Alan

You are right Rom.  I didn't check them all carefully enough!

"what if it was a rational number like -5.2 or something, would that still be an integer? "

No!  An integer is a whole number.  Nothing after the decimal point.  No fractional part. 

I get 19999.

19999/101 = 198.00990099... ; ceil(19999/101) = 199;   ceil(19999/101) + 1 = 200;   19999/100 = 199.99

ceil(20000/101)+1 = 200;    20000/100 = 200

.

10/3 is not equal to 3.3333333333333333 whatever your calculator might say!  It is equal to 3.3333..... where the 3's repeat without end. This means that when you multiply this by 3 you get 9.9999.... where the 9's repeat without end.

Now if we let

x = 9.99999....               (where the 9's repeat without end), then

10x = 99.99999.....      ((where the 9's repeat without end).

Subtract the first from the second to get:

9x = 90          (as all the 9's behind the decimal point cancel each other)

So, dividing both sides by 9 we get x = 10.   That is:  3*3.33333..... = 10

Similar reasoning applies to your second example.

If tc/tf =max(3.3-1.1*0.104-0.34*4.5, 1.1) then tc/tf = max( 1.6556, 1.1), since 3.3-1.1*0.104-0.34*4.5 = 1.6556;  so tc/tf = 1.6556

Multiply both sides by tf:    tc = 1.6556*tf  or  tc = 1.6556*2.9 or tc = 4.8012   or tc ≈ 4.8

.

On any one day Bob will expect to roll once 5/6 of the time, twice 5/6² of the time, three times 5/6³ of the time, ...etc.

Expected number for one day = 1*5/6 + 2*5/6² + 3*5/6³ + ...  = 6/5

Expected number for a non-leap year = 365*6/5 = 438

.

Here's one solution:

(Of course, you don't have to use base 60; any base would do - but I only thought of this after doing the above!). 

.

Like so:

.

Melody, for "the back to front e thingy" read "such that".

You are correct!