The figure, not drawn to scale, is made up of 3 circles. The ratio of the area of the smallest circle to the largest circle is 2:5 while the shaded area is 3/7 of the unshaded area. What is the ratio of the shaded area to the area of the smallest circle?
+118609
A,B and C represent the areas
\(\frac{C}{A+B+C}=\frac{2}{5}\qquad (1)\\~\\ B=\frac{3}{7}(A+B)\\ \frac{7B-3C}{3}=A \qquad (2)\\ sub \;\;2\;\; into \;\;1 \\ \frac{C}{\frac{7B-3C}{3}+B+C}=\frac{2}{5}\\ \frac{3C}{7B-3C+3B+3C}=\frac{2}{5}\\ \frac{3C}{10B}=\frac{2}{5}\\ \frac{C}{B}=\frac{20}{15}\\ \frac{C}{B}=\frac{4}{3}\\ \frac{B}{C}=\frac{3}{4}\\\)
LaTex:
\frac{C}{A+B+C}=\frac{2}{5}\qquad (1)\\~\\
B=\frac{3}{7}(A+B)\\
\frac{7B-3C}{3}=A \qquad (2)\\
sub \;\;2\;\; into \;\;1 \\
\frac{C}{\frac{7B-3C}{3}+B+C}=\frac{2}{5}\\
\frac{3C}{7B-3C+3B+3C}=\frac{2}{5}\\
\frac{3C}{10B}=\frac{2}{5}\\
\frac{C}{B}=\frac{20}{15}\\
\frac{C}{B}=\frac{4}{3}\\
\frac{B}{C}=\frac{3}{4}\\
+128475 Let C = area of smallest circle
Let B = area of next largest circle
Let A = area of largest circle
Shaded area = B - C
Unshaded area = C + ( A - B)
C / A = 2 / 5 ⇒ A = (5/2)C
Shaded Area / Unshaded Area =
(B - C) / [ C + A - B ] = 3/ 7
( B - C ) / [ C + (5/2)C - B ] = 3/7
(B - C) / [ (7/2)C - B ] = 3/7
3 [(7/2)C - B ] = 7 (B - C)
(21/2)C - 3B = 7B - 7C
(21/2)C + 7C = 7B + 3B
(35/2)C = 10B
(35/20) C = B
(7/4)C = B
C = (4/7)B
Unshaded Area / Smallest circle
(B - C) / C = [ (B - 4/7)B ] / [ (4/7)B ] = (3/7B) / [ 4/7) B = 3 / 4 [ As Melody found !!! ]
