Arithmetic+ 08: UNIZOR.COM - Math+ & Problems - Arithmetic

Notes to a video lecture on http://www.unizor.com

Arithmetic+ 08

Try to solve this problem without any writing, just in your head.

Problem A

There are certain number of students in the class and certain number of desks.
If 2 students sit at a desk, 5 desks will remain unoccupied.
If 1 student sits at a desk, 5 students will have no place to sit at.
How many students should sit by 2 at the desk and how many should sit by 1 at the desk, so all students will have seats and all desks occupied?

Solution A
If a single seating arrangement results in 5 students without a desk, we can put these students as doubles, which results in a combination of 5 doubles plus unknown singles needed to occupy all desks.
If double seating results in 5 desks unoccupied, let's convert 5 doubles into singles to have 5 students to occupy previously unoccupied 5 desks. Then the number of singles will be 10 (5 doubles converted into singles and 5 previously unoccupied desks having only one student each).
So, to occupy all desks with double or single we should use 5 doubles and 10 singles.
Incidentally, it makes the number of desks 15 and the number of students 20.

Answer A
5 desks should accommodate two students each and 10 desks should sit single student each.


Problem B

A large water tank is filled to the brim.
It's capacity is unknown.
There are two pipes used to empty this tank.
If only the first pipe is open, the tank will be empty in 14 minutes.
If both pipes are open, the tank will be empty in 10 minutes.
Assume that the speed of the water flow is always constant and depends only on the pipe's diameter.
How long would it take to empty this tank, if only the second pipe is used?

Solution B
An easy way to solve this problem without pen and paper is to see how much water different pipes can let through in the same amount of time.
Let's choose a number divisible by both 14 and 10, like 70.
If a tank can be emptied through pipe #1 in 14 minutes, in 70 minutes this pipe can empty 5 tanks.
Two pipes can let through the water of one tank in 10 minutes. Therefore, in 70 minutes two pipes can empty 7 tanks of water.
So, the pipe #1 in 70 minutes can empty 5 tanks. Both pipes in the same amount of time (70 minutes) can empty 7 tanks of water. Therefore, pipe #2 alone in 70 minutes can empty 7−5=2 tanks of water.
Therefore, one tank the pipe #2 can empty in 70/2=35 minutes.