Draw your triangles on a piece of paper. It is easier to see the problem if you separate inscribed triangles and draw them individually in a row with the similar angles in the same places. For example: if you draw a 30-60-90 triangle with the 90 degree angle in the bottom left corner you should draw your similar triangle or triangles with their 90 degree angles in the bottom left corner.
Label the pieces of the triangles you have been given. Since your triangles are drawn in a row with corresponding sides in the same places, it is easy to see which sides you have to make your ratio. For example, if you have the hypotenuse of both triangles labeled, these are the sides you will use.
Make a ratio of your two sides. If the hypotenuse of the first triangle is 6 and the hypotenuse of the second triangle is 12 your ratio is 6:12, or reduced to 1:2. This ratio will be the same for all sides of the triangle.
Find the missing sides using the ratio. If you have the short leg of the small triangle in this example it is 3; to find the short leg of the large triangle, you set up the ratios, 1:2 equals 3:x. Cross multiply 1 * x = 3 * 2. So x = 6. The short leg of your large triangle is 6.
Repeat step 4 to solve for the other missing leg.