Questions: Questions about …

The Waiter

Three men in a cafe order a meal the total cost of which is $15. They each contribute $5. The waiter takes the money to the chef who recognizes the three as friends and asks the waiter to return $5 to the men.

The waiter is not only poor at mathematics but dishonest and instead of going to the trouble of splitting the $5 between the three he simply gives them $1 each and pockets the remaining $2 for himself.

Now, each of the men effectively paid $4, the total paid is therefore $12. Add the $2 in the waiters pocket and this comes to $14…..where has the other $1 gone from the original $15?

Algebra Questions-1

This is first question in the series of Algebra Questions. The numerals in this question are replaced by letter codes. Replace the same characters by the same numerals so that the mathematical operations are correct.

CODED QUESTION

ABCB - DEFC = GAFB
      :         +         -
   DH x    AB  =    IEI
——————————
GGE  + DEBB = DHDG

In other words, find the parameters A, B, C, D, E, F, G, H, and I.

Hole in a Sphere
“Hole in a Sphere” question may be solved using calculus or more easily using logic.
A 6-inch hole is drilled through a sphere. What is the volume of the remaining portion of the sphere?
Clarifications:

1. the hole is a circular cylinder of empty space whose axis passes through the center of the sphere - just as a drill would make if you aimed the center of the drill at the center of the sphere and made sure you drilled all the way through.
2. the length of the hole [6 inches] is the height of the cylinder that forms the inside surface once the hole is drilled. picture the inside surface as viewed from inside the hole and measure the length of that surface in the direction of the axis of the drill.

in this sense, you could for example drill a 6-inch hole through the earth. the diameter of the hole would be huge, and you’d just have a tiny remnant of the earth left. but if you could set it on a table [a big table] it would be 6 inches high.
You of course could not drill a 6-inch hole through a sphere whose diameter was less than 6 inches. This fact leads to the logical answer.
The hard way involves calculus. The easy way uses logic.