Units of Mass Anna and Bill notice that there are 50.0 mg of sodium in the soda they bought to go with their lunch. How many grams of sodium are present in the can of soda? How many pounds?

Question

Units of Mass

Anna and Bill notice that there are 50.0 mg of sodium in the soda they bought to go with their lunch. How many grams of sodium are present in the can of soda? How many pounds?

Accepted Answer

One way to solve this problem uses the dimensional-analysis approach. Consult Math Toolbox 1.3 for details. The general approach to solving the first part of the problem can be summarized by the following diagram:

[latex]\fbox{Mass in milligrams}\overset{?}{\longrightarrow } \fbox{Mass in grams }[/latex]

The mass in milligrams has to be converted to the mass in grams. We need to find a relationship between these two quantities: 10^–3g = 1 mg or 1 g = 1000 mg (obtained from Table 1.4). Using this relationship, we get the following conversion:

[latex]\fbox{Mass in milligrams}\overset{1000 mg=1 g}{\longrightarrow } \fbox{Mass in grams }[/latex]

TABLE 1.4 | Metric Conversions

Prefix	Factor	Symbol
giga	10⁹	G
mega	10⁶	M
kilo	10³	k
deci	10^–1	d
centi	10^–2	c
milli	10^–3	m
micro	10^–6	μ
nano	10^–9	n
pico	10^–12	p

We use the equivalence to set up possible conversion ratios:

[latex]\frac{1 g}{1000 mg} ext{ and } \frac{1000 mg}{1 g}[/latex]

To convert milligrams to grams, we can multiply 50.0 mg by the ratio (conversion factor) that will allow like units to cancel:

[latex] ext{Mass in mg }= 50.0 \ \cancel{mg} imes \frac{1 \ g}{1000 \ \cancel{mg}}= 0.0500 \ g[/latex]

Note that the milligram units cancel to leave the appropriate unit of grams. Also notice that the answer is reported to three significant figures because the measured quantity (50.0 mg) is reported to three significant figures and the other numbers in the calculation (1 g and 1000 mg) are exact quantities. (Consult Math Toolbox 1.2 for details about significant figures.) For convenience, we could report the answer in scientific notation: [latex] 5.00 × 10^{–2} [/latex]. (See Math Toolbox 1.1 for a discussion of scientific notation.)

The second part of the question asks you to convert milligrams to pounds:

[latex]\fbox{Mass in milligrams}\overset{?}{\longrightarrow } \fbox{Weight in pounds }[/latex]

There isn’t a direct relationship between milligrams and pounds listed in Tables 1.4 and 1.5. However, Table 1.5 lists a relationship between pounds and grams: 1 lb = 453.6 g. We can convert the grams we found in the first part of this example to pounds using the relationship summarized in the following diagram:

[latex]\fbox{Mass in grams}\overset{453.6 g=1 lb}{\longrightarrow } \fbox{Weight in pounds }[/latex]

TABLE 1.5 | Some English-Metric Conversions

English Unit	Metric Unit
1 lb = 16 oz	453.6 g
1 in	2.54 cm (exactly)
1 yd	0.9144 m
1 mi	1.609 km
1 fluid oz	29.57 mL
1 qt	0.9464 L
1 gal	3.785 L
1 ft³	28.32 L

The ratios for converting between grams and pounds are

[latex]\frac{1 lb}{453.6 g} ext{ and } \frac{453.6 g}{1 lb}[/latex]

To convert grams to pounds, we can multiply 0.0500 g by the ratio (conversion factor) that will allow like units to cancel:

[latex] ext{weight in pounds }= 0.0500 \cancel{g} imes \frac{1 lb}{453.6 \cancel{g} }=1.10 imes 10^{-4} lb[/latex]

Does this answer make sense? Yes, it does. There are a lot of grams (453.6) in a pound, so we would expect the answer to be very small. Would the answer 22.7 lb make sense? No.

Without a single conversion from milligrams to pounds, the problem we just solved involves multiple steps:

[latex]\fbox{Mass in milligrams}\overset{1000 mg=1 g}{\longrightarrow } \fbox{mass in grams }\overset{453.6 g=1 lb}{\longrightarrow } \fbox{Weight in pounds }[/latex]

The sequence of steps can be summarized as:

[latex] ext{weight in pounds }= 50.0 \cancel{mg} imes \frac{1 \cancel{g}}{1000 \cancel{mg} } imes \frac{1 lb}{453.6 \cancel{g}} =1.10 imes 10^{-4} lb[/latex]

Question 1.5: Units of Mass Anna and Bill notice that there are 50.0 mg of...

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