Alan

    6 = 2*3

  18 = 3*6

  72 = 4*18

360 = 5*72

 xxx = 6*360

 yyy = ?*xxx

I'm sure you can work out what xxx is.

Looking at the pattern, what do you think ? is?  With it and xxx you can work out what yyy is.

It doesn't!

But it does equal zero:

$${{\mathtt{e}}}^{\left({i}{\mathtt{\,\times\,}}{\mathtt{\pi}}\right)}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}} = {\mathtt{0}}$$

This is a famous equation known as Euler's equation. 

Take logs of both sides:  ln(2^x3^x-1)=ln(5^x-1)         ln is the natural, or Napierian logarithm.

Using a property of logarithms (see the Formulary on this site) we can write ln(2^x)+ln(3^x-1) = ln(5^x-1)

Using another property of logarithms (again, see the Formulary) this becomes x*ln(2) + (x-1)*ln(3) = (x-1)*ln(5)

Expand the bracketed terms and collect:  x*(ln(5) - ln(2) - ln(3)) = ln(5) - ln(3) or x*ln(5/6) = ln(5/3) so x = ln(5/3)/ln(5/6) 

$${\mathtt{x}} = {\frac{{ln}{\left({\frac{{\mathtt{5}}}{{\mathtt{3}}}}\right)}}{{ln}{\left({\frac{{\mathtt{5}}}{{\mathtt{6}}}}\right)}}} = {\mathtt{x}} = -{\mathtt{2.801\: \!784\: \!016\: \!923\: \!930\: \!8}}$$

It has no solutions.  Draw a graph of each side of the equation against x and you will see they never intersect.

Let the price he paid for the first item be f, and the price he paid for the second item be s.

We are told that:  f + s = 90  ...(1)

What he got for the first item is: 1.05*f and on the second he got 1.5*s. The sum of these two equals what he got in total, which we are told is 1.4*90, so:   1.05*f + 1.5*s = 1.4*90   ...(2)

Rearrange eqn (1) to get:  f = 90 - s.  Put this in eqn (2) to get 1.05*(90 - s) + 1.5*s = 1.4*90 or

1.05*90-1.05*s+1.5*s = 1.4*90

See if you can take it from here.  Having found s, you can substitute this back into eqn (1) to find f.

$$Starting with:
$$\frac{1-\sqrt{3}}{2\sqrt{3}}+\frac{\sqrt{3}}{2}$$

Multiply left term by $\sqrt{3}$ top and bottom:
$$\frac{(1-\sqrt{3})\sqrt{3}}{6}+\frac{\sqrt{3}}{2}$$

Multiply out the bracketed term:
$$\frac{\sqrt{3}-3}{6}+\frac{\sqrt{3}}{2}$$

Multiply right term by 3 top and bottom:
$$\frac{\sqrt{3}-3}{6}+\frac{3\sqrt{3}}{6}$$

Collect terms:
$$\frac{4\sqrt{3}-3}{6}$$

Rewrite as:
$$-\frac{1}{2}+\frac{2\sqrt{3}}{3}$$$$

so a is -(1/2), b is 2 and c is 3

For the second part:

At the end there is 225 L  (the value given)

At the start there is 306.67 L (the value of w that we've just worked out)

So we end up with less than we started with.  

Express end as a percentage of start by dividing 225 by 306.67 and multiplying by 100:

$$\left({\frac{{\mathtt{225}}}{{\mathtt{306}}}}\right){\mathtt{\,\times\,}}{\mathtt{100}} = {\mathtt{73.529\: \!411\: \!764\: \!705\: \!882\: \!4}}$$

At the end we have about 73.5% of what we started with.  Therefore we have a reduction of (100-73.5)% or 26.5%

"i understand everything in your answer alan but all untill the part that you say to multiplay it all by 10/9, why 10/9 if its 0.9? is it because i switch the sides of the 0.9?"

Sorry, I should have explained more carefully!  0.9 is the same as 9/10, so by multiplying by 10/9 we eliminate it from the left-hand side of the equation.  Perhaps I should have simply said divide both sides by 0.9.

Start by letting the original volume of water in the pool be, say, w litres. 

First step is to add 160 to this: w+160

Reducing the result by 10% is to be left with 90%: (w+160)*0.9      (0.9 is 90% as a fraction).

Take out 195 litres:  (w+160)*0.9 - 195, and this equals 225, so:

(w+160)*0.9 - 195 = 225

Add 195 to both sides:  (w+160)*0.9 = 420

Multiply both sides by 10/9:  w+160 = 4200/9 = 466.67  (actually 466 and two-thirds)

Subtract 160 from both sides to get: w = 306.67

If you draw a graph of this function you will see there are two real roots at x = 2.561553 and x = 3.117176 (approximately).