Question -
Answer -
Given:┬аAmount tobe counted is тВ╣ 10710
We need tofind┬аtime taken by man to count the entire amount.
He counts at the rateof тВ╣ 180 per minute for half an hour or 30 minutes.
So, Amount to becounted in an hour = 180 ├Ч 30 = тВ╣ 5400
Amount left = 10710 тАУ5400 = тВ╣ 5310
Sn┬а=5310
After an hour, rate ofcounting is decreasing at тВ╣ 3 per minute. This rate will form an A.P.
A.P. is 177, 174,171,тАжтАж
Here a = 177 and d =174 тАУ 177 = тАУ3
By using the formula,
Sn┬а=n/2 [2a + (n тАУ 1)d]
5310 = n/2 [2(177) +(n-1) (-3)]
5310 = n/2 [354 -3n +3]
5310 ├Ч 2 = n [357 тАУ3n]
10620 = 357n тАУ 3n2
10620 = 3n(119 тАУ n)
10620/3 = n(119 тАУ n)
3540 = 119n тАУ n2
n2┬атАУ119n + 3540 = 0
n2┬атАУ59n тАУ 60n + 3540 = 0
n(n тАУ 59) тАУ 60(n тАУ 59)= 0
(n тАУ 59) (n тАУ 60) = 0
n = 59 or 60
We shall considervalue of n = 59. Since, at 60th┬аmin he will count тВ╣ 0
тИ┤ The total time takenby him to count the entire amount = 30 + 59 =┬а89 minutes.