A man is employed to count Rs 10710. He counts at the rate of Rs 180 per minute for half an hour.

Solution:

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×">××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.

$5310=\frac{n}{2}[2 \times 177+(n-1) \times-3]$

$\Rightarrow 10620=354 n-3 n^{2}+3 n$

$\Rightarrow 3 n^{2}-357 n+10620=0$

$\Rightarrow n^{2}-119 n+3540=0$

$\Rightarrow n^{2}-59 n-60 n+3540=0$

$\Rightarrow n(n-59)-60(n-59)=0$

$\Rightarrow(n-59)(n-60)=0$

$\therefore n=59$ or 60

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.