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Question -

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



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.

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