Q18

Where is the shaded region?

Q19

In a span of 12 days, Rick had gone from "learning five times as long as Nic" to "learning twice as long as Nic".

We assume that they study at the same rate.

Let Nic's initial number of days studied be t days (during two weeks ago).

Rick's initial number of days studied is 5t days.

Twelve days later (i.e. two days ago),

Nic: (t + 12) days
Rick: (5t + 12) days

As it turned out, Rick ended up learning only twice as much as Nic, so

5t + 12 = 2 (t + 12)
5t + 12 = 2t + 24
3t = 12
t = 4

So two weeks ago, Nic studied for 4 days and Rick studied for 20 days.

Today,
Nic: 18 days
Rick: 34 days
Total: 52 days

This question can be done using ratio approaches also, by converting the ratios 5 : 1 and 2 : 1 into 5 : 1 and 8 : 4 respectively to reflect the case where the difference (of 3 units studied) remains unchanged.

Hi, for qn18 the shaded area is where the isosceles triangles overlap one another. Thank you

Q20

I believe that when 4 : 3 is being scaled, it will get "stuck" at 12 : 9 (because the limit for the 16 : 9 screen is 9 units tall)

So, 4/16 of the screen, or 1/4 of the screen, is black.

Similarly, when 16 : 9 is being scaled, it will get stuck at 4 : 2.25 (because the limit for the 4 : 3 screen is 4 units wide)

So, 0.75/3 of the screen, or 1/4 of the screen, is black.

The required ratio is 1/4 : 1/4, or 1 : 1.

Thank you very much, Happy Chinese New Year!