  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 then the preceding minute. Find the entire time taken by him to count the entire amount.

A
56 minutes  B
70 minutes  C
89 minutes  D
103 minutes  Solution

The correct option is D $$89$$ minutesMoney counted in half an hours=$$180\times 30=$$ Rs. $$5400$$then money left for counting =$$10710-5400=$$ Rs. $$5310$$After half an hour he counted $$3$$ less per minute.$$\therefore$$ rate of counting per minute $$=$$ $$(180-3),(180-3-3).180-3-3-3).....$$$$=177,174,171..$$This form an A.P. whose first term $$=177$$ and difference $$=-3$$Sum of A.P $$=5310$$ $$\Rightarrow \dfrac{n}{2}[2a+(n-1)d]=5310$$$$\Rightarrow \dfrac{n}{2}[2\times 177+(n-1)(-3)]=5310$$$$\Rightarrow n[354-3n+3]=5310\times 2$$$$\Rightarrow 357n-3n^2=10620$$$$\Rightarrow 3n^2-357n+10620=0$$ $$\Rightarrow n^2-119n+3540=0$$$$\Rightarrow n^2-59n-60n+3540=0$$$$\Rightarrow n(n-59)-60(n-59)=0$$$$\Rightarrow (n-59)(n-60)$$$$\Rightarrow n=60$$ or $$n =59$$but $$59<60$$$$\therefore n=59$$Therefore, the time taken to count entire amount $$=$$ $$30+59=89$$ minute.Mathematics

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