Question: A shop stores x kg of rice. The first customer buys half this amount plus half a kg of rice. The second customer buys half the remaining amount plus half a kg of rice. Then the third customer also buys half the remaining amount plus half a kg of rice. Thereafter, no rice is left in the shop. Which of the following best describes the value of x?
Explanation:
The initial quantity of rice is x kg.
The first customer buys half the total rice in the store, and another half kg.
∴ Rice purchased by the first customer = x 2 + 1 2 = x + 1 2
∴ Remaining rice = x - x + 1 2 = x - 1 2
Now, the second customer buys half of this, and another half kg.
∴ Rice purchased by the second customer = x - 1 4 + 1 2 = x + 1 4
∴ Remaining rice = x - 1 2 - x + 1 4 = x - 3 4
Now, the third customer buys half the remaining rice, and another half kg.
∴ Rice purchased by the third customer = x - 3 8 + 1 2 = x + 1 8
Since after this purchase, there is no rice left in the store, we conclude that:
x - 3 4 - x + 1 8 = x - 7 8 = 0
∴ x = 7
Alternately ,
If the shopkeeper had x kg rice before a customer comes, the amount of rice left after every time a customer buys half the amount of rice + half a kg more = x 2 - 1 2
For 3rd customer.
Now, x 2 - 1 2 = 0 [No rice is left after third customer]
⇒ x = 1 kg.
Now, the shopkeeper had 1 kg rice when 3rd customer came, hence after 2nd customer left he had 1 kg of rice.
For 2nd customer.
x 2 - 1 2 = 1
⇒ x = 3 kg.
Now, the shopkeeper had 3 kg rice when 2nd customer came, hence after 1st customer left he had 3 kg of rice.
For 1st customer.
x 2 - 1 2 = 3
⇒ x = 7 kg.
Now, the shopkeeper had 7 kg rice when 1st customer came.
Hence, option (b).