A shopkeeper bought a certain number of notebooks for Rs. 200, five of which were stolen. He sold the rest for Rs.4 more than he paid for each. Thus he made a profit of Rs.40. How many notebooks did he buy?

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Solution

Step 1:
Obtain a quadratic equation:
Let the number of notebooks bought by the shopkeeper be $x .$
The cost of $x$ books $= Rs . 200$
Cost of one notebook $= Rs . \frac{200}{x}$
As $5$ books were stolen,
Therefore, the number of left notebooks $= x - 5$
As he sold the rest for $Rs . 4$ more than he paid for each,
Therefore, the selling price of each notebook $= Rs . (\frac{200}{x} + 4)$
The selling price of $x - 5$ notebooks $= Rs . [\frac{200}{x} + 4] \times (x - 5)$
According to the question, the profit made by the shopkeeper $= Rs . 40$
Now,
$Selling price - cost price = Profit$
$\Rightarrow [\frac{200}{x} + 4] \times (x - 5) - 200 = 40 \Rightarrow [\frac{200 + 4 x}{x}] \times (x - 5) - 200 = 40 \Rightarrow [\frac{200 + 4 x}{x}] \times (x - 5) = 40 + 200 \Rightarrow [\frac{200 + 4 x}{x}] \times (x - 5) = 240 \Rightarrow (200 + 4 x) \times (x - 5) = 240 x \Rightarrow 4 (50 + x) \times (x - 5) = 240 x [Take 4 common] \Rightarrow (50 + x) \times (x - 5) = 60 x \Rightarrow (x + 50) \times (x - 5) = 60 x \Rightarrow x^{2} - 5 x + 50 x - 250 = 60 x \Rightarrow x^{2} + 45 x - 60 x - 250 = 0 \Rightarrow x^{2} - 15 x - 250 = 0$
Step 2:
Determine the number of notebooks bought:
Solve the obtained quadratic equation.
$x^{2} - 15 x + 250 = 0 \Rightarrow x^{2} - 15 x + 250 = 0 \Rightarrow x^{2} - 25 x + 10 x + 250 = 0 \Rightarrow x (x - 25) + 10 (x - 25) = 0 \Rightarrow (x - 25) (x + 10) = 0 \Rightarrow x - 25 = 0 or x + 10 = 0 \Rightarrow x = 25 or x = - 10$
Since the number of notebooks cannot be negative, we reject $x = - 10$ .
$∴ x = 25$
Hence, the shopkeeper bought $25$ notebooks.

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