CPhill

He travels   (m / h) miles in one hour = Rate

And

Distance  / Rate =  Time

So

h  / (m / h)    = 

( h / 1) * (h / m)  =

h^2 / m   (hours)

18% per month.....were can I invest???

Anyway.....we have

150 ( 1.18)^18   =   $ 2950.99

Show that it is true for n = 1

(1)^3  - 1   =   0         and this is divisible by 3

Assume that it is true for    n = k

That is     k^3 - k     is divisible by 3

Note that we can write this as   k (k^2 - 1)  = k (k + 1) (k - 1) = (k - 1)(k)(k + 1)

Prove that it is true for    k + 1

(k + 1)^3  -   (k + 1)       factor

(k + 1) [ (k + 1)^2 - 1]

( k + 1) ( k^2 + 2k + 1  - 1 ]

(k + 1) [ k^2 + 2k ]

(k + 1) [ k ( k + 2) ] =

k (k + 1) (k + 2) =

k(k + 1) ( k + 3 - 1) =

k(k + 1) ( k - 1 + 3)  =

k(k + 1) (k - 1) + 3k(k+1)

(k-1) (k) (k + 1) + 3k(k + 1)

And we assumed that the first term was divisible by 3

And the second term is a multiple of 3, so it is also divisible by 3

So.... (k + 1)^3 - (k + 1) is divisible by 3

If  y varies directly with x, then    y / x    must be the same constant

So

11 / 7  ≈ 1.57

But

13 / 8   = 1.625

So.....there is no direct variation

 ln [ (x-3)(x+1) ]  = 2ln3       we can write

ln [ ( x - 3) ( x + 1) ] = ln 3^2      forget the natural logs

(x - 3) ( x + 1) =  3^2       simplify

x^2 - 2x - 3  = 9    subtract 9 from both sides

x^2 - 2x - 12 =  0         add 12 to both sides

x^2 - 2x    =  12        take (1/2) of 2 = 1....square this = 1...add to both sides

x^2 - 2x + 1  =    13      factor the left side

(x - 1)^2  =  13     take both roots

x - 1   = ±√13     add 1 to both sides

x = ±√13  + 1

Well sorry.....I can't promise "INSTANT" results.....

Each nickel is wort 5 cents, and each dime is worth 10 cents

So

4 nickels * 5cents   +   7(4) dimes * 10 cents =

20 cents +   28 * 10 cents =

20 cents + 280 cents =

300 cents =

$3.00

Let the number of nickles = N

Then....the number of dimes = 7N

And

No. of nickles * their value each  + No. of dimes * their value each =  total value

So we have

N(5) + 7N(10 )  = 300        simplify

5N + 70N     =  300

75N =  300        divide both sides by 75

4 = N     ⇒    [ 4 nickles ]

She would first pay    20%  of  [ 43000 - 11000]  =  .20 [ 32000] =  £ 6400

Additionally  she would pay 40% of   [ 45600 - 43000 ] = .40 [ 2600] =  £ 1040

So....her total tax bill is

£ [ 6400 + 1040 ] =      £ 7440

1x^2 + 3x - 5   = 0

We have the form   ax^2 + bx + c   = 0

Where a = 1     b = 3      and    c   = -5

The quadraic formula   is

[  - b ± √ [  b^2  - (4 * a * c) ]   ]     /    [ 2 * a ]

Plugging in for a, b, c......we have

[ - 3 ± √ [ 3^2 - (4 * 1 * -5) ] ]  /   [ 2 * 1 ]       simplify

[ -3  ± √ [ 9 - ( -20) ] } /  2

[ - 3  ± √ 29   ]    /   2

And the two solutions as are tertre found

x = [ -3  + √ 29   ]    /   2        and       x =    [- 3  - √ 29   ]    /   2