Question

A man is employed to count Rs 10710. He counts at the rate 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 1

In the given problem, the total amount = Rs 10710.

For the first half and hour (30 minutes) he counts at a rate of Rs 180 per minute. So,

The amount counted in 30 minutes = (180) (30) = 5400

So, amount left after half an hour  = 10710 - 5400 = 5310

After 30 minutes he counts at a rate of Rs 3 less every minute. So,

At 31^st minute the rate of counting per minute = 177.

At 32^nd minute the rate of counting per minute = 174.

So, the rate of counting per minute for each minute will form an A.P. with the first term as 177 and common difference as −3.

So, the total time taken to count the amount left after half an hour can be calculated by using the formula for the sum of n terms of an A.P,

`S_n = n/2 [ 2a + (n-1)d]`

We get,

          `5310 = n/2 [ 2 (177) + (n-1) (-3) ] `           ..............(1)

      5310(2) = n [354-3n + 3] 

         10620 =  n (357 - 3n) 

          10620 = 357n - 3n²

So, we get the following quadratic equation,

3n² - 357n + 10620 = 0

   n² - 119n + 3540 = 0 

Solving the equation by splitting the middle term, we get,

n² - 60n - 59n + 3540 = 0 

n ( n - 60 ) - 59 ( n - 60) =0

So,

n - 59 = 0

       n = 59

Or

n - 60 = 0 

       n = 60

\[Now let n = 60 then finding the last term, we get\]
\[ S_n = \frac{n}{2}\left[ a + l \right]\]
\[5310 = \frac{60}{2}\left[ 177 + l \right]\]
\[177 = 177 + l\]
\[l = 0\]
\[\text{ It means the work will be finesh in 59th minute only because 60th term is 0 }  . \]
\[\text{ So, we will take n = 59 } \]

Therefore, the total time required for counting the entire amount  = 30 + 59 minutes = 89 minutes 

So, the total time required for counting the entire amount is 89 minutes .