Question 746229
please help me solve this equation ; the circle is divided into 4 parts a,b,c,d where part a & c form a semicircle and part b&d form the other semicircle area of part a to area of part c is in the ratio 1: 3 the area of part b to the area of part d is in the ratio 1:2. the area of part c is bigger than part b by 20 SquareCentimeters find the area of the whole cirle.
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Ok, let solve it one by one,
Let,
At = Total area of circle
Aa = Area of a
Ab = Area of b
Ac = Area of c
Ad = Area of d
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part a & c form a semicircle...part b&d form the other semicircle:
{{{At/2 = Aa + Ac}}}, and {{{At/2 = Ab + Ad}}}
part a to area of part c is in the ratio 1:3...{{{Aa/Ac=1/3}}}, or {{{Ac=3Aa}}}
part b to the area of part d is in the ratio 1:2...{{{Ab/Ad=1/2}}}, or {{{Ad=2Ab}}}
the area of part c is bigger than part b by 20...{{{Ac = Ab + 20}}}
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Ok, here is my favorite part:
We have 5 unknown (At, Aa, Ab, Ac, and Ad), so we have to form 5 equations to solve it. The 4 equations are ready above, 
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since a & c and b & d are semi circle, {{{At/2=At/2}}}, equation 5
we have the 2 equations for that above, substituting here, {{{Aa + Ac = Ab + Ad}}}, or {{{-Aa + Ab - Ac + Ad = 0}}}, then add this to the other 3 remaining equation: {{{(-Aa + Ab - Ac + Ad) + (3Aa-Ac) + (2Ab-Ad) + (-Ab + Ac - 20) = 2Aa + 2Ab - Ac - 20=0}}}
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but {{{ Aa = Ac/3}}} and {{{Ab=Ac-20}}}, so, {{{2(Ac/3) + 2(Ac-20)- Ac - 20=0}}},
So, {{{Ac = 36 cm^2}}}, {{{Ab = 16 cm^2}}}, {{{Ad=32 cm^2}}}, and {{{Aa=12 cm^2}}}
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Therefore, {{{At = Aa + Ab + Ac + Ad = 96 cm^2}}}

Check it, the value of individual area must satisfy the given ratio.