The man counts 180 per minute for 1/2 hr. or 30 minutes. Therefore he counts till 5400. The amount left is 5310.
From the 31st minutes he counts 3 less every minute than he did in the previous. This is a arithmetic progression a+nd where a=177 and d=-3. Let assume he needs n minutes to count 5310. The amount counted in the nth minute is 177-3(n-1). So the sum of all the terms from 177 to 177-3(n-1) is 5310.
The sum of n terms of an AP is (T1+Tn)*(n/2)
Here it is [177+177-3(n-1)]*(n/2).
As this is equal to 5310 we get [177+177-3(n-1)]*(n/2)=5310. Eliminate 3 from both the sides, 2*1770=n*[59+59-n+1]
This gives the quadratic equation n^2-119n+3540=0
Or n=either 59 or 60
Substituting 59 in the expression 119n-n^2 we get 3540. Therefore 59 more minutes are required after the initial 30 minutes. This makes the total time equal to 59+30=89 minutes.
Let n be the time taken in minutes, after half an hour.
Then in half an hour he counts @180 /min an amount = 30*180 and left out amount = 10710 - 30*180 = 5310.
Now he counts 180-3 i1st min = 177.
In the 2nd min he counts 180-2*3 = 174 .
In the nth minute he cnts = 180 -3n .
Therefore the sum of the money in n minutes = 177+176+173 +.... 180-3n which should be equal to 5310
LHS = 3 (59+58+57+....60-n) , n terms = 5310.
LHS is an AP with common difference -1. So sum = (1st term +last term)/2
3 (59+60-n)n/2 = 5310
(110-n)n = (5310*2/3 = 3540
110n-n^2 = 3540
n^2 -110n +3540 = 0
(n-60)(n-59) = 0
n = 60 Or n = 59.
So it take 59 minutes after half an hour when the sum becomes 5310 . And in the 60th minutes after 1/2 hour he does mot have any thing to count and his speed of counting reduces to 0 per minute.
So the total time require = (1/2) hour+59 min = 89 min.