Alan

Rewrite as x^2 - 6x + k  = 0

This has at least one real solution as long as the discriminant is greater than or equal to zero.

i.e.  6^2 - 4k >= 0

4k <= 36

k<= 9

So largest value of k is 9.

Cheetah does C pages/min;   

Monkey does M pages/min;

Sloth does S pages/min.

Total number of pages = P

(C + M)*45 = P

(C + S)*60 = P

(M + S)*90 = P

Solve these for C, M and S in terms of P

C = P/72;   M = P/120;  S = P/360

So (C + M + S)*time = P   or  (P/72 + P/120 + P/360)*time = P

I'll let you finish.

1 does not equal 0.999

However, it does equal 0.9999..... etc.  i.e. an infinite number of 9s.

To see this, let  

x = 0.9999.....   etc.

10x = 9.9999..... etc

10x - x = 9

9x = 9

x = 1

There are 8 vans and 52 non-vans.  Given that a non-van leaves, there are 8 vans and 51 non-vans, or 59 vehicles in total.

Hence, the probability that a van leaves next is 8/59.

Just by writing the first few terms the pattern becomes clear.  In general \(f_n(x)=\frac{x}{nx+1}\)

Like so:

As follows:

Assuming Shamique is asking the question, then, in equation form the information is:

S = 2b           (1)              where S is Shamique's age, b is her brother's age

S = s - 5        (2)              where s is her sister's age

s + 5 = 2(b + 5)              However, this is not independent information as it can be derived from equations (1) and (2), so we ignore it!

S + 7 = (3/4)(s + 7)    (3)

We can now solve equations (1), (2) and (3) for S, b and s.

Can you take it from here?

(You should get S = 8, b = 4 and s = 13).

I think there are three values:

(Just noticed that there should be a dot where the steepest line meets the x-axis!).

Like so