White shapes and black shapes are used in a game. Some of the shapes are circles. All the other shapes are squares. The ratio of the number of white shapes to the number of black shapes is 5 11 The ratio of the number of white circles to the number of white squares is 3 7 The ratio of the number of black circles to the number of black squares is 3 8 Work out what fraction of all the shapes are circles.

The best way to approach this problem is to assign values that fit the ratios.

First we consider the ratio of white and black shapes. Since it's [tex]5:11[/tex], let's assume that there are [tex]500[/tex] white shapes and [tex]1,100[/tex] black shapes.

Next we look at the ratio of white circles to white squares which is [tex]3:7[/tex]. This means that [tex] \frac{3}{10} [/tex] of the white shapes are circle and [tex] \frac{7}{10} [/tex] are square. This will lead us to have [tex] \frac{3}{10}*500=150 [/tex] white circles and [tex] \frac{7}{10}*500=350 [/tex] white squares.

Then, we look at the ratio of black circles to black squares ([tex]3:8[/tex]). We can get at the number of black circles and black squares by performing the same step as before:
[tex] \frac{3}{11}*1100=300 [/tex] black circles
[tex] \frac{8}{11}*1100=800 [/tex] black squares

Now that we have all the assumed values, we just proceed to get the fraction of circles by counting the assumed number of circles and dividing it by the total number of assumed shapes. (We can get the total number of assumed shapes by adding the number of black and white shapes).

[tex] \frac{150+300}{500+1100} = \frac{450}{1600}=\frac{9}{32} [/tex]

ANSWER: [tex] \frac{9}{32} [/tex] of all shapes are circles.

(This method will work since we have maintained the given ratios.)