Alan

Looks good to me Melody!

90% means 90 out of every 100, so: 90% of 630,000 is

$${\frac{{\mathtt{90}}}{{\mathtt{100}}}}{\mathtt{\,\times\,}}{\mathtt{630\,000}} = {\mathtt{567\,000}}$$

10% off means they will pay 90% of the basic cost.  90% of 420 is

$${\frac{{\mathtt{90}}}{{\mathtt{100}}}}{\mathtt{\,\times\,}}{\mathtt{420}} = {\mathtt{378}}$$   so they pay $378

Alternatively, you can take 10% of $420 (that is, one-tenth of $420), namely $42, and subtract this from $420.

$420 - $42 = £378

You didn't say what x is; however, it becomes fairly obvious from the following:

Methinks there is no positive integer solution to this! See below:

3.  If there were a net force on the plane it would accelerate (or decelerate) - i.e. its speed would be changing - but the plane is travelling at constant speed so there must be zero net force on it.

4.  Think of light going from, say, glass to air.  Total internal reflection can occur.  The refractive index of glass is greater than that of air.

5.  273K (approx).

6.  This is already done in the picture!

Use the chain rule to get:

$$\frac{d\ln{p}}{dt}= \frac{d\ln{p}}{dp}\frac{dp}{dt}=\frac{1}{p}\frac{dp}{dt}$$

Can find this from:

$$\sum_{k=1}^nk^3=\frac{n^2(n+1)^2}{4}$$

$${\frac{{\left({\mathtt{2\,014}}{\mathtt{\,\times\,}}{\mathtt{2\,015}}\right)}^{{\mathtt{2}}}}{{\mathtt{4}}}} = {\mathtt{4\,117\,267\,101\,025}}$$