Geometry Puzzle – Challenge 67

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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