misconceptions-involving-percents-%E2%80%93-part-ii

Cheers,
Brent

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Post Tue Feb 28, 2017 9:11 am
Quote:
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
A fast way to answer this question is by plugging in the answer choices, which I'll leave to you to try.

Here's an algebraic approach:

Let x = the total BEFORE the gratuity.

A 15% gratuity = 0.15x
So, the TOTAL cost = x + 0.15x
In other words, the TOTAL cost = 1.15x
So, we can write 1.15x = 207
So, x = 207/1.15

NOTE: The GMAT often rewards people for using fractions rather than decimals. The reason for this is that most people are very uncomfortable working with fractions.

So......
207/1.15 = 207/(1 3/20) [I converted the decimal to a fraction]
= 207/(23/20)
= 207 x (20/23)

IMPORTANT: At this point, the calculation looks very hard. But first notice that (23)(10) = 230. In other words, 230 = the sum of ten 23's
Also notice that 207 is 23 less than 230
So, 207 = the sum of nine 23's
In other words 207 = (23)(9)
So, we have . . .
= (23)(9) x (20/23)
= 9 x 20 [the 23's cancel out]
= 180

Yes, these calculations look cumbersome, but that's only because I added a lot of extra information. In reality, students with strong fraction skills can perform most (if not all) of these calculations in their heads.

Since the total PRE-GRATUITY price = $180, the price PER PERSON = 180/15 = 12 = B

Cheers,
Brent

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Post Tue Feb 28, 2017 9:15 am
marat_isr wrote:
Why is it wrong to get the price before the gratuity by simply multiplying 207*0.85 ?
X is 15% greater than Y does NOT imply that Y is 15% less than X or that Y is 85% of X.

In the problem above, let P = the final price with gratuity and C = the total cost for 15 people without gratuity.

Since the final price is 15% greater than the cost without gratuity, we get:
P = (115/100)C
P = (23/20)C
C = (20/23)P.

Thus, the cost without gratuity is equal to 20/23 -- not 85% -- of the final price with gratuity.

Since P = 207, we get:
C = (20/23)(207) = 180.
Since there are 15 people, the cost per person without gratuity = 180/15 = 12.

Alternate approach:

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 -- a value 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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Post Tue Feb 28, 2017 11:49 am
Hi marat_isr,

In these types of situations, we have to think about the "timeline" of the math....

Here, we had a bill, THEN the 15% gratuity was added, then we had a total:

X + .15X = 207

1.15X = 207

This is the equation that will get us the value of X...

X = 207/1.15

At NO POINT was the idea of 85% mentioned, so that should not be in your original calculation.

GMAT assassins aren't born, they're made,
Rich

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Post Tue Feb 28, 2017 9:24 pm
marat_isr wrote:
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) $11.73
b) $12.00
c) $13.80
d) $14.00
E) $15.87

Why is it wrong to get the price before the gratuity by simply multiplying 207*0.85 ?
Hi marat_isr,

Let us take a simple example.

Say, you dined in a restaurant and the food bill was $100. A gratuity of 15% was charged on it, making the final bill equals to $100 + 100*15% = $115.

Quote:
Now you wish to know only the food bill from the only two information: 1. Bill incl. gratuity = $115 and 2. Percent of gratuity = 15%.
If you do the following:

Gratuity = $115*15% = 115*0.15 = $17.25.

As per this, only the food bill = $115 - $17.25 = $97.75, which is not equal to the correct value $100.

you should do the following.

Only food bill = (Bill incl gratuity) / (100% + gratuity %) = 115 / (100% + 15%) = 115 / 115% = 115*(100/115) = 100.

Hope this makes sense.

Relevant book: Manhattan Review GMAT Math Essentials Guide

-Jay
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