# Geometry Puzzle – Challenge 67

Time for another mind-blowing math problem to tease your brain and improve your logic skills!

## Challenge:

In the figure above, there are three connected wheels. The ratio of the radius of wheel B to the radius of wheel A is 1:3 and the ratio of the radius of wheel A to radius of wheel C is 2:3. When wheel B makes 18 revolutions, how many revolutions does wheel C make?

A- 2

B- 4

C- 8

D- 54

E- 64

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Since, the three wheels are connected; we need to find the ratio of the circumference of all wheels:
The circumference of a circle = 2πr
The ratio of the radius of wheel B to wheel A is 1 to 3. Let $$r_{A }$$ be the radius of the wheel A and $$r_{B}$$ be the radius of the wheel B. Then:
$$r_{A }= 3r_{B}$$
Circumference of the wheel $$A = 2π r_{A }$$
Replace $$r_{A }$$ with $$r_{B }$$. Then:
Circumference of the wheel $$A = 2π r_{A } = 2π (3r_{B })= 6π r_{B }$$
The ratio of the circumference of wheel B to A is:
$$2π r_{B} : 2πr_{A } → 2π r_{B } : 6π r_{B} → 2 : 6 → 1 : 3$$
Therefore, the ratio of the circumference of wheel B to A is 1 to 3. (Similar to their radius)
With the same method, the ratio of the circumference of the wheel A to C is 2 to 3.
Therefore, the ratio of the wheels B, A and C is:
B : A =1 : 3 = 2 : 6
A : C = 2: 3 = 6: 9 → B : A : C = 2 : 6 : 9
The ratio of the circumference of the wheel B to the wheel C is 2 to 9 or:
The circumference of the wheel B $$= \frac{2}{9}$$ the circumference of the wheel C.
B = 18 → C $$= 18 (\frac{2}{9}) = 4$$
When wheel B makes 18 revolutions, wheel C makes 4 revolutions.

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