Question

A machine takes 1 minute to full 6 cartons of eggs. At this rate, how long will it take to fill 420 cartons?

Answer 1

Hint: We use the concept of unitary system. We find the time that is required to fill 1 carton in the form of a fraction. Then we multiply with 420 and we simplify the fraction to find the solution.

Complete step-by-step solution:
It’s given that the machine takes 1 minute to full 6 cartons of eggs.
We have to find the time that is required to fill 420 cartons.
We use the concept of unitary system. The unitary method is a method in which we find the value of a unit and then the value of a required number of units. The unitary method is also used to find the value of a single unit from a given multiple.
To fill 1 carton of egg, we need $\dfrac{1}{6}$ minutes.
Therefore, for 420 cartons we need $\dfrac{420}{6}$ minutes.
We need to find the simplified form of the proper fraction $\dfrac{420}{6}$.
Simplified form is achieved when the G.C.D of the denominator and the numerator is 1.
For any fraction $\dfrac{p}{q}$, we first find the G.C.D of the denominator and the numerator. If it’s 1 then it’s already in its simplified form and if the G.C.D of the denominator and the numerator is any other number d then we need to divide the denominator and the numerator with d and get the simplified fraction form as $\dfrac{{}^{p}/{}_{d}}{{}^{q}/{}_{d}}$.
For our given fraction $\dfrac{420}{6}$, the G.C.D of the denominator and the numerator is 6.
$\begin{align}
  & 2\left| \!{\underline {\,
  6,420 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3,210 \,}} \right. \\
 & 1\left| \!{\underline {\,
  1,70 \,}} \right. \\
\end{align}$
The GCD is $2\times 3=6$.
Now we divide both the denominator and the numerator with 6 and get $\dfrac{{}^{420}/{}_{6}}{{}^{6}/{}_{6}}=\dfrac{70}{1}$.
Therefore, the time that is required to fill 420 cartons is 70 minutes.

Note: We can also solve the problem using the concept of proportionality. This denotes the ratio of two different amounts being equal. For our given problem the ratio of the number of cartons to the time will be equal in two different cases. If we assume the required time is $x$, we get $1:6::x:420$ which in fraction form gives $\dfrac{1}{6}=\dfrac{x}{420}$. The solution is $x=\dfrac{420}{6}=70$.