Question

A man is employed to count Rs 10710. He counts at the rate of Rs 180 per minute for half an hour. After this he counts at the rate of Rs 3 less every minute than the preceding minute. Find the time taken by him to count the entire amount.

Answer 1

It is given that the man counts Rs 180 per minute for half an hour.
∴ Sum of money the man counts in 30 minutes = Rs 180$\times$30 = Rs 5400

Total money counted by the man = Rs 10710
∴ Money left for counting after 30 minutes = Rs (10710 − 5400) = Rs 5310

It is given that after 30 minutes, he counts at the rate of Rs 3 less every minute than the preceding minute.

Therefore, it would be an A.P. where a = 177 and d = −3.
Let the time taken to count Rs 5310 be n minutes.
$$\begin{gathered} 5310=\frac{n}{2}\left[2\times 177+\left(n-1\right)\times -3\right] \\ \Rightarrow 10620=354n-3n^2+3n \\ \Rightarrow 3n^2-357n+10620=0 \\ \Rightarrow n^2-119n+3540=0 \\ \Rightarrow n^2-59n-60n+3540=0 \\ \Rightarrow n\left(n-59\right)-60\left(n-59\right)=0 \\ \Rightarrow \left(n-59\right)\left(n-60\right)=0 \\ \therefore n=59\mathrm{or}60 \end{gathered}$$

Thus, the time taken to count Rs 5310 would be 59 minutes or 60 minutes.

Hence, the total time taken to count Rs 10710 would be (30 + 59) minutes or (30 + 60) minutes, i.e. 89 minutes or 90 minutes, respectively.