## 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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The correct answer is B.

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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