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# A reduction of 20% in the price of the sugar enables a purchaser to obtain $2\dfrac{1}{2}kg$ more for Rs 160. Find the original price per kg of the sugar.[a] Rs 16[b] Rs 20[c] Rs 12[d] Rs 24  Hint: Assume that the initial price of one kilogram of sugar be Rs x. Hence calculate the final price of the sugar in terms of x after the reduction by 20%. Calculate the amount of sugar the purchaser could obtain initially and finally for Rs. 160. Equate the difference of the two to be equal to $2\dfrac{1}{2}kg$. Hence form an equation in x. Solve for x and hence find the original price per kg of the sugar.

Let the original price per kg of the sugar be x.

Hence the final price per kg of the sugar $=x-20%\text{ of }x=x-0.2x=0.8x$

The amount of sugar(in Kgs) which can be purchased initially for Rs 160 $=\dfrac{160}{x}$

The amount of sugar(in kgs) which can be purchased after price reduction by 20% is $=\dfrac{160}{0.8x}=\dfrac{200}{x}$

Since finally we can purchase $2\dfrac{1}{2}kg$ of sugar more, we have

$\dfrac{200}{x}-\dfrac{160}{x}=2\dfrac{1}{2}=\dfrac{5}{2}$

Taking x as L.C.M, we get

$\dfrac{200-160}{x}=\dfrac{5}{2}\Rightarrow \dfrac{40}{x}=\dfrac{5}{2}$

Cross multiplying, we get

$5x=80$

Dividing both sides by 5, we get

$x=\dfrac{80}{5}=16$

Hence the original price of the sugar was 16/kg

Hence option [a] is correct.

Note: Verification:

Amount of sugar which can be purchased initially for Rs 160 $=\dfrac{160}{16}=10kg$

Amount of sugar which can be purchased finally for Rs 160 $=\dfrac{160}{12.8}=\dfrac{1600}{128}=\dfrac{25}{2}kg$

Hence the extra amount of sugar which can be finally purchases for Rs 160 $=\dfrac{25}{2}-10=\dfrac{5}{2}=2\dfrac{1}{2}kg$

Hence our answer is verified to be correct.

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