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Is this GPT answered? Because usually GPT, Siri, and Alexa start with "I'm happy to help you with...."

Note that \(0.0\overline2 = \frac2{90}\), and \(0.1=\frac1{10}\), so the decimal is \(\frac2{90} + \frac1{10} = \frac{11}{90}\).

By the way, since the flavors can be repeated there would be more than just \(\binom n3\) selections. 

Good job, plaintainmountain !!!!!

They probably forgot to copy the LaTeX from their homework 

This would mean that \(1+\frac1n \implies \frac1n\) is an integer, so the only possible value of \(n\) would be \(1\).

The concept of \(e\) comes from compound interest. Suppose a very generous bank offers you a \(100\%\) interest rate, which would give you twice as much as what you put in every year. If you compound the interest quarterly, you would recieve \(25\%\) of your \(\bf{current}\) amount every three months, eventually gaining around \((1.25)^4\approx 2.44\) times of what you invested. 

So suppose that you compounded your money even further, recieving interest every day, hour or even second. Would you achieve infinte wealth? Sadly, no. \(e\) is defined as the limit of this growth, which is 
$$\lim_{n\to \infty}\left(1+\frac1n\right)^n = 2.71828182846\dots$$

Despite its apparent disconnection from other areas of mathematics, \(e\) turns up quite a lot in calculus and complex numbers. For example, you may have heard of the famous equation \(e^{i\pi}=-1\). 

Some other facts about \(e\):

• It is both irrational and transcendental, that is, it cannot be expressed as \(\frac mn\) where \(m\) and \(n\) are relatively prime integers. Nor is it the root of any polynomial with rational coefficients. 
• The growth of the function \(e^x\) is exactly itself, that is, the slope of the tangent line at any point \(\left(x,e^x\right)\) is exactly equal to \(e^x\), or in other words \(\frac d{dx}\left(e^x\right)=e^x\). You will learn more about it when you get to calculus. 

Let the smaller digit =  d

Let the larger digit = 2d - 1

So the larger number can be expressed as  10(2d-1) + d

And the smaller number can be expressed as 10d + 2d -1

And the difference  between these = 36 .....so......

[10(2d -1) + d]  - [ 10d + 2d -1] =  36          simplify

20d -10 + d - 10d -2d + 1  = 36

9d - 9 = 36

9d = 36 + 9

9d =45

d = 45/9 = 5

2d -1  =  9

Her favorite number =  95   and the other is 59  and their difference = 36

"e" is NOT an "imaginary" number....  is the  base for the  natural logarithm  (just as 10 is the base  for the  "normal" log)

It equals  ≈ 2.71828.....   (it is irrational so it has a non-repeating, non-terminating decimal part )

This number  comes in very handy for many scientific and financial applications  (such as growth/decay functions)

Textot provides a nice solution here :

https://web2.0calc.com/questions/terminating-decimals_4

Divide it into triangles and rectangles and add the area of those up

From above figure given measurements are,

12 mi, 5 mi, 9 mi, 5 mi, 3 mi, 3 mi, 6 mi, 3 mi and 5 mi

5 - 3 = 2 mi

9 + 3 = 12 mi

5 - 2 = 3 mi

5 + 3 = 8 mi

12 mi - (3 + 6) mi

12 - 9 = 3 mi

here we using formulas:

Area of right triangle = 1/2 * base * height and, Area of rectangle = length * width

Here Total area of the figure 

= Area of triangle + Area of rectangle + Area of rectangle + Area of rectangle + Area of triangle

= 1/2 * 12 * 8 + 9 * 3 + 9 * 2 + 12 * 3 + 1/2 * 3 * 3

= 48 + 27 + 18 + 36 + 9/2 

= 129 + 4.5

= 133.5

Total area of the figure = 133.5 mi2

Call the original  amount in his savings, S

Amt he spent on the watch =  (1/4)(3/5)S = (3/20)S

He has (3/5)S - (3/20)S =  (9/20)S   left

Amt he spent on the DVD player =  (5/6) ( 9/20)S =  (9/24)S

So

(3/5)S - amt on watch - amt on DVD player  = 30           ......in equation form......

(3/5)S -  (3/20)S - (9/24)S = 30

[ 72 - 18 - 45 ] / 120 S   = 30

[ 9] /120 S  =  30

S = (120 * 30) / 9 =  $400 = amt of original savings

Amt spent on watch = (3/20)(400)  =   $60

This is known as a "see-saw" problem

3 *2  =  5 * x            where x is the  time in hours for 5 people to cut the property

6 =  5 * x                divide both sides by 5

6/5  = x

It will take 5 people 6/5 of an hour to  cut the property = 1 + 1/5 hrs =  =  1 hr  12 min

Total value = Price per geode * Number of geodes
Total value = $2.87 * 16
Total value = $45.92

So, Gavin's maximum bid for the box of geodes should be $45.92.
 

If his mean score was 91 on two tests....his  combined  score  must have been  91 * 2  =  182

Starting off 

Dave  had   D  game cards

Perry had D + 72

Perry gave (1/3) of his cards to Dave

Dave now has  (1/3)(D + 72) + D  = (4/3)D + 24

Perry now has (2/3) (D + 72)  = (2/3)D + 48

Dave then  gave 3/8 of his cards to Perry

Dave now has (5/8) [ (4/3)D +24 ] =  (5/6)D + 15

Perry now has  (2/3)D + 48 +  (3/8)[(4/3)D + 24] =  (2/3)D + (1/2)D + 48 +9  = (7/6)D + 57

And the difference at the end was that Perry had 114 more cards than Dave

So

  Perry's  - Dave's   =  114           in equation  form we have

[ (7/6)D + 57]  -  [  (5/6)D + 15 ]  =  114    simplify

(2/6)D + 57 -15  = 114

(1/3)D  + 42  = 114

(1/3)D = 114 - 42

(1/3)D =  72     multiply through  by 3

D = 216 =  what Dave started with

D + 72 = 216 + 72 =  288  = what Perry started with

I believe you have left something out, as you did not state the total amount of flavors. For your reference, you can find the answer by finding C(total amount of flavors, 3)

aha! thanks!