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.
Best Answer
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.