Question

White shapes are 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 7:3
The ratio of the number of white circles to the number of white squares is 2:7
The ratio of the number of black circles to the number of black squares is 1:2

Work out what fraction of all the shapes are circles.
Give your answer as a fraction in its simplest form.

Answer 1

23/90.


——
White Circles 2:7 has a sum of 9.
2:7 = 2/7, where the 2 is circles.
2/9 = the ratio or amount of white circles of all white shapes.

Black Circles 1:2 has a sum of 3.
1:2 = 1/2, where the 1 is circles.
1/3 = the ratio or amount of black circles of all black shapes.


Now, back to the ratio of white to black shapes 7:3. Sum: 7+3=10 shapes.
7/10, where the 7 is white and 10 is black.

Multiply the total shapes by the circle ratios:
The total number of white circles of all the shapes is (2/9) x (7/10).
This simplifies to 2*7 / 9*10 = 14/90 (don’t simplify yet).

The total number of black circles of all the shapes is (1/3) x (3/10).
This simplifies to
1*3 / 3*10 = 3/30 (don’t simplify, instead multiply by 3 for the common denominator 90). (3/30) x 3 = 9/90.

Now we have 2 fractions; one for the white circles and black circles, respectively.

Add them together: 9/90 + 14/90 = 23/90.
This doesn’t simplify further, so the fraction of the shapes that is circles = 23/90.