1.10: Solving Multistep Conversion Problems


Multiple Conversions

Sometimes you will have to perform more than one conversion to obtain the desired unit. For example, suppose you want to convert 54.7 km into millimeters. We will set up a series of conversion factors so that each conversion factor produces the next unit in the sequence. We first convert the given amount in km to the base unit, which is meters. We know that 1,000 m =1 km.

Then we convert meters to mm, remembering that $$1\; \rm{mm}$$ = $$10^{-3}\; \rm{m}$$.

Concept Map

Calculation

\begin{align*} 54.7 \; \cancel{\rm{km}} \times \dfrac{1,000 \; \cancel{\rm{m}}}{1\; \cancel{\rm{km}}} \times \dfrac{1\; \cancel{\rm{mm}}}{\cancel{10^{-3} \rm{m}}} & = 54,700,000 \; \rm{mm} \\ &= 5.47 \times 10^7\; \rm{mm} \end{align*} \nonumber

In each step, the previous unit is canceled and the next unit in the sequence is produced, each successive unit canceling out until only the unit needed in the answer is left.

Example $$\PageIndex{1}$$: Unit Conversion

Convert 58.2 ms to megaseconds in one multi-step calculation.

Solution
Solution for Example 2.7.1
Unit Conversion
Identify the "given" information and what the problem is asking you to "find."

Given: 58.2 ms

Find: Ms

List other known quantities

$$1 ms = 10^{-3} s$$

$$1 Ms = 10^6s$$

Prepare a concept map.
Calculate.

\begin{align} 58.2 \; \cancel{\rm{ms}} \times \dfrac{10^{-3} \cancel{\rm{s}}}{1\; \cancel{\rm{ms}}} \times \dfrac{1\; \rm{Ms}}{1,000,000\; \cancel{ \rm{s}}} & =0.0000000582\; \rm{Ms} \nonumber\\ &= 5.82 \times 10^{-8}\; \rm{Ms}\nonumber \end{align}\nonumber

Neither conversion factor affects the number of significant figures in the final answer.

Example $$\PageIndex{2}$$: Unit Conversion

How many seconds are in a day?

Solution
Solution for Example 2.7.2
Unit Conversion
Identify the "given" information and what the problem is asking you to "find."

Given: 1 day

Find: s

List other known quantities.

1 day = 24 hours

1 hour = 60 minutes

1 minute = 60 seconds

Prepare a concept map.
Calculate. $1 \: \text{d} \times \dfrac{24 \: \text{hr}}{1 \: \text{d}}\times \dfrac{60 \: \text{min}}{1 \: \text{hr}} \times \dfrac{60 \: \text{s}}{1 \: \text{min}} = 86,400 \: \text{s} \nonumber$
Exercise $$\PageIndex{1}$$

Perform each conversion in one multi-step calculation.

1. 43.007 ng to kg
2. 1005 in to ft
3. 12 mi to km
$$4.3007 \times 10^{-11} kg$$
$$83.75\, ft$$
$$19\, km$$