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Cold RATE for 1 bath = 1/3 b/m   b = bath   m=minutes

Hot  RATE for 1 bath = 1/2 b/m

Rate x time = volume

We want volume = 1  bath       (1/3 b/m + 1/2 b/m) *x = 1 b         b= bath     x = time to fill (in minutes)

(1/3+1/2)x =1

(2/6 + 3/6) x = 1

5/6 x = 1

x = 6/5 min       =   1 min 12 sec

Guest i think you should explain your method. 

If you cannot explain it then that means you do not understand why it works.

I'm impressed too Gavin :)

Here is one of the answers Gavin found.     

The angles he is talking about have to be angles at a point as I will show in this diagram.

Here is another simple way:

1/3 + 1/2 =5/6. Then flip 5/6 to 6/5 =1.2 minutes to fill the bathtub.

If it takes the cold water 3 minutes to fill the bathtub, then in 1 minute it will fill 1/3 of the bathtub.
Similarly, the same applies to hot water, or in 1 minute it will fill 1/2 of the bathtub.
So, the 2 taps combined in 1 minute will fill: 1/3 of the bathtub + 1/2 of the bathtub=5/6 of the tub in 1 minute.
So, the time it takes to fill the entire bathtub will be: 1 / (5/6) =1 x 6/5 =6/5 =1.2 minutes to fill the entire bathtub.

"Their internal angles must sum to 360"

why?

EDIT: thanks for the diagram melody :) now i understand. I thought all polygonals must surround a specific point, like a point contained within a circle. 

A cold water tap takes 3 minutes to fill the bath. A hot water tap takes 2 minutes to fill the bath. How long does it take to fill the bath when both taps are open at the same time?

It is easiest just to let the bath be a given size.

My little bath will be 12 litres

cold water  4L/min          hot water = 6L/min      

together 10L/1 min

10*(12/10) L per  1*(12/10) minutes

12L per 1.2min 

12L per 1min and 12sec

It will take 1 minute and 12 seconds to fill the bath.

The answer is correct.

85+x^2=(x-17)^2

85+x^2=x^2-34x+289

        85=-34x+289

     -204=-34x

          6=x

 THE NUMBER IS *drumroll*6! im answering my own problem, pls help me, show me wrong parts, help me out here, work with me.

In a certain pentagon, the interior angles are   a,b,c,d,e       where  are integers strictly less than 180

If the median of the interior angles is  61 degrees and there is only one mode, then what are the degree measures of all five angles?

Angle sum of a pentagon is  180*3 = 540 degrees. 

540-61= 479

479/4=119.75      since 2 are very little,  2 must be really big.

179+179=358

60+60=120

358+120=478  which is one too little

60,60,61,179,179 = 539    1 to little and there is 2 modes

60,61,61,179,179 = 540     right size but 2 modes

61,61,61,178,179 = 540     right size and only one mode.    

Same as Tertre so that is great :)

\(5^{12}-2 \cdot 10^6+2^{12}\\ =5^{12}-2\cdot 5^6 \cdot 2^6+2^{12}\\ =(5^{6})^2-2\cdot 5^6 \cdot 2^6+(2^{6})^2\\ =(5^6-2^6)^2\\ =[(5^3)^2-(2^3)^2]^2\\ =[(5^3-2^3)(5^3+2^3)]^2\\ =[(5-2)(5^2+5\cdot 2+2^2)(5+2)(5^2-5\cdot 2+2^2)]^2\\ =[3(25+10+4)\cdot 7(25-10+4)]^2\\ =[3\cdot 39\cdot 7\cdot 19]^2\\ =[3\cdot 3\cdot 13\cdot 7\cdot 19]^2\\ \)

So the highest prime divisor is 19.

\(i^{4k+2}+i^{4k+4}=0 \\ i^{4k+1}+i^{4k+3}=0\)

\(\text{we can rewrite the sum as}\\ (i + i^3) + (i^2 + i^4) + (i^5 + i^7) + (i^6 + i^8) + \dots + (i^{98}+i^{100}) +(i^{97}+ i^{99}) - i^{100} = -i^{100} \\ -i^{100} =- i^{4(25)} = -1\)

Someone actually gave you a point for this bulllshit. How can you NOT see logarithmic and exponential relationships in a simple question like this, when you answer interest rate questions all the time?

It's the green dot! So, -0.5, (C), the third option. 

Is it the last option? 40

x-50=41, x=91, x-50=59, x=109. So, (D). 

I think it is: 61, 61, 61, 178, 179, and no angle can be < 61 degrees.

First Quartile= Quartile 1, 25, 35, 45, 50, 60, 80, 90, 95, 100- median is 60, 35+45=80/2=40, (A)?

The middle one should be easy.  I'm going to give hints.

How do you find the roots?

   you have to simultaneously solve y=0 with that parabola. That will give the points where the p[arabola cuts the x axis.

The vertex is on the axis of symmetry and the axis of symm is half way between the roots.

Once you have the x value you can find the y with substitution.

the roots are 

\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)      seperate those 2 out and you will see that they are symmetrical about a very easy to find point. 

if you start with one layer and double the number of layers with each fold, and noting that

\(2^{10}=1024\)

it's clear that 10 folds is 1024 layers

if you wanted some number of layers that was enormous say N you would find

\(folds = \left \lceil \log_2(N) \right \rceil\)

where

\(\lceil x \rceil = \text{ the least integer }k,~\ni k > x\)

1. 

Let \(a = x^2 - 10x - 29\)

\(\frac{1}{a} + \frac{1}{a - 16} - \frac{2}{a - 40} = 0.\\  (a - 16)(a - 40) + a(a - 40) - 2(a)(a - 16) = 0 \\ -64a + 40 \times 16 = 0, \Rightarrow a = 10. \\\)

\(10 = x^2 - 10x - 29 \Longleftrightarrow 0 = (x - 13)(x + 3)\) The positive root is \(\boxed{13}.\)

2. 

https://artofproblemsolving.com/wiki/index.php?title=1990_AIME_Problems/Problem_5

I hope this helped,

Gavin

125 ?   Not sure how to make a damascus blade....but if you fold it in half several times to complete it, the whole number 125 will work

125    fold in half  = 250  fold in half =  500     fold in half = 1000

Just sequentially divide 1000 by 2 until you get no further whole numbers

1000 /2=500

500/2=250

250/2=125    <--------!

125/2=62.5   XXXXXX

it would be nice to do all of them but its ok to just do one  :)

12% / 12 = 1%, or the first on your list.

Look under the question just above this one where I calculated the payment for you. Copy it exactly as I calculated it. The EXACT payment is $115.36, or B on your list.

First, you calculate the monthly payment using the formula given in your question as follows:

M = Monthly payment

P = Present value of the loan or $2,500

m=monthly interest rate =10% =0.10 / 12 =0.008333....

a = Number of years

n = Number of compounding periods in 1 year =12

M =2,500 x 0.008333.. x [1 + 0.008333]^(2*12) / [1 + 0.008333]^(2*12) - 1

M =20.8333...x [1.220391] / [1.220391 - 1]

M =20.8333...x               5.53739116

M =$115.36 - The monthly payment on the loan.

$115.36  x  24 months =$2,768.64 - Total payback for $2,500 loan @ 10%.

Note: The discrepancy of about $1 is due to rounding off.