(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.
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Step-by-Step Solution
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 x1800.
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 x+151800. 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=2a−b±b2−4ac 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.