Alan

Assuming no-one had both pasta and chicken

1/3 + 5/12 → 4/12 + 5/12 → 9/12 → 3/4  i.e. 3/4 of the guests ate the pasta and chicken 

Assuming the remaining 1/4 ate the beef, then  (1/4)N = 24   (where N is total number of guests).

 N = 24*4  or N = 96  i.e. there are 96 guests in total.

See https://web2.0calc.com/questions/help_24903 now.

Here's #1 for you:

Collect the terms on right hand side over a common denominator:

\(\frac{A}{x-2}+\frac{B}{x+1}\rightarrow\frac{A(x+1)+B(x-2)}{(x-2)(x+1)}\rightarrow\frac{(A+B)x+A-2B}{x^2-x-2}\)

Compare this with \(\frac{x+7}{x^2-x-2}\)

To make them the same compare the numerators.  It is clear we must have:

A + B = 1   and

A - 2B = 7

Subtract the second from the first to get: 3B = -6  so B = -3

Put this back into the first to get A - 3 = 1  so  A = 4

Hence (A, B) = (4, -3)

For #2  

Set x = each term in turn and solve for x.  if the result lies in the range given for each term then that result is valid.  If not it isn't.

For #3

Can't see the whole expression!

For #4

First plug sqrt(2) into h to get, say m = h(sqrt(2))

Then plug m into h to get h(m) which is h(h(sqrt(2))) 

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Simply add up all the numbers in the f column to get a total, then divide each number in the f column by that total to get the individual p values.

f(x) = x² + 12

f(m) = m² + 12

f(3m) = (3m)² + 12 → 9m² + 12

f(3m) = 3f(m)

9m² + 12 = 3(m² + 12)

9m² + 12 = 3m² + 36

6m² = 24

m² = 4

m = 2  (or -2) 

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Yes and yes (I confess I didn’t notice the “per wheel” bit!).

the number of eggs collected each day on a farm.  Yes, if it’s a large egg farm

the number of yolks in randomly selected eggs.   No, the number of yolks in each egg will only be a small number like 1 (mostly), a few with 2 and possibly the occasional 3.

the weights of eggs in the kitchen of a restaurant.  Yes, if there are a large number of eggs.

the number of eggs in cartons sold at a supermarket. No, this will be a few discrete numbers like, 6, 12 etc.

≈≈≈Soccer balls:

Number of standard deviations above the mean = (0.45 - 0.43)/0.01 → 2

A proportion of ≈ 0.023 lie above 2 standard deviations (from tables)

0.023*800 ≈ 18

Ambulances:

Stdevs below mean = (400 - 450)/50 = -1

Stdevs above mean = (500 - 450)/50 = 1

Proportion between these: 0.843 - (1-0.843) = 0.686

Expect 0.686*97 = 67

Here are parts (3) and (4):

I'll leave you to put in a value for g and hence work out k (make sure your units are consistent).

Here are parts (1) and (2)

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