# Quantitative Reasoning – Lockers – Open and Closed!

PSA 2014_XI_Q16. Lockers for storing small items are found in some schools. The lockers in a school were numbered 1 to 100. These were all opened for cleaning. Manoj came past and closed every 5th locker. Arvi then came past and closed every even numbered locker that was open, and opened every even numbered locker that Manoj had closed.
How many lockers were left open after Manoj and Arvi had both gone past ?

(1) 60

(2) 10

(3) 40

(4) 50

Ans: Manoj came and closed every 5th locker ( 5th, 10th, 15th …., 95th, 100th lockers etc.). Total lockers closed= 20.

Arvi came and acted upon every even numbered lockers; Arvi closes the opened lockers, and opened the closed ones.

If there were no lockers closed, Arvi would have closed 50 lockers but in this scenario she would have closed 40

lockers only (2nd, 4th, 6th, 8th, 12th, 14th, 16th, 18th, 22nd, and so on) and opened 10 closed ones (10th, 20th, 30th…100th). So, total lockers closed = 20 – 10 + 40 = 50. Option (4).

Second method:

This could also be solved using only 10 lockers.

Similarly for 100 lockers; 50 would be opened and other 50 closed.

The number 10 is a carefully chosen subset. Not all the numbers can be taken as a subset to arrive at the result.