Vincen wrote: ↑

Sat Nov 27, 2021 4:38 am

A number of people shared a meal, intending to divide the cost evenly among themselves. However, several of the diners left without paying. When the cost was divided evenly among the remaining diners, each remaining person paid \(\$12\) more than he or she would have if all diners had contributed equally. Was the total cost of the meal, in dollars, an integer?

(1) Four people left without paying.

(2) Ten people in total shared the meal.

Answer: A

Source: Veritas Prep

Target question: Was the total cost of the meal, in dollars, an integer?
This is a great candidate for rephrasing the target question

Given: A number of people shared a meal, intending to divide the cost evenly among themselves. However, several of the diners left without paying. When the cost was divided evenly among the remaining diners, each remaining person paid $12 more than he or she would have if all diners had contributed equally.

Let c = TOTAL cost of the meal
Let n = number of people who SHARED the meal
Let d = the number of deadbeats who left before paying
So, n - d = number of people who PAID for the meal

Cost per person with all n people = c/n
Cost per person with n-d people = c/(n - d)

Word equation: (cost per person with original diners) + 12 = cost per person with reduced number of diners
Equation: c/n + 12 = c/(n - d)
Eliminate fractions by multiplying both sides by (n)(n - d) to get: c(n - d) + 12(n)(n - d) = cn
Expand: cn - cd + 12n² - 12nd = cn
Subtract cn from both sides: -cd + 12n² - 12nd = 0
Add cd to both sides: 12n² - 12nd = cd
Divide both sides by d to get: 12n²/d - 12n = c
Factor: n(12n/d - 12) = c

Since n(12n/d - 12) equals the total cost of the meal, we can REPHRASE the target question....
REPHRASED target question: Is n(12n/d - 12) an integer?

Statement 1: Four people left without paying
In other words, statement 1 tells us that d = 4
Plug d = 4 into the REPHRASED target question to get: Is n(12n/4 - 12) an integer?
n(12n/4 - 12) = n(3n - 12), and n(3n - 12) is definitely an integer.
How do we know this?
Well, we know that n is an integer.
So, 3n is an integer, which means 3n - 12 is an integer, which means n(3n - 12) is an integer.
Since we can answer the REPHRASED target question with certainty, statement 1 is SUFFICIENT

Statement 2: Ten people in total shared the meal
In other words, statement 2 tells us that n = 10
Plug n = 10 into the REPHRASED target question to get: Is (10)(120/d - 12) an integer?
There are several values of d that yield conflicting answers to the target question. Here are two:
Case a: if d = 2, then (10)(120/d - 12) = (10)(120/2 - 12) = 480, and 480 IS an integer
Case b: if d = 7, then (10)(120/d - 12) = (10)(120/7 - 12), and this does NOT evaluate to be an integer
Since we cannot answer the REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A