A man is employed to count rs.10710. he counts at the rate of rs.180per minute for half an hour. after this, he counts at the rate of rs.3 less every minute than the preceding minute. find the total time taken by him to count the entire amount.

The man has to count Rs 10710

Initial Counting rate = Rs 180/min

Initial Time = 30 min

? Initial counting = 180�30 = Rs 5400

Rs left to be counted = 10710-5400 = Rs 5310

Let us assume time t=0 at the end of first thirty minutes.

Now the man is counting Rs 3 less every minute

at any time t, counting rate of the man will be = 180-3t Rs/min

So, at any time t, the money man would have counted will be

= (180-3*1) + (180-3*2) + (180-3*3) + (180-3*4) + (180-3*5)+…+(180-3*t)

= (180+180+180+…t times) - 3(1+2+3+4+…t)

= 180t - 3t(1+t)/2

This should be equal to the remaining money

Hence,

180t - 3t(1+t)/2 = 5310

360t - 3t(1+t) = 10620

120t - t - t^{2} = 3540

t^{2} - 119t +3540 = 0

t^{2} - 60t - 59t +3540 = 0

t(t - 60) - 59 (t-60) = 0

(t-59) (t-60) = 0

Hence t = 59 or 60

but for t = 60, his counting rate will become 180 - 3*60 = 0 , so he can not count for t = 60

Hence t = 59 min is the right value.

Now total time = initial time + 59 min = 59 + 30 = 89 min

Time taken to count entire amount is 89 minutes.