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 bulk, and a register ton is equal to 20 barrels bulk. A barrel bulk is another measure of volume: $\mathbf{1}\mathbf{}\mathbf{barrel}\mathbf{}\mathbf{bulk}\mathbf{}\mathbf{=}\mathbf{}\mathbf{0}\mathbf{.}\mathbf{1415}\mathbf{}{\mathbf{m}}^{\mathbf{3}}$. Suppose you spot a shipping order for “73 tons” of$\mathbf{M}\mathbf{\&}\mathbf{M}$candies, and you are certain that the client who sent the order intended “ton” to refer to volume (instead of weight or mass, as discussed in Chapter 5). If the client actually meant displacement tons, how many extra U.S. bushels of the candies will you erroneously ship if you interpret the order as (a) 73 freight tons and (b) 73 register tons? ($\mathbf{1}{\mathbf{m}}^{\mathbf{3}}\mathbf{=}\mathbf{28}\mathbf{.}\mathbf{378}$U.S. bushels.)

Short answer

(a). When the order is interpreted as 73 freight tons, the extra U.S. bushels are 293 U.S. bushels.

(b). When the order is interpreted as 73 register tons, the extra U.S. bushels are $\mathbf{3}\mathbf{.}\mathbf{81}\mathbf{\times }{\mathbf{10}}^{\mathbf{3}\mathbf{}}\mathbf{U}\mathbf{.}\mathbf{S}\mathbf{.}\mathbf{}\mathbf{bushels}$.

Step-by-step solution

Given data

$$\begin{gathered} 1\mathrm{displacement}\mathrm{ton}=7\mathrm{barrel}\mathrm{bulk} \\ 1\mathrm{freight}\mathrm{ton}=8\mathrm{barrel}\mathrm{bulk} \\ 1\mathrm{register}\mathrm{ton}=20\mathrm{barrel}\mathrm{bulk} \end{gathered}$$

Understanding the conversion factor

Unit conversion is the conversion between different units of measure for the same quantity, usually through multiplier conversion factors. Here, displacement ton, freight ton, register ton and barrel bulk and bushels are the different units for the quantity volume.

Determination of the number of U.S. bushels for displacement ton

To calculate the number of extra U.S. bushels, you must first convert all the tons into their equivalent U.S. bushels.

$$\begin{gathered} \mathbf{1}\mathbf{}{\mathbf{m}}^{\mathbf{3}}\mathbf{=}\mathbf{28}\mathbf{.}\mathbf{378}\mathbf{}\mathbf{U}\mathbf{.}\mathbf{S}\mathbf{}\mathbf{bushels} \\ \mathbf{1}\mathbf{}\mathbf{barrel}\mathbf{}\mathbf{bulk}\mathbf{}\mathbf{=}\mathbf{}\mathbf{0}\mathbf{.}\mathbf{1415}\mathbf{}{\mathbf{m}}^{\mathbf{3}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{1415}\mathbf{}{\mathbf{m}}^{\mathbf{3}}\mathbf{\times }\frac{\mathbf{28}\mathbf{.}\mathbf{378}\mathbf{}\mathbf{U}\mathbf{.}\mathbf{S}\mathbf{}\mathbf{bushel}}{\mathbf{1}\mathbf{}{\mathbf{m}}^{\mathbf{3}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{4}\mathbf{.}\mathbf{0155}\mathbf{}\mathbf{U}\mathbf{.}\mathbf{S}\mathbf{}\mathbf{bushel} \end{gathered}$$

Therefore, $\mathbf{1}\mathbf{}\mathbf{barrel}\mathbf{}\mathbf{bulk}\mathbf{}$is equivalent to $\mathbf{4}\mathbf{.}\mathbf{0155}\mathbf{}\mathbf{U}\mathbf{.}\mathbf{S}\mathbf{}\mathbf{bushel}$.

$$\begin{gathered} \mathbf{1}\mathbf{}\mathbf{displacement}\mathbf{}\mathbf{ton}\mathbf{}\mathbf{=}\mathbf{}\mathbf{7}\mathbf{}\mathbf{barrel}\mathbf{}\mathbf{bulk}\mathbf{} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{}\mathbf{7}\mathbf{}\mathbf{barrel}\mathbf{}\mathbf{bulk}\mathbf{}\mathbf{\times }\frac{\mathbf{4}\mathbf{.}\mathbf{0155}\mathbf{}\mathbf{U}\mathbf{.}\mathbf{S}\mathbf{}\mathbf{bushels}}{\mathbf{1}\mathbf{}\mathbf{barrel}\mathbf{}\mathbf{bulk}\mathbf{}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{28}\mathbf{.}\mathbf{108}\mathbf{}\mathbf{U}\mathbf{.}\mathbf{S}\mathbf{}\mathbf{bushel}\mathbf{} \end{gathered}$$

Therefore, $\mathbf{1}\mathbf{}\mathbf{displacement}\mathbf{}\mathbf{ton}$ is equivalent to $\mathbf{28}\mathbf{.}\mathbf{108}\mathbf{}\mathbf{U}\mathbf{.}\mathbf{S}\mathbf{}\mathbf{bushel}\mathbf{}$.

Determination of the number of U.S. bushels for freight ton

Now, use the relationship between freight ton and barrels bulk to find the relationship between freight ton and U.S. bushels.

$$\begin{gathered} \mathbf{1}\mathbf{}\mathit{f}\mathit{r}\mathit{e}\mathit{i}\mathit{g}\mathit{h}\mathit{t}\mathbf{}\mathit{t}\mathit{o}\mathit{n}\mathbf{}\mathbf{=}\mathbf{}\mathbf{8}\mathbf{}\mathit{b}\mathit{a}\mathit{r}\mathit{r}\mathit{e}\mathit{l}\mathbf{}\mathit{b}\mathit{u}\mathit{l}\mathit{k}\mathbf{} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{}\mathbf{8}\mathbf{}\mathit{b}\mathit{a}\mathit{r}\mathit{r}\mathit{e}\mathit{l}\mathit{s}\mathbf{}\mathit{b}\mathit{u}\mathit{l}\mathit{k}\mathbf{}\mathbf{\times }\frac{\mathbf{4}\mathbf{.}\mathbf{0155}\mathbf{}\mathbf{U}\mathbf{.}\mathbf{S}\mathbf{}\mathbf{b}\mathbf{u}\mathbf{s}\mathbf{h}\mathbf{e}\mathbf{l}\mathbf{s}}{\mathbf{1}\mathbf{}\mathbf{b}\mathbf{a}\mathbf{r}\mathbf{r}\mathbf{e}\mathbf{l}\mathbf{}\mathbf{b}\mathbf{u}\mathbf{l}\mathbf{k}\mathbf{}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{32}\mathbf{.}\mathbf{124}\mathbf{}\mathbf{U}\mathbf{.}\mathbf{S}\mathbf{}\mathit{b}\mathit{u}\mathit{s}\mathit{h}\mathit{e}\mathit{l}\mathit{s}\mathbf{} \end{gathered}$$

Therefore, $\mathbf{1}\mathbf{}\mathbf{freight}\mathbf{}\mathbf{ton}\mathbf{}$is equivalent to $\mathbf{32}\mathbf{.}\mathbf{124}\mathbf{}\mathbf{U}\mathbf{.}\mathbf{S}\mathbf{}\mathbf{bushels}\mathbf{}$.

Determination of the number of U.S. bushels for a registered ton

Now, use the relationship between register ton and barrels bulk to find the relationship between register ton and U.S. bushels.

$$\begin{gathered} \mathbf{1}\mathbf{}\mathbf{register}\mathbf{}\mathbf{ton}\mathbf{}\mathbf{=}\mathbf{20}\mathbf{}\mathbf{barrel}\mathbf{}\mathbf{bulk}\mathbf{} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{20}\mathbf{}\mathbf{barrel}\mathbf{}\mathbf{bulk}\mathbf{}\mathbf{\times }\frac{\mathbf{4}\mathbf{.}\mathbf{0155}\mathbf{}\mathbf{U}\mathbf{.}\mathbf{S}\mathbf{}\mathbf{bushels}\mathbf{}}{\mathbf{1}\mathbf{}\mathbf{barrel}\mathbf{}\mathbf{bulk}\mathbf{}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{80}\mathbf{.}\mathbf{31}\mathbf{}\mathbf{U}\mathbf{.}\mathbf{S}\mathbf{.}\mathbf{}\mathbf{bushels}\mathbf{} \end{gathered}$$

Therefore, $\mathbf{1}\mathbf{}\mathbf{register}\mathbf{}\mathbf{ton}\mathbf{}$ is equivalent to $\mathbf{80}\mathbf{.}\mathbf{31}\mathbf{}\mathbf{U}\mathbf{.}\mathbf{S}\mathbf{.}\mathbf{}\mathbf{bushels}\mathbf{}$
.

(a) Determination of the extra U.S. bushels when interpreted as freight tons.

The extra U.S. bushels can be given as:

$\mathbf{Extra}\mathbf{}\mathbf{bushels}\mathbf{}\mathbf{=}\mathbf{}\mathbf{73}\mathbf{}\mathbf{\times }\left(\mathrm{freight}\mathrm{tons}-\mathrm{displacement}\mathrm{tons}\right)$.

Substitute the values in the above expression.

$$\begin{gathered} \mathbf{Extra}\mathbf{}\mathbf{bushels}\mathbf{}\mathbf{=}\mathbf{}\mathbf{73}\mathbf{\times }\left(32.124U.S.\mathrm{bushels}-28.108U.S.\mathrm{bushels}\right) \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{293}\mathbf{.}\mathbf{168}\mathbf{}\mathbf{U}\mathbf{.}\mathbf{S}\mathbf{.}\mathbf{}\mathbf{bushles} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\approx }\mathbf{293}\mathbf{U}\mathbf{.}\mathbf{S}\mathbf{.}\mathbf{}\mathbf{bushles} \end{gathered}$$

Thus, when the order is interpreted as 73 freight tons, the extra U.S. bushels are 293 U.S. bushels.

(b) Determination of the extra U.S. bushels when interpreted as register tons

The extra U.S. bushels can be given as:

$\mathbf{Extra}\mathbf{}\mathbf{bushels}\mathbf{}\mathbf{=}\mathbf{}\mathbf{73}\mathbf{}\mathbf{\times }\left(\mathrm{freight}\mathrm{tons}-\mathrm{displacement}\mathrm{tons}\right)$

Substitute the values in the above expression.

$$\begin{gathered} \mathbf{Extra}\mathbf{}\mathbf{bushels}\mathbf{}\mathbf{=}\mathbf{}\mathbf{73}\mathbf{\times }\mathbf{(}\mathbf{80}\mathbf{.}\mathbf{31}\mathbf{}\mathbf{U}\mathbf{.}\mathbf{S}\mathbf{.}\mathbf{}\mathbf{bushels}\mathbf{}\mathbf{-}\mathbf{}\mathbf{28}\mathbf{.}\mathbf{108}\mathbf{}\mathbf{U}\mathbf{.}\mathbf{S}\mathbf{.}\mathbf{}\mathbf{bushels}\mathbf{)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{3810}\mathbf{.}\mathbf{74}\mathbf{}\mathbf{U}\mathbf{.}\mathbf{S}\mathbf{.}\mathbf{}\mathbf{bushles} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\approx }\mathbf{3}\mathbf{.}\mathbf{81}\mathbf{\times }{\mathbf{10}}^{\mathbf{3}}\mathbf{U}\mathbf{.}\mathbf{S}\mathbf{.}\mathbf{}\mathbf{bushles} \end{gathered}$$

Thus, when the order is interpreted as 73 register tons, the extra U.S. bushels are $\mathbf{3}\mathbf{.}\mathbf{81}\mathbf{\times }{\mathbf{10}}^{\mathbf{3}}\mathbf{U}\mathbf{.}\mathbf{S}\mathbf{.}\mathbf{}\mathbf{bushles}$.