+0

Circle

0
102
4

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?

Mar 3, 2022

#1
+117162
+1

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:

B=\frac{3}{7}(A+B)\\
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}\\

Mar 3, 2022
#2
+122390
+1

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

(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

(B - C)  / C  =  [ (B - 4/7)B ]  / [ (4/7)B ]   =  (3/7B) / [ 4/7) B =   3 / 4  [ As Melody found !!!  ]

Mar 3, 2022