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A number of people shared a meal, intending to divide the cost evenly

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A number of people shared a meal, intending to divide the cost evenly [#permalink]

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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.
Spoiler: OA

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Re: A number of people shared a meal, intending to divide the cost evenly [#permalink]

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lmuenzel wrote:
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.


Hi,

A GOOD Q..

Info from Q
1) Rs 12 extra was paid by remaining people.
so the Q asks us if \(\frac{12*r}{l}\) is integer.
It will be possible ONLY if l is factor 12*r....
r is remaining people and I is people who left w/o paying

Let's see the statements..
1) l is 4..
So \(\frac{12*r}{l}=\frac{12*r}{4}=3*r\)
Now r is integer so 3*r will be integer. This is what each pays.
When this is multiplied by TOTAL number of persons, it will still be integer.
Suff

2) r is 10..
12*10/l will depend on l whether it is integer or not..
If l is a factor of 120 such as 10,12,2,3,4,6,30,60, it will be integer.
If it is 7,9,50etc it will not be an integer.
Insufficient

A
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Re: A number of people shared a meal, intending to divide the cost evenly [#permalink]

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lmuenzel wrote:
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.


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

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A number of people shared a meal, intending to divide the cost evenly [#permalink]

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New post 07 Feb 2017, 19:24
lmuenzel wrote:
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.


Official solution from Veritas Prep.

Following our normal approach to word problems, we want to label our unknowns and then translate the statements in the problem to algebra. What makes this problem feel more difficult is simply the sheer number of unknowns. But don’t panic – it’s fine to have statements containing mostly unknowns, and the algebra will always work itself out in the end.

Let’s call the total cost of the meal \(C\). The number of diners who ended up paying will be \(P\), and then number who left without paying will be \(L\). The total number of diners is therefore \(P+L\). (We chose the variables in this way to match the information provided in the two additional premises, but we’re better off not actually plugging in those premises until much later.)

The price that each diner “should have paid” for the meal is \(\frac{C}{(P+L)}\). The price that each remaining diner actually paid, however, is \(\frac{C}{P}\). The difference can be expressed as

\(\frac{C}{P}−\frac{C}{(P+L)}=12\)

Taking the common denominator gives

\(\frac{C*(P+L)}{P*(P+L)}−\frac{CP}{P*(P+L)}=12\)

\(\frac{(CP+CL–CP)}{P(P+L)}=12\)

\(CL=12P(P+L)\)

Since the question asks whether the total cost was an integer, we’ll solve for that:

\(C=\frac{12P(P+L)}{L}\)

At this point, we can see that whether the total cost \(C\) is an integer depends most of all on \(L\) – whether the number of people who left divides evenly into \(12\), \(P\), and/or \(P+L\). Since statement 1 answers this question in the affirmative (4 divides evenly into \(12\), making \(C\) an integer), this statement is sufficient alone. Since statement \(2\) provides no information of relevance, that statement is insufficient. The answer is \(A\).
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Re: A number of people shared a meal, intending to divide the cost evenly [#permalink]

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Re: A number of people shared a meal, intending to divide the cost evenly   [#permalink] 13 May 2018, 06:58
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