HighSchoolCalculus

Well if you need to move at least two bottles during your three moves then here is my order of moves:

Some symbols to simplify things

Upside down = Ω

Rightside up = Σ

You start off with...

 Ω Σ Ω 

FIRST MOVE: In this move I'll make the right side up upside down:

Ω Ω Ω

SECOND MOVE: In this move I'll now move ANOTHER bottle rightside up since we need to move more than ONE bottle:

Σ Ω Ω

ΤΗΙRD MOVE: I can just flip the bottle (Σ) back to being upside down (Ω):

Ω Ω Ω

I think I did this correctly, someone tell me if I did something wrong. The instructions were a bit wonky to read.

Hope this helped c:

Alright Guest, in order to solve this problem you need to get rid of that mixed number, in order do so, do as what follows:

You start off with:

7 x 1 \(\frac{3}{5}\)

Let's ignore the seven for now and focus on that mixed number: 

\(\frac{3}{5}\)

1 3/5

Now what you want to do is multiply the denominator (5) by the whole number (1) and add that answer to the numerator, WHILE KEEPING IT AS A FRACTION if done correctly you should get:

8/5

Now put that seven back into play:

7 x 8/5

If you'd like you can think of the seven as 7/1 multiply the numerators by the numerators and multiply the denominators by the denominators.

7/1 x 8/5

After multiplying, you should obtain the solution of:

56/5

Because the induction proof is FORMAL and i don't know why but formal proofs make you do that sorta crud. I'm actually doing those as we speak. Wish me luck :c

Hey Katie nice to see you back again.

The trick to these kind of problems is to factor the numerator so that it could become two binomials, like so:

(x^2 - x - 6)  =  (x + 2) (x - 3)

    (x + 2)              (x + 2)

Since there is an "(x + 2)" on both the numerator and denominator you cancel them out leaving you with you answer:

(x - 3)

Easy my friend, just punch it into a calculator you should get...   0.3125

You start off with:

2/3 x 15/14

Off of that you get:

30/42

Simplify...

5/7