# A gear system has a driving wheel of radius $2\,{\text{cm}}$ and driven wheel of radius $20\,{\text{cm}}$. If the driven wheel has 40 teeth, find the number of teeth in the driving wheel.A. $4$B. $8$C. $6$D. $2$

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Hint:Use the formula for gear ratio in terms of radius of the driving wheel and radius of the driven wheel and determine the gear ratio for the system of given driving and driven wheels. Also use the formula for gear ratio in terms of the number of teeth in the driving wheel and number of teeth in the driving wheel. Substitute the value of gear ratio and number of teeth in the driven wheel and calculate the required answer.

Formulae used:
The gear ratio of the wheels is given by formula
${\text{Gear ratio}} = \dfrac{{{\text{Radius of driving wheel}}}}{{{\text{Radius of driven wheel}}}}$ …… (1)
The gear ratio of the wheels is also given by formula
${\text{Gear ratio}} = \dfrac{{{\text{Number of teeth driving wheel}}}}{{{\text{Number of teeth driven wheel}}}}$ …… (2)

We have given that the radius ${r_A}$ of the driving wheel is $2\,{\text{cm}}$ and the radius ${r_B}$ of the driven wheel is $20\,{\text{cm}}$.
${r_A} = 2\,{\text{cm}}$
$\Rightarrow{r_B} = 20\,{\text{cm}}$
Let us calculate first the gear ratio.We can determine the gear ratio using equation (1).Substitute ${r_A}$ for radius of the driving wheel and ${r_B}$ for radius of the driven wheel in equation (1).
${\text{Gear ratio}} = \dfrac{{{r_A}}}{{{r_B}}}$
Substitute $2\,{\text{cm}}$ for ${r_A}$ and $20\,{\text{cm}}$ for ${r_B}$ in the above equation.
${\text{Gear ratio}} = \dfrac{{2\,{\text{cm}}}}{{20\,{\text{cm}}}}$
$\Rightarrow {\text{Gear ratio}} = \dfrac{1}{{10}}$
Hence, the gear ratio for the given system of the driving wheel and driven wheel is $\dfrac{1}{{10}}$.We have given that the number of teeth in the driven wheel is 40.
${n_B} = 40$.We have asked to determine the number of teeth ${n_A}$ in the driving wheel.Substitute ${n_A}$ for the number of teeth in the driving wheel and ${n_B}$ for the number of teeth in the driven wheel in equation (2).
${\text{Gear ratio}} = \dfrac{{{n_A}}}{{{n_B}}}$
Rearrange the above equation for ${n_A}$.
${n_A} = {n_B}\left( {{\text{Gear ratio}}} \right)$
Substitute 40 for ${n_B}$ and $\dfrac{1}{{10}}$ for ${\text{Gear ratio}}$ in the above equation.
${n_A} = 40\left( {\dfrac{1}{{10}}} \right)$
$\therefore {n_A} = 4$
Therefore, the number of teeth in the driving wheel are 4.

Hence, the correct option is A.

Note: The students should keep in mind that if the wheels are not specified as the driving wheel and the driven wheel then one should keep in mind that the wheel with more teeth or larger radius is the driven wheel and the wheel with the less teeth or smaller radius is the driving wheel. This will make the process of identifying the wheels easy.