Explanation of double answer

Question

A man is employed to count the total sum of $10710$ . He counts at the rate of $180$ per minute for half an hour. After that he counts at the rate of $3$ less every minute than the preceding minute. Find the time it takes to count the entire amount.

How I solved this:

let the time taken after 30min be $y$ and the total time be $30+y$

the rate for first $30$ min was $180$ i.e. in $1$ min he counts $180$ so in $30$ minutes he will count $5400$. now taking the remaining amount $10710-5400=5310$.. In the $31$st minute his rate decreases by $3$ so in this minute his rate is $177$. During the $32$nd minute his rate is $177-3 = 174$. During the $33$rd minute his rate is $177+(2)(-3)= 177-6=171$ so a time will come when the amount will become $0$ and this also forms an $AP$

so applying this thing I got:
$5310-S_n=0$
$5310=n/2(2(177)+(n-1)(-3))=> 10620=n(354+3-3n)$
$10620= 357n-3n^2=> 10620-357n+3n^2=0=>n^2+119n+3540=0$

after solving this quadratic equation by quadratic formula I got the value of $n=59,60$ so the total time taken is $89', 36''$

I have the two answer and only one is possible: either $89$ or $90$. $89$ is the correct answer. I explained this double answer to myself thinking of the fact that $S_{90}=S_{89}$. You see that for $n=59$ the general term is $3$ and for $n=60$ it is $0$, so the partial sums coincide. The question asked the total time taken to count the entire amount. Shouldn't it be $90$ minutes? At the $89$th minute $3$ still need to be counted.

Answer 1

Using the formula for sum of n terms of an Arithmetic Progression, I get (Your equation is wrong, probably due to typos; refer to mine, and figure out where you made mistakes)... $$ 5310 = \frac{n(2(177)-3(n-1))}{2}$$ [Double answer is due to a quadratic equation arising.]

Solving for n yields n = 59 or n = 60 => 89 or 90 minutes.

Now realise, that he counts 3 dollars IN THE $89^{th}$ minute. He counts zero dollars in the $90^{th}$ minute, because by the end of the $89^{th}$ minute, he has finished counting. (Because the $89^{th}$ term of the sequence is 3, a.k.a, when rephrasing the answer to the question, in the $89^{th}$ minute, he counts the last 3 dollars).

Therefore it is, in fact, 89 minutes.

Answer 2

Your logic is correct he's done counting at the end of the the 89th minute.

From the question: "... After that he counts at the rate of $3 less every minute than the preceding minute."

So on the 31st minute he counts 177 dollars (180-3*(31-30)), on the 90th minute he counts $0 dollars (180-3*(90-30))=0. Lucky for you the total sum has been counted by the end of the 89th minute because when the clock strikes 90 minutes that guy will not work ;)

I guess you could say it takes 90 minutes to count it, because he spends the full time up to the end of the 89th minute but does not spend any time after the clock reaches 90 minutes.