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

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

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

Solution:

Given that the amount to be counted is ₹ 10710
We have to find the time taken by man to count the entire amount.
He counts the amount at the rate of ₹ 180 per minute for 30 minutes.
Amount to be counted in an hour $=180 \times 30=$ ₹ 5400
Amount left $=10710-5400=$ ₹ 5310
$S_{n}=5310$
Rate of counting is decreases at ₹ 3 per minute after an hour. This rate will form an A.P.
A.P. is $177,174,171, \ldots \ldots$
Here $a=177$ and $d=174-177=-3$
Using the formula,
$\begin{array}{l} S_{n}=n / 2[2 a+(n-1) d] \\ 5310=n / 2[2(177)+(n-1)(-3)] \\ 5310=n / 2[354-3 n+3] \\ 5310 \times 2=n[357-3 n] \\ 10620=357 n-3 n^{2} \end{array}$
$\begin{array}{l} 10620=3 n(119-n) \\ 10620 / 3=n(119-n) \\ 3540=119 n-n^{2} \\ n^{2}-119 n+3540=0 \\ n^{2}-59 n-60 n+3540=0 \\ n(n-59)-60(n-59)=0 \\ (n-59)(n-60)=0 \\ n=59 \text { or } 60 \end{array}$
Consider the value of $n=59 .$ As, at $60^{\text {th }}$ min he will count ₹ 0
As a result, the total time taken by him to count the entire amount $=30+59=89$ minutes.