Question

A coffee container is \[\dfrac{3}{5}\] full of beans. These beans are then put into another container having volume thrice of the first one. What fraction of the large box is filled with beans?

Answer 1

Hint: Let us assume the volume of the coffee container is equal to V. Let us assume the volume of beans in the container is equal to \[{{V}_{1}}\]. From the question, we were given that a coffee container is \[\dfrac{3}{5}\] full of beans. Now we were given that the volume of another container is equal to thrice of the coffee container. Let us assume the volume of another container is equal to \[{{V}_{2}}\]. According to these assumptions, we can find the fraction of the large box is filled with beans.

Complete step by step answer:
First of all, let us assume the volume of the coffee container is equal to V. Let us assume the volume of beans in the container is equal to \[{{V}_{1}}\]. From the question, we were given that a coffee container is \[\dfrac{3}{5}\] full of beans. So, now we get the ratio of \[{{V}_{1}}\] and V is equal to \[\dfrac{3}{5}\]. So, we get
\[\Rightarrow \dfrac{{{V}_{1}}}{V}=\dfrac{3}{5}\]
Now by using cross multiplication, then we get
\[\Rightarrow {{V}_{1}}=\dfrac{3V}{5}...(1)\]
Now we were given that the volume of another container is equal to thrice of a coffee container. Let us assume the volume of another container is equal to \[{{V}_{2}}\].
\[\Rightarrow \dfrac{{{V}_{2}}}{V}=3\]
Now by using cross multiplication, we get
\[\Rightarrow \dfrac{{{V}_{2}}}{3}=V....(2)\]
Now let us substitute equation (2) in equation (1), then we get
\[\begin{align}
  & \Rightarrow {{V}_{1}}=\dfrac{3\left( \dfrac{{{V}_{2}}}{3} \right)}{5} \\
 & \Rightarrow {{V}_{1}}=\dfrac{{{V}_{2}}}{5} \\
 & \Rightarrow \dfrac{{{V}_{1}}}{{{V}_{2}}}=\dfrac{1}{5}...(3) \\
\end{align}\]

So, from equation (3), we can say that in the second container \[\dfrac{1}{5}\] fraction of the large box is filled with beans. So, it is clear that there are \[\dfrac{1}{5}\] fraction of beans in second coffee container.

Note: Students may have a misconception that \[\dfrac{{{V}_{2}}}{V}=\dfrac{1}{3}\]. But we know that \[\dfrac{{{V}_{2}}}{V}=3\]. If this misconception is followed, then we cannot get the correct answer. So, we should be able to avoid this misconception to have a correct answer. Students should also be careful in the calculation part. If a small mistake is done, then we cannot get the exact answer.