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All 30 students in a class study at least one of the two subjects History and Geography.
Twice as many study History as Geography. 8 students study only Geography.
By drawing a Venn diagram, or otherwise, find the number of students who study both History and Geography.
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Let x students study only History;
y students study only Geography;
z students study both subjects.
Then we have these three equations
x + y + z = 30 (1)
x + z = 2*(y+z) (2)
y = 8 (3)
Substutitute y= 8 from equation (3) into equations (1) and (2)
x + 8 + z = 30 (1')
x + z = 2*(8+z) (2')
Simplify
x + z = 22 (1'')
x - z = 16 (2'')
From equations (1'') and (2'') you get by adding 2x = 22+16 = 38, x = 38/2 = 19.
From equations (1'') and (2'') you get by subtracting 2z = 22-16 = 6, z = 6/2 = 3.
Thus 19 students study only History; 8 students study only Geography, and 3 students study both subjects.
ANSWER. 3 students study both subjects.
Solved.
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With no doubts, it can be solved by other ways, too, but my goal was to present an Algebra solution
in the most clear form without explicit using of Venn diagram.