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I'm not sure that I have interpreted your writing correctly, but:

If Gregory has 2/3 rds of an entire pizza and if he gives 1/4 th of an entire pizza to Marlene,

then he has given (1/4) / (2/3)  =  (1/4) · (3/2)  =  3/8 of his partial pizza to Marlene.

There are 5 integers.

If you solve these 5 equations, you will get the 5 integers.

1)  n - 2  =  2n

2)  n - 2  =  n/2

3)  n + 2  =  2n 

4)  n + 2  =  n/2

5)  2n  =  n/2

3)  

18 cm on a blueprint, with a scale of 3 cm to 4 ft, represents 24 ft.    (18 / 3  =  6  and 6 x 4  =  24)

The area =  pi · r².

If the diameter is 24 feet, its radius is 12 ft.

Area  =  pi · 12²  =  452.4 square feet.

The two smallest primes greater than 20 are 23 and 29.

The product of 23 and 29 is ...

V  =  pi · r² · h

V  =  pi · 2² · 60

V  =  753.6 cu cm  for each rod

192 rods  =  192 · 753.6  =  144,691.2 cu cm

In factored form:  3774  =  2 · 3 · 17 · 37

Creating two two-digit numbers:  3 · 17  =  51  and 2 · 37  =  74.

So, the two numbers are:  51  and  74.

The smaller is ...

If the glass holds  x  ounces of water and if it is  3/4s  full, then  (3/4)·x  =  6

Multiply both sides by 4:    3x  =  24.

Divide both sides by 3:        x  =  8.

See https://web2.0calc.com/questions/help_86416.

See https://web2.0calc.com/questions/help_79256.

There are three distinct integers n: 0, 2, 4.

There are 10 vertical angles.

The fourth number is -2 - 7i.

For your answer to part b, the graph you showed me. When you said "3 ways all the same" or "6 ways all the same". Is that assuming that the postcards were the same? 

Thanks!

Jon is hitting baseballs. When he tosses the ball into the air, his hand is 5 feet above the ground. He hits the ball when it falls back to a height of 4 feet. The height of the ball is given by h = - 16x^2 + 6t + 5 where in seconds How much time will pass before Jon hits the ball? What is the maximum height the ball attains?

I just redid part B my own way and it agrees with CPhil's answer ... but his is much easier.

Anyway, here is my logic

(b) I also bought 7 different postcards. How many ways can I send the postcards to my 3 friends, so that each friend gets at least one postcard?

The original answer was 15, that was if all the cards were the same.

So my answer agrees with CPhill's      1806 ways

12 c 4 = 495          11 c 4 = 330

You will need 12

Area of triangle = 1/2 b h     in a 45 90 45 triangle   b  and h are the same value

1/2 b^2 =32

b^2 =64

b = 8        she will nee two 8 ft pieces = 16 ft   PLUS the lenght of the hypotenuse

sqrt (8^2 + 8^2)    = hyp = 11.31 ft      (via pythag theorem)

11.31 + 16 = 27.3 ft of fence ~~ 27 ft

I answered these questions recently by the way.  I've never actually heard of stirling numbers before but i see Chris has answered part B easily using stirling numbers so maybe I should do some homework.   

Part A is fairly easy just using stars and bars which is a method that I often use.

(a) While travelling abroad, I bought 7 identical bags of candy for 3 friends. How many ways can I distribute the candy to my 3 friends, so that each friend gets at least one bag of candy?

I'd give one bag of candy to each friend leaving 4 bags to distribut between the 3 of them.

I'd use stars and bars method now.

there are 4 bags which are the stars.

I want them divided into 3 groups which means inerting 2 bars somewhere within the 4 stars

Person one gets the bags at the front, person 3 gets the bags at the end and person 2 gets the bags in the middle

eg

*|***|           person1 =1bag   person 2 = 3 bags     person 3=0 bags.

There will be 6C2 = 15 ways to do this.

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Part b is much harder and I was not sure if my answer last time was correct.

1 1/3 is to 1/5    as   x  is to 1

4/3 / 1/5  = x / 1

4/3 * 5 = x

20/3 = x

x = 6 2/3 cups wheat cereal

This is in Quadrant 4    cos and secant will be positive

sqrt(10^2 + (-4)^2 ) = sqrt116 = 2 sqrt 29    (hypotenuse)

Sec = 1/cos =  hyp/adj   =  2 sqrt 29 / 10         sqrt29/5

h = 4 = -16t^2 + 6t + 5

0 = -16t^2+6t +1                  Use Quadratic formula to find t = .5 s  or   -.125 (throw out)

max = - b/2a =  -6/-32 = .1875 sec

        -16(.1875^2) + 6 (.1875) + 5  = 5.56 ft 

Hi MG it isa great to see you on the forum again