Let's analyze the problem step-by-step:
Problem Restatement:
A person sells a book for Rs. 11. He gains a percentage equal to the cost price (in rupees). We need to find the cost price.
Let:
- Cost price = Rs. xxx
- Gain% = xxx% (since gain% is equal to the cost price)
Using the formula for selling price:
Selling Price=Cost Price+Profit\text{Selling Price}=\text{Cost Price}+\text{Profit}Selling Price=Cost Price+Profit
Profit is x100×x=x2100\frac{x}{100}\times x=\frac{x^2}{100}100x×x=100x2. So,
11=x+x210011=x+\frac{x^2}{100}11=x+100x2
Multiply both sides by 100 to clear the denominator:
1100=100x+x21100=100x+x^21100=100x+x2
Rearranged as a quadratic equation:
x2+100x−1100=0x^2+100x-1100=0x2+100x−1100=0
Solve the quadratic equation:
Using the quadratic formula:
x=−b±b2−4ac2ax=\frac{-b\pm \sqrt{b^2-4ac}}{2a}x=2a−b±b2−4ac
Where a=1a=1a=1, b=100b=100b=100, and c=−1100c=-1100c=−1100. Calculate the discriminant:
Δ=1002−4×1×(−1100)=10000+4400=14400\Delta =100^2-4\times 1\times (-1100)=10000+4400=14400Δ=1002−4×1×(−1100)=10000+4400=14400
14400=120\sqrt{14400}=12014400=120
So,
x=−100±1202x=\frac{-100\pm 120}{2}x=2−100±120
Two possible solutions:
- x=−100+1202=202=10x=\frac{-100+120}{2}=\frac{20}{2}=10x=2−100+120=220=10
- x=−100−1202=−2202=−110x=\frac{-100-120}{2}=\frac{-220}{2}=-110x=2−100−120=2−220=−110 (not possible since cost price can't be negative)
Final answer:
10\boxed{10}10
The cost price of the book is Rs. 10.
