White shapes and black shapes are used in a game. some of the shapes are circles. all the other sahpes 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.

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Step-by-step explanation:

BC = black circle

BS = black square

WC = white circle

WS = white square

Given:

(WC + WS) / (BC + BS) = 5 / 11

WC / WS = 3 / 7

BC / BS = 3 / 8

Find: (BC + WC) / (BC + BS + WC + WS)

Solve for WC and BC in the last two equations, then substitute into the first:

WC = 3/7 WS

BC = 3/8 BS

WC + WS = 5/11 (BC + BS)

3/7 WS + WS = 5/11 (3/8 BS + BS)

10/7 WS = 5/8 BS

WS = 7/16 BS

Therefore:

WC = 3/7 WS

WC = 3/16 BS

Substitute:

(BC + WC) / (BC + BS + WC + WS)

(3/8 BS + 3/16 BS) / (3/8 BS + BS + 3/16 BS + 7/16 BS)

(9/16 BS) / (2 BS)

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