A reduction of $10\%$...

Question

A reduction of $10\%$ in the price of sugar would enable a man to buy $2\,\rm{kg}$ of sugar more for Rs. $125$. Find the reduced price per kg.

My attempt: Let the initial price of sugar be Rs. $x$ kg. Price after $10\%$ reduction: $$x-10\%\ x=\frac{9x}{10}$$ Now, I could not understand the other conditions.

Answer 1

Let $P_0$ denote the initial price per kg and $P_F$ denote the price after the reduction in $10\%$, then

$$ P_F = 0.9 P_0. \tag1$$

Let $Q_0$ denote the number of kgs that the man could buy when the price was $P_0$ and $Q_F$ denote the number of kgs that the man can buy when the price is changed to $P_F$. We know three things about this:

\begin{align} Q_F - Q_0 = 2 \tag2\\[2ex] P_0 Q_0 = 125 \tag3\\[2ex] P_F Q_F = 125 \tag4 \end{align}

Substituting $(2)$ into $(4)$ we get

$$ P_F ( Q_0 +2)= 125 \;\; \Rightarrow \;\; P_F ( Q_0 +2)= 125 ,$$

now, substitute $Q_0$ from $(3)$ to get

$$ P_F \left( \frac{125}{P_0} +2 \right)= 125 \;\; \Rightarrow \;\; 125\frac{P_F}{P_0} +2P_F= 125 .$$

Finally use $(1)$ to get

$$ 0.9*125 +2P_F= 125 \;\; \Rightarrow \;\; P_F= \frac{0.1*125}{2} .$$

Answer 2

The price is Rs/kg. If you pay attention to the units it can guide you. He can initially buy $\frac {125}x$ kg of sugar. After the reduction, how much can he buy? The difference of these is $2$ kg.