(a) If Nidhi were 7 years younger than what she actually is, then the square of her age (in years) would be 1 more than 5 times her actual age. What is her present age ? OR (b) A shopkeeper buys a number of books for ₹ 1,800. If he had bought 15 more books for the same amount, then each book would have cost him ₹ 20 less. Find how many books he bought initially.

Step 1: Define variables and set up the equation

Let $x$ be the initial number of books the shopkeeper bought. The total cost for these books is given as ₹1800. We need to find the initial number of books.

Step 2: Express the cost per book

The cost per book can be found by dividing the total cost by the number of books. So, the initial cost per book is $\frac{1800}{x}$.

Step 3: Formulate the equation based on the given condition

If the shopkeeper had bought 15 more books, the new number of books would be $x+15$. The new cost per book would then be $\frac{1800}{x+15}$. The problem states that each book would have cost ₹20 less, so the difference between the initial cost per book and the new cost per book is ₹20.

Step 4: Simplify the equation

To simplify the equation, we first find a common denominator and combine the terms on the left side. Then, we multiply both sides by the common denominator to eliminate the fractions. After expanding and rearranging, we get a quadratic equation in standard form.

Step 5: Solve the quadratic equation using the quadratic formula

We use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to solve for $x$. Here, $a=1$, $b=15$, and $c=-1350$. We substitute these values into the formula and calculate the discriminant.

Step 6: Calculate the possible values for x and choose the valid solution

We get two possible values for $x$. Since the number of books cannot be negative, we discard $x = -45$. Therefore, the initial number of books is 30.

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