A man is employed to count Rs.10710. He counts at there of Rs.180 per

1.	A man is employed to count Rs.10710. He counts at there of Rs.180 per minute for half an hour. After this, he counts at the rate of Rs.3 less every minute than the preceding minute. Find the time taken by him to count the entire amount.
Answer» Total amount for counting = Rs. 10710 In 1 min he counts =Rs. 180 In half an hour (30 min) he count = Rs.180×30 =Rs.5400 Amount left count after half an hour, S_n= Rs.10710 – Rs. 5400 = 5310 Now, in 31^stmin he count 3 less than preceding minute = Rs.180 – Rs.3 = Rs.177 In 32^nd min he count 3 less than preceding minute = Rs. 177 –Rs. 3 =Rs. 174 Then, Arithmetic progression formed is 177, 174,… Here, difference between minutes = 174 – 177 = -3 We know, sum of n terms, S_n= (n/2) [2a + (n – 1) d] 5310 = (n/2) [2(177) + (n – 1) -3] 5310 × 2 = n [354 – 3n + 3] 10620 = 354n – 3n² + 3n 10620 = 357n – 3n² ⇒ 3n² – 357n + 10620 = 0 On taking 3 common from the complete equation, we get, ⇒ 3 (n² – 119n + 3540) = 0 ⇒ n² – 119n + 3540 = 0 Now, from factoring by splitting the middle term method, we get, ⇒ n² – 60n – 59n + 3540 = 0 ⇒ n (n – 60) – 59 (n -60) = 0 ⇒ (n – 59) (n – 60) = 0 Therefore, either n = 59 or n = 60 We will use 59 as these are minutes So The total time to calculate whole amount is 59 + 30 = 89 min = 1 hr 29 min

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

Answer»

Total amount for counting = Rs. 10710

In 1 min he counts =Rs. 180

In half an hour (30 min) he count = Rs.180×30 =Rs.5400

Amount left count after half an hour, S_n= Rs.10710 – Rs. 5400 = 5310

Now, in 31^stmin he count 3 less than preceding minute = Rs.180 – Rs.3 = Rs.177

In 32^nd min he count 3 less than preceding minute = Rs. 177 –Rs. 3 =Rs. 174

Then, Arithmetic progression formed is 177, 174,…

Here, difference between minutes = 174 – 177 = -3

We know, sum of n terms, S_n= (n/2) [2a + (n – 1) d]

5310 = (n/2) [2(177) + (n – 1) -3]

5310 × 2 = n [354 – 3n + 3]

10620 = 354n – 3n² + 3n

10620 = 357n – 3n²

⇒ 3n² – 357n + 10620 = 0

On taking 3 common from the complete equation, we get,

⇒ 3 (n² – 119n + 3540) = 0

⇒ n² – 119n + 3540 = 0

Now, from factoring by splitting the middle term method, we get,

⇒ n² – 60n – 59n + 3540 = 0

⇒ n (n – 60) – 59 (n -60) = 0

⇒ (n – 59) (n – 60) = 0

Therefore, either n = 59 or n = 60

We will use 59 as these are minutes So The total time to calculate whole amount is 59 + 30 = 89 min

= 1 hr 29 min