Question

In a car factory, $6$ machines can make $6$ wheels in $6$ minutes.

How long will it take $30$ machines to make $30$ wheels?

Answer 1

Calculating the total time needed:

Step-1: Calculating the time need to make $6$ wheels by $30$ machines

Given that $6$ machines can make $6$ wheels in $6$ minutes.

Let $30$ machines take $t_1$ minutes time to make $6$ wheels. So, here the number of wheels is unchanged.

Hence, in tabular form, the above problem can be represented as:

Machines(No.)	$6$	$30$
Time(min)	$6$	$t_1$

Now, the number of machines is inversely proportional to time. So, if the time taken to make $6$ wheels by $30$ machines be $t_1$, then we must get:

$$\begin{gathered} 6\times 6=30t_1 \\ \Rightarrow t_1=\frac{36}{30} \\ \therefore t_1=\frac{6}{5} \end{gathered}$$

Therefore, $30$ machines take $\frac{6}{5}$ minutes time to make $6$ wheels.

Step-2: Calculating the total time needed to make $30$ wheels:

Let $30$ machines take $t_1$ minutes time to make $30$ wheels. So, here the number of machines is unchanged.

Hence, in tabular form, the above problem can be represented as:

Wheels(No.)	$6$	$30$
Time(min)	$\frac{6}{5}$	$t$

Now, the number of wheels is directly proportional to time. Hence, we must get:

$$\begin{gathered} \frac{6}{{\displaystyle \frac{6}{5}}}=\frac{30}{t} \\ \Rightarrow 6t=30\times \frac{6}{5} \\ \Rightarrow t=\frac{36}{6} \\ \therefore t=6 \end{gathered}$$

Therefore, $30$ machines will take $6$ minutes to make $30$ wheels.