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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
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B
70 minutes
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C
89 minutes
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D
103 minutes
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Solution

The correct option is D $$89$$ minutes
Money 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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