A shopkeeper buys some books for `80. If he had bought 4 more books for the same amount, each book would have cost ` 1 less. Find the number of books he bought.

Or

The sum of two numbers is 9 and the sum of their reciprocals is . Find the number

We need to find the number of books he bought, so let the number of books he bought be x.

According to the question, the shopkeeper bought 4 more books for the same amount.

It is depicted mathematically as, x + 4

The amount for which these books are bought by the shopkeeper = Rs.80

‘If he had bought 4 more books for the same amount, each book would have cost Rs.1 less’ is depicted as,

⇒

⇒ 80x + 320 – 80x = x² + 4x

⇒ x² + 4x – 320 = 0

⇒ x² + 20x – 16x – 320 = 0

⇒ x(x + 20) – 16(x + 20) = 0

⇒ (x – 16)(x + 20) = 0

⇒ x – 16 = 0 or x + 20 = 0

⇒ x = 16 or x = -20

Since, books cannot be in a negative number so number of books bought is 16.

Or

Let one number be x and the other number be y.

∵, sum of two numbers be 9 ⇒ x + y = 9

⇒ y = 9 – x

Also, sum of their reciprocals be 1/2.

⇒

⇒

⇒ [∵, y = 9 – x]

⇒ 18 = 9x – x²

⇒ x² – 9x + 18 = 0

⇒ x² – 6x – 3x + 18 = 0

⇒ x(x – 6) – 3(x – 6) = 0

⇒ (x – 3)(x – 6) = 0

⇒ x – 3 = 0 or x – 6 = 0

⇒ x = 3 or x = 6

When x = 3, y = 9 – 3 ⇒ y = 6

When x = 6, y = 9 – 6 ⇒ y = 3

Thus, the numbers are 3 and 6 or the numbers are 6 and 3.