Help: Easier questions from 2007 AMC (1 Viewer)

[HSC2014]

14. Two rectangular garden beds have a combined area of 40 m^2. The larger bed has twice the perimeter of the smaller and the larger side of the smaller bed is equal to the smaller side of the larger bed. If the two beds are not similar, and if all edges are a whole number of metres, what is the length, in metres, of the longer side of the larger bed?

If I let the larger rectangle have sides a,x and smaller rectangle b,x
then I have the following equations
ax + bx = 40 and
a+x = 2(b+x)

However I can't seem to solve for a using simultaneous equations - is this the wrong approach?

Also,
15. I take a two digit positive number and add to it the number obtained by reversing the digits. For how many two-digit numbers is the result of this process a perfect square?

I have no idea where to go from this and I feel like I'm missing some critical theory on perfect squares or something

 

[nightowl]

14. Two rectangular garden beds have a combined area of 40 m^2. The larger bed has twice the perimeter of the smaller and the larger side of the smaller bed is equal to the smaller side of the larger bed. If the two beds are not similar, and if all edges are a whole number of metres, what is the length, in metres, of the longer side of the larger bed?

If I let the larger rectangle have sides a,x and smaller rectangle b,x
then I have the following equations
ax + bx = 40 and
a+x = 2(b+x)

However I can't seem to solve for a using simultaneous equations - is this the wrong approach?

Also,
15. I take a two digit positive number and add to it the number obtained by reversing the digits. For how many two-digit numbers is the result of this process a perfect square?

I have no idea where to go from this and I feel like I'm missing some critical theory on perfect squares or something

You have

and

Using the second equation express x in terms of a and b.

Factorise the first equation and sub in for x. Now think about possible factors of 40.

Second question: ab+ba (where a and b are digits of your two digit number).

Your sum is then 10a+b + 10b+a=11(a+b)...