Problem Solving

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

OG Answer B

My Answer A.

15% of 207 is 31.05, and the total amount WITHOUT gratuity is 207-31.05 = 175.95
and the individual price per person is 175.95/15 = 11.73

Please explain what is wrong with the above method.

umasarath52 wrote: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.

15% of 207 is 31.05, and the total amount WITHOUT gratuity is 207-31.05 = 175.95
and the individual price per person is 175.95/15 = 11.73

Please explain what is wrong with the above method.

The step in red implies that the gratuity is equal to 15% OF THE FINAL PRICE ($207).
Not so.
The gratuity is equal to 15% OF THE BASE PRICE ($180, as shown in my solution above).

Also:
ALWAYS VERIFY THAT YOUR ANSWER CHOICE MAKES SENSE.
If the base price per person = $11.73, then the gratuity per person = 15% of 11.73 = $1.7595.
A gratuity that includes $.0095 is not viable.

In our case, the cost of the food is x. When we add a 15% tip, we've got x + (15% of x), or 1.15x.

You've done it as 15% of the TOTAL BILL, but that leads to a pricier meal!

Here's a friendlier example. Suppose I buy a phone for $100, then sell it for 50% more than I paid. That means I sold it for $150, right? Of course.

But if we KNOW that I paid $150 for the phone, and work from that total, we'd take 50% of $150 and get $75, implying that I paid $75, then added $75 to the price ... clearly not the right answer!