Problem Solving

Q: 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

So,

$207.00*85%=x ; x=price of lunch EXCLUDING gratuity
x=$175.95
avg. price per person = $175.95/15 = $11.73 (A)

Where am I going wrong? The correct answer is B

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 -- yields 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.

Algebraic approach:

Let x = the cost per person WITHOUT gratuity.
Cost for 15 people without gratuity = 15x.
Cost for 15 people with gratuity = 115% of the cost without gratuity = (115/100)(15x).
Since the total cost with gratuity is 207, we get:
(115/100)(15x) = 207
x = (207 * 100) / (115 * 15).

Here is one way to simplify:

Divide both 100 and 15 by 5:
(207 * 100) / (115 * 15)
(207 * 20) / (115 * 3).

Divide 3 into 207; divide both 20 and 115 by 5:
(207 * 20) / (115 * 3)
(69 * 4) / 23.

Divide 23 into 69:
(69 * 4) / 23
3 * 4
12.

Thank you!

I guess I am curious why I can't start with the total price of lunch with gratuity.

So, $207. Since this includes the 15% gratuity, then to get the price without gratuity, multiply the total price by 85% (1-gratutity (15%).

why does this not give the same answer?

chgordon wrote:Thank you!

I guess I am curious why I can't start with the total price of lunch with gratuity.

So, $207. Since this includes the 15% gratuity, then to get the price without gratuity, multiply the total price by 85% (1-gratutity (15%).

why does this not give the same answer?

The step in red is the equivalent of subtracting 15% OF THE FINAL PRICE.
But the gratuity is not equal to 15% of the final price; the gratuity is equal to 15% OF THE BASE PRICE.
The two values are not the same.

When do I know, whether to use price/1.15 or when to multiply price*0.85 ?