Question

A ton is a measure of volume frequently used in shipping but that use requires some care because there are at least three types of tons: a displacement ton is equal to 7 barrels bulk, a freight ton is equal to 8 barrels and a register ton is equal to 20 barrels bulk (1 barrel bulk\[=0.1415{{m}^{3}}\]). Suppose you got a shipping order for 73tons of M & M candies and you are certain that the client who sent the order intended to refer to volume (instead of weight or mass). If the client actually meant displacement tons, how many extra U.S bushels of the candies will you erroneously ship if you interpreted the order as 73 freight tons?

Answer 1

Hint: We are given three different units which name the same. We need to convert them all to the same unit and then they can be compared to get an idea of how much of one unit is the other. We can convert all the given units to U.S Bushels in this situation.

Complete answer:
We are all familiar with the usage of the term ton in our daily life. But it will be exciting to know that there are three types of ton.
We can summarise the different units as –
\[\begin{align}
  & \text{1 displacement ton = 7barrels bulk} \\
 & \text{1 freight ton = 8 barrels bulk} \\
 & \text{1 register ton = 20 barrels bulk} \\
\end{align}\]
We know that in the given situation 73 displacement tons is misunderstood as 73 freight tons. We can convert all the units to the U.S Bushels and find the difference involved.
\[\text{1 Barrel bulk = 4}\text{.0155U}\text{.S Bushel = 0}\text{.1415}{{\text{m}}^{3}}\]
Let us convert all the ‘ton’ units into U.S Bushels as –
\[\begin{align}
  & \text{1 displacement ton = 7barrels bulk}\times \text{4}\text{.0155 = 28}\text{.1085U}\text{.S Bushels} \\
 & \text{1 freight ton = 8 barrels bulk}\times \text{4}\text{.0155 = 32}\text{.124U}\text{.S Bushels} \\
 & \text{1 register ton = 20 barrels bulk}\times \text{4}\text{.0155 = 80}\text{.31U}\text{.S Bushels} \\
\end{align}\]
Now, we can calculate the excess of candies shipped due to the misunderstanding.
\[\begin{align}
  & \text{The change in volume = 73freight ton - 73displacement ton} \\
 & \Rightarrow \text{ }\Delta \text{V = (73}\times \text{32}\text{.124 - 73}\times \text{28}\text{.1085 )U}\text{.S Bushels} \\
 & \Rightarrow \text{ }\Delta \text{V = 73}\times 4.0155\text{ U}\text{.S Bushels} \\
 & \Rightarrow \text{ }\Delta \text{V = 293}\text{.1315 U}\text{.S Bushels} \\
\end{align}\]
So, the excess amount of candies shipped will be about 293.1315 U.S Bushels.

Note:
We understand from the description in the question that there are different measures of ton which need to be dealt carefully. We need to analyse a situation knowing which type of quantity is used to avoid unnecessary confusions and mistakes.