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Help please

done

35 * (1/7)   =  35/7  =  5

Any even integer is divisible by 2

If  the sum of the digits of an integer  is divisible by 3, then the integer itself is also divisible by 3

Any integer ending in 5 or 0   is divisible by 5

I didn't work it through, but your answer indicates that

2m = 3n     and   2m = 2020p        so.....

n  = (2/3)m       and  p  = 2m / (2020)    =    m/ 1010

Subbing this back into your first first generated equation we get that

2m =   1/m + 1/  [ (2/3)m]  + 1/ (m/1010)       simplify

2m  = 1/m  + 3/(2m) + 1010/m     multiply through by m

2m^2   = 1 + (3/2) + 1010

2m^2   =  2.5 + 1010

2m^2 =  1012.5

m^2 = 1012.5 /2       

m^2 = 506.25     take both roots

m  = ±22.5

When

m =  22.5  =  45/2                        m =  -45/2

n = (2/3)(45/2)   =   15                 n =   -15

p  = (45/2)/ 1010 =  9/404            p  =  - 9/404

BTW....I verified these solutions with WoframAlpha

Good work   !!!!

60 (1/3)(1/4)  = 

60  / 12   =

5  have birds

I think we can use  similar triangles to solve this

Area of smaller circle  =  pi * radius^2

1 = pi * radius^2

1 / pi  = radius^2

√[1/pi]  = radius  of smaller circle

Call the radius of the larger circle, R

And  we have this

      3√(1/pi)              3√(1/pi) + R

      _____  =       _____________   simplify    

        √(1/pi)                R

3R   =   3√(1/pi)  + R

2R =  3√(1/pi)

R  = (3/2)√(1/pi)

So  the area of the larger circle  =

pi [( 3/2) √(1/pi) ] ^2  =     

9/4  units^2 = 

 2.25 units^2

AB => a = 3

BC => b = 4

CA => c = 5

Area of triangle is:                  A = ab/2 = 6 u²

Area of AB semicircle is:         A = [(a/2)² * pi] /2  = 3.534291735 u²

Area of BC semicircle is:         A = [(b/2)² * pi] /2  = 6.283185307 u²

Area of AC semicircle is:         A = [(c/2)² * pi] /2  = 9.817477043 u²

Tota area is:                             A_t = A + A + A = 15.81747704 u²

Area of a shaded region is:      A_s = A_t - A = 6 u² 

These areas are equal:          A = A_s    

Call the original number of bags, B

After  1/6  of these were eaten, there  were   B  - (1/6)B  =  (5/6)B left

After  50% of these were eaten.....there were (1/2)(5/6)B   =  (5/12)B left

And  this  = 30     

So

(5/12)B   = 30     multiply  both sides by  12/5

B = 30(12/5)

B =  360 / 5

B = 72

Thanks Melody and Dragan. 
I will post other questions in a new post. 

This would be about 1.5212011806014482 which would round to 1.5212.

Hope this helped!

Let the  number  of cats  = x

Then the number of dogs  = 17  - x

And we have this

cost of care per cat * number  +  cost of care per dog * number  = total revenue

       3x       +   5 (17 - x)  =   69      simplify

       3x  + 85  -  5x   =  69

       -2x  + 85   =  69             subtract 85 from both sides

        -2x =   -16               divide both sides by - 2

          x =   8    (cats) 

I'll do the first one......the other two should then be easy

projection of   u =  ( 2,4,2)  onto  v =  (1,2,1)

Calculate  the dot product of the two vectors  =

2*1 + 4*2  + 2*1  =    2 + 8 + 2   =  12

Calculate  the length  of  the vector  v   = sqrt [ 1^2 + 2^2 + 1^2 ] =  sqrt [ 6]

Square this  = 6

So

                        dot product of u * v

proj_v u    =     __________________  *   vector v    =

                        (  length of v  )^2

                              12

                            ____    *    ( 1,2,1)        =

                               6

                                    2 * (1,2,1)  =

                                    (2, 4 , 2 )

This doesn't  work much differently than in 2 space.....we can still use the Law of Cosines

BC  = sqrt [ (2 - 4)^2  + (-2 -4)^2  + (2 - 4)^2 ]  = sqrt [ 4 + 36 + 4]  = sqrt (44)

AC = sqrt [ (1 - 4)^2 + (0 - 4)^2 + (2 - 2)^2] = sqrt ( 9 + 16)  = sqrt (25)   =5

AB = sqrt [ (1 - 2)^2 + (-2 - 0)^2 + ( 4 - 2)^2 ]  = sqrt [ 1 + 4 + 4 ] =  sqrt (9)  = 3

So....we have  that

BC^2  = AC^2 + AB^2  - 2 ( AC)(AB) cos (BAC)

44 = 25 + 9  - 2 (5)(3) cos (BAC)

[44 - 25 - 9]  / [ -2 * 5 * 3 ]  = cos (BAC)

cos (BAC)  =   10 / [ - 30 ]  =  -1/3

The type of triangle must be specified! We can only assume that this is an isosceles triangle.  

No it is not clear.

How many shots does Mia have?

If 3 go down with one shot do they all go down together?  What order do they fall in? 

The question is not clear at all.

Nice, CTG  !!!!

done

Either the American wins the gold, silver or bronze   (3 ways)

And in each case, we want to choose 1 of 4 Americans to win one of the medals  and  any 2 of the other 11 to receive the other two medals

So...we have

3 * C(4,1) * ( 11, 2)   = 

660  ways

done

Welcome aboard CTG    

I am glad that I could help you.

 I hope you enjoy it here and learn a lot while you are at it :)

@Melody, I think the problem is just saying that every time Mia shoots a duck, the ducks next to it fall down. Then, how many ways can Mia shoot three ducks? This should be clear now.

yes it is.

Maybe if you have to bring down 3 ducks win one go then there are just 8 ways. 

I do not think that the rules are clear.

Order counts. If she knocks over duck number 2 then ducks one and duck 3 also fall. What order do they fall?

Also how many turns does Mia get?

Do misses pass as turns? 

If she hits any duck from 2 to 9 then 3 ducks will go over.   That is 8 possibilities.

If she hits duck one or duck 10 with the first shot then that is 2 down, the next shot that does not miss altogether will bring down at least 2 more, that is four or 5 in total.

So yea ... I do not know what is being asked.