Concentric circles are drawn with radii 1, 2, 3 … 100. The interior of th

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Concentric circles are drawn with radii 1, 2, 3 … 100. The interior of the smallest circle is colored black and the annular regions are colored alternately red and black, so that no two adjacent regions are the same color. The total area of the red regions divided by the area of the largest circle is

Options:

A . 12

B . 51100

C . 101200

D . 50101

Answer: Option C
:
C
Areas of circles follow the sequence 1,4,9,16,…………100

^{2}

Areas of Black & Red annuals – 1,3,5,7,9……………199 – General term 2n – 1
Red annuals – 3, 7, 11, …………….. 199 Sum =

(\frac{50}{2})

*(3 + 199)
Largest circle’s area = 100

^{2}

Ratio =

\frac{101}{200}

Note : Ignore pi as it is a ratio of areas
Shortcut: Reverse gear approach –
We know that the denominator is 100

^{2}

and num is an integer, hence the denominator cannot be 101 as it is in no-way a factor of 10000. Hence option d) has been eliminated.
We need to find

\frac{R}{(R + B)}

. R + B = 10000. Each of the red annual is greater in area by 2 units than the subsequent black annual and there are 50 such cases. Hence R = 5050 and B = 4950.

\frac{R}{(R + B)}

. =

\frac{5050}{10000}

=

\frac{101}{200}

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