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15 cookies and brownies were sold. Each cookie was sold at $1.50 and each brownie was sold at $2.25. $ 26.25 was the total. How many cookies were sold? How many brownies were sold?

15 Kekse und Brownies wurden verkauft. Jeder Cookie wurde zu 1,50 USD und jeder Brownie zu 2,25 USD verkauft. 26,25 $ war die Summe. Wie viele Cookies wurden verkauft? Wie viele Brownies wurden verkauft?

Hello Guest!

\(c+b=15\\ 1.5c+1.5b=22.5\\ \underline{$1.50\cdot c+$2.25\cdot b=$26.25}\\ 0.75b=3,75 \)

\(b=5\\ c=10\)

5 brownies and 10 cookies were sold.

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Write an equation of a line that is perpendicular to y=2x-3 and has a y intercept of 10

Hello Guest!

The equation is called  \(y=-\frac{1}{2}x+10\)

  !

How many interior diagonals does an icosahedron have?

(An icosahedron is a 3-dimensional figure with 20 triangular faces and 12 vertices, with 5 faces meeting at each vertex.

An interior diagonal is a segment connecting two vertices which do not lie on a common face.)

Icosahedron :

\(\begin{array}{lcll} F=faces = 20 \\ V=vertices=12 \\ E=edges=\ ? \end{array} \)

For convex 3-dimensional polyhedra Euler's formula: \(\mathbf{V + F - E = 2}\) or 

\(\begin{array}{|rcll|} \hline E &=& V+F-2 \\ E &=& 12+20-2 \\ \mathbf{E} &=& \mathbf{30} \\ \hline \end{array}\)

Interior diagonals :

\(\dbinom{\text{vertices}}{2} - \text{edges} \quad | \quad \text{vertices}=12,\ \text{edges} = 30\)

\(\begin{array}{|rcll|} \hline && \mathbf{\dbinom{\text{vertices}}{2} - \text{edges}} \quad | \quad \text{vertices}=12,\ \text{edges} = 30 \\\\ &=& \dbinom{\text{12}}{2} - 30 \\\\ &=& \dfrac{12}{2}*\dfrac{11}{1} - 30 \\\\ &=& 66 - 30 \\ &=& \mathbf{36} \\ \hline \end{array}\)

An icosahedron has 36 interior diagonals

The vertices of a triangle are A(4, 1), B(9, 1), C(2, 5).

What is the area of this triangle.

\(\begin{array}{|rcll|} \hline \text{The area of this triangle is } & \dfrac{5*4}{2} = \mathbf{10}\\ \hline \end{array} \)

If 3 is added to twice a number and this sum is multiplied by 6​,

the result is the same as if the number is multiplied by 5 and 4 is added to the product.

What is the​ number?

\(\begin{array}{|rcll|} \hline 6(3+2x) &=& 5x+4 \\ 18+12x &=& 5x+4 \\ 12x-5x &=& 4- 18 \\ 7x &=& -14 \\ \mathbf{x} &=& \mathbf{-2} \\ \hline \end{array}\)

The number is \(\mathbf{-2}\)

If 8 is added to twice a number and this sum is multiplied by 3,
the result is the same as if the number is multiplied by 3 and 9 is added to the product.
What is the number?

\(\begin{array}{|rcll|} \hline 3(8+2x) &=& 3x+9 \\ 24+6x &=& 3x+9 \\ 6x-3x &=& 9- 24 \\ 3x &=& -15 \\ \mathbf{x} &=& \mathbf{-5} \\ \hline \end{array}\)

The number is \(\mathbf{-5}\)

What is the least four-digit whole number that is both a perfect square and a perfect cube?

The number must be perfect 6th power (\(2\times 3\))

\(\begin{array}{|c|rcl|cc|} \hline n & n^6 && &\text{four-digit}\\ \hline 1 & 1^6 &=& 1 \\ 2 & 2^6 &=& 54 \\ 3 & 3^6 &=& 729 \\ 4 & 4^6 &=& 4096 & \checkmark & 4096=64^2,\ 4096=16^3 \\ 5 & 5^6 &=& 15625 \\ \hline \end{array}\)

Find three consecutive integers such that twice the greatest integer is 3 less than 3 times the least integer

Three consecutive integers: \((n-1),\ n,\ (n+1)\)

Twice the greatest integer is 3 less than 3 times the least integer:

\(\begin{array}{|rcll|} \hline 2(n+1) &=& 3(n-1)-3 \\ 2n+2 &=& 3n-3-3 \\ 2n+2 &=& 3n-6 \\ 2n+8 &=& 3n \\ 8 &=& 3n-2n \\ 8 &=& n \\ \mathbf{n} &=& \mathbf{8} \\ \hline \end{array}\)

The three consecutive integers are: \(\mathbf{7,\ 8,\ 9}\)

I started with Alan's technique.  

Thanks Alan I would not have thought of it myself.

I am thinking of the numbers as four distinct zeros, 4 distinct ones and 4 distinct twos.

I am going to say it is 3 columns and 4 rows with each row adding to a multiple of 3 because that will be much easier for me to work with.

Here is one possibility

0,0,0,0     4! = 24

1,1,2,2     4*3*4*3  * 4!/(2!2!) = 144*6 = 864

1,1,2,2     4!                     =24

So far I get 24*864*24 = 497664 ways

But the rows can be in any order. I think that is 3 ways =3 ways

So that is  497664*3 = 1 492 992 ways.

Another possibility is

2,2,2,0    4*3*2*4*4 = 384 ways

1,2,0,0    4*1*4*3*(4!/2!) = 48*12 = 576 ways

1,1,1,0    3*2*4 = 24 ways

384*576*24 = 5308416

Times by 6 = 31 850 496 ways

Another possibility is 

1,2,0,0    4*4*4*3*(4!/2!) =   193*12 =  2304

1,2,0,0    3*3*2*1*12      =    18*   12=  216

1,2,1,2     2*1*2*1*(4!/4) =     4*    6 =     24

2304*216*24= 11 943 936

Times by 3 = 35 831808

1 492 992+31 850 496+35 831808 = 69 175 296 ways

This does seem like a lot and maybe I have double counted

but if there were no restrictions at all then there would be 12! ways = 479 001 600.

69175296/497001600 = 0.1391852581561106

So my answer is only 14% of all possible answers with no conditions at all.

So it may be correct. It passes my reasonableness test.

Thanks Alan,

I am still not understanding this statement

"The non-remainder part vanishes when x=0, 1 or 2, "

Could you expand upon that please?

a) On each action, there are 15 possible inputs: 5 inputs consisting of a single button, and 10 inputs consisting of two different buttons.

the answer is 60.

this question is not original. 

the answer can be found on 

https://brilliant.org/problems/question-19/

180 m / 10 hr   = 18 m/hr

180 m / 5 hr = 36 m/hr            ( 18 +36 )/2 = 27    speed of boat         9 = current

27 for the rate of the boat

C=9 for the current

https://www.algebra.com/algebra/homework/word/travel/Travel_Word_Problems.faq.question.735649.html

credited to checkley79

D=RT
180=(R-C)10
180=10R-10C DIVIDE BY 2 & ADD.
180=(R+C)5
180=5R+5C
90=5R-5C
---------------
270=10R
R=270/10
R=27 MPH. ANS. FOR THE RATE OF THE BOAT.
180=(27-C)10
180=270-10C
10C=270-180
10C=90
C=90/10
C=9 MPH. ANS.FOR THE CURRENT.
PROOF:
180(27+9)5
180=36*5
180=180

credited to checkley79

https://web2.0calc.com/questions/help_90245

8y2 / 3x

40 miles/1/2 hr  = 80 m/hr

40 miles/1/3 hr = 120 m/hr              airplane speed = 100    wind = 20 m/hr

192 x^2 = 16

192 = 16/x^2

192/16 = 1/x^2

12 = 1/x^2                square both sides

144 = 1/x^4

-17 (c-d)

n = nickels

p=  pennies =   2n -5

It is 22/51 because the first time he has a 2/3 chance of brown, the second time he has a 11/17 chance of getting it. Multiply them you get 22/51.

P.S. Can somebody answer my question, it's directly after this question.

the mean is 8 apples/bag  

12 bags  x  8 apples/bag = 96 apples

        so far you have 11 bags and    6 + 28  + 16  + 27 + 10 = 87 apples   

                                                                you'll need to have 9 apples in the 12th bag to have an average of 8 app/bag

n = nickels                    value = .05 n

d = dimes = (50-n)        value = .10 (50-n)        values summed = 3.40

.05n   +  .1 (50-n) = 3.4      solve for 'n' the number of nickels, then dimes = 50-n