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
= (180-31) + (180-32) + (180-33) + (180-34) + (180-35)+…+(180-3t)
= (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 - t2 = 3540
t2 - 119t +3540 = 0
t2 - 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.