Many thanks to Michael from Worth School who has sent us a solution to this difficult problem. Michael says:

The area of triangle 1 is a half of the top left hand quarter:
Area 1 = 1/4 x 1/2 = 1/8
The area of triangle 2 is a half of a half of a half of a quarter:
Area 2 = 1/4 x 1/2 x 1/2 x 1/2 = 1/32
And so on:
Area 3 = 1/4 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 = 1/128
Area 4 = 1/4 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2= 1/512
Area 5 = 1/4 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2= 1/2048
Area (1+2) = 1/8 + 1/32 = 5/32 = 0.15625
Area (1+2+3) = 1/8 + 1/32 + 1/128 = 21/128 = 0.1640625
Area (1+2+3+4+5) = 341/2048 = 0.16650390625

1/6 = 0.166666666

The area seems to be tending towards 1/6.

Zeno's Paradox states that motion is impossible because in order to get from A to B, one must first pass through the mid-point C. But to get from A to C, you must first pass through the midpoint D, and so on passing through midpoints E, F, G, etc.
This is similar in to the triangle problem, since there are an infinite number of triangles, most of which have tiny areas. It is for this reason that we say that the area 'tends towards 1/6th'.
The way to resolve these two problems is very similar. In the triangle question we have:
1/8 + 1/32 + 1/128 + 1/512 + 1/2048 + ... = 1/6

In Zeno's Paradox, assuming that you can cover this infinite amount of small distances, you travel:
1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ... = 1

This means that there is not really any problem, since the sum of the tiny distances equals 1.

A very clear explanation, thank you Michael.

Bryony, Louise and Sarah from Caldicot School in South Wales sent in images in Word documents which show their solutions. Thank you girls - this makes it much easier to see the fractions. Here is Bryony's , here is Louise's and here is Sarah's .