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A rise of 25 % in the price of sugar, compels a man to buy 5 kg sugar less for Rs. 200. Calculate the original price per kg of the sugar.

Answer Verified Verified
Hint: In the above question, we will first assume the original cost of the sugar to be a variable. Also, we will assume the original amount of sugar to be another variable. Since, the total cost remains the same, we will equate the original cost to the new cost.

Complete step-by-step answer:
Let us suppose the original cost of the sugar to be Rs. C/kg.
Since it is given that there is a rise of 25 % in the price of sugar.
So the new cost is equal to 25 % increment in the original cost.
New cost,
\[\begin{align}
  & =\left( C+25\%\times C \right) \\
 & =C+\dfrac{25}{100}\times C \\
 & =C+\dfrac{1}{4}C \\
 & =\dfrac{5}{4}C \\
\end{align}\]
Now let us suppose the original amount of sugar bought be x kg.
As it is given that the man buys 5 kg sugar less.
Then, new amount of sugar \[=(x-5)kg\].
Also, original cost of sugar \[=Rs.\left( C\times x \right)\].
New cost of sugar = new amount \[\times \] new cost
\[\begin{align}
  & =(x-5)\times \dfrac{5}{4}C \\
 & =Rs.\dfrac{5}{4}C(x-5) \\
\end{align}\]
Since we know that the total cost remains the same, we have as follows:
\[C\times x=\dfrac{5}{4}C(x-5)\].
We also have been given that the total cost is Rs. 200.
\[C\times x=\dfrac{5}{4}C(x-5)=200....(1)\]
Now considering \[C\times x=\dfrac{5}{4}C(x-5)\].
\[\Rightarrow Cx=\dfrac{5}{4}Cx-\dfrac{25}{4}C\].
By taking \[\dfrac{25}{4}C\] to the left hand side of the equal sign, we get as follows:
\[Cx+\dfrac{25}{4}C=\dfrac{5}{4}Cx\]
Again, by taking the term ‘Cx’ to the right hand side of the equal sign, we get as follows:
\[\begin{align}
  & \dfrac{25}{4}C=\dfrac{5}{4}Cx-Cx \\
 & \dfrac{25}{4}C=Cx\left( \dfrac{5}{4}-1 \right) \\
 & \dfrac{25}{4}C=Cx\left( \dfrac{5-4}{4} \right) \\
 & \dfrac{25}{4}C=Cx\times \dfrac{1}{4} \\
\end{align}\]
On multiplying the above equation by ‘4’, we get as follows:
\[\begin{align}
  & 4\times \dfrac{25}{4}C=\dfrac{Cx}{4}\times 4 \\
 & 25C=Cx \\
\end{align}\]
We know that the value of \[Cx=200\] from equation (1).
So, by substituting the value of \[Cx=200\] in the above equation, we get as follows:
\[\begin{align}
  & 25C=Cx \\
 & 25C=200 \\
 & C=\dfrac{200}{25} \\
 & C=8 \\
\end{align}\]
Therefore, the original price per kg of the sugar is equal to Rs. 8.

Note: Be careful while doing calculation and take care of the sign while solving the equation. We can also solve this question by considering the other two expressions from the equation (1).

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