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

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 \(A\).

Bunuel or chetan2u

Trying to understand if my reasoning (Especially for statement 2) is correct. Would appreciate if you can provide come feedback.

Statement 1:
Let's say N is the original number of people and P is the original price they shared.

Total=NP

Now we have 4 people who left us. So remaining number of people is (N-4). New price shared among people is (P+12)

Total = (N-4)(P+12)

Total in both cases the total paid remains the same so
-> NP=(N-4)(P+12)
-> NP= NP+12N-4P-48
-> 48=12N-4P
-> 48=4(3N-P)
-> 12=3N-P
-> (12+P)/3=N
-> 4 + P/3 = N

N has to be an integer because it's the number of people. For N to be an integer, P has to be an integer and a multiple of 3. This means that both N and P are integers so our total amount paid NP must also equals an integer.

Statement 2: We only know that the number of people is 10. Total amount paid is 10(12+x) where x was the original amount everyone would've shared if no one left.
-> Total Amount = 120 + 10*x

The result can be an integer or it might not (if x was a decimal i.e. 1.75?)

So B is Insufficient

\(\frac{t}{n}+12=\frac{t}{n-x}…12=\frac{t}{n-x}-\frac{t}{n}…12=\frac{tn-tn+tx}{n(n-x)}…12=\frac{tx}{n(n-x)}…t=\frac{12n(n-x)}{x}\)

(1) Four people left without paying: sufic.
\(t=\frac{12n(n-x)}{x}…t=\frac{12n(n-4)}{4}…t=3n(n-4)=integer\)

(2) Ten people in total shared the meal: insufic.
\(t=\frac{12(10)(10-x)}{x}…t=\frac{120(10-x)}{x}…t=\frac{(1200-120x)}{x}…\)
if x=1, t=integer… if x=7, t=non-integer

Answer (A)

The key to this question is understanding clearly, that question just wants us to know if C is an integer and nothing more.

From official solution above we know

\(C=\frac{12P(P+L)}{L}\) (C=Cost, P=# of members who paid , L= # of members who left)

So all we need to know is whether 12/L is an integer? Do we have a sure shot Yes/No for this question

St 1) 12/4 --> Yes, definiltely ..>SUFF
St 2) P+L=10, now L may be 1,2,3,4,0...etc...--> C may or may not be an integer--> INSUFFICIENT

Hence A is the answer here