Question

By selling an article for Rs. 48, a trader loses as much percent as half of the cost price of the article. Calculate the cost price and loss amount of the article.

Hint:

Always test both positive roots of a quadratic equation in word problems.
Often, both roots are physically realistic and represent two perfectly valid solutions!

Answer 1

Step 1: Set up the equation using the loss amount directly.
Let the cost price be $x$. The loss percent is $\frac{x}{2}\%$, so the actual loss in rupees is $\frac{x}{2}\%$ of $x$, which is $\frac{x^2}{200}$. Since selling price = cost price minus loss, $48 = x - \frac{x^2}{200}$.
Step 2: Clear the fraction to form a standard quadratic.
Multiplying through by $200$: $9600 = 200x - x^2$, which rearranges to $x^2 - 200x + 9600 = 0$.
Step 3: Solve using the quadratic formula instead of factorising.
$x = \frac{200 \pm \sqrt{200^2 - 4(9600)}}{2} = \frac{200 \pm \sqrt{40000-38400}}{2} = \frac{200 \pm \sqrt{1600}}{2} = \frac{200 \pm 40}{2}$, giving $x = 120$ or $x = 80$.
Step 4: Find the loss amount for each cost price.
If $x=120$: loss $\% = 60\%$, loss $= 0.6 \times 120 = \text{Rs. }72$ (check: $120-72=48$, correct). If $x=80$: loss $\% = 40\%$, loss $= 0.4 \times 80 = \text{Rs. }32$ (check: $80-32=48$, correct).
\[ \boxed{\text{CP} = \text{Rs. }120 \text{ (loss Rs. }72\text{) or CP} = \text{Rs. }80 \text{ (loss Rs. }32\text{)}} \]