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प्रश्न
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.
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उत्तर
It is given that the man counts Rs 180 per minute for half an hour.
∴ Sum of money the man counts in 30 minutes = Rs 180
\[\times\] 30 = Rs 5400
Total money counted by the man = Rs 10710
∴ Money left for counting after 30 minutes = Rs (10710 − 5400) = Rs 5310
It is given that after 30 minutes, he counts at the rate of Rs 3 less every minute than the preceding minute.
Therefore, it would be an A.P. where a = 177 and d = −3.
Let the time taken to count Rs 5310 be n minutes.
\[5310 = \frac{n}{2}\left[ 2 \times 177 + \left( n - 1 \right) \times - 3 \right]\]
\[ \Rightarrow 10620 = 354 n - 3 n^2 + 3n\]
\[ \Rightarrow 3 n^2 - 357n + 10620 = 0\]
\[ \Rightarrow n^2 - 119n + 3540 = 0\]
\[ \Rightarrow n^2 - 59n - 60n + 3540 = 0\]
\[ \Rightarrow n\left( n - 59 \right) - 60\left( n - 59 \right) = 0\]
\[ \Rightarrow \left( n - 59 \right)\left( n - 60 \right) = 0\]
\[ \therefore n = 59 \text { or } 60\]
Thus, the time taken to count Rs 5310 would be 59 minutes or 60 minutes.
Hence, the total time taken to count Rs 10710 would be (30 + 59) minutes or (30 + 60) minutes, i.e. 89 minutes or 90 minutes, respectively.
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