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

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

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 multistep calculation.

SOLUTION

Steps for Problem Solving

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

Steps for Problem Solving

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

\[1 \: \text{d} \times \frac{24 \: \text{hr}}{1 \: \text{d}}\times \frac{60 \: \text{min}}{1 \: \text{hr}} \times \frac{60 \: \text{s}}{1 \: \text{min}} = 86,400 \: \text{s} \nonumber\]

Exercise \(\PageIndex{1}\)
Perform each conversion in one multistep calculation.

43.007 ng to kg
1005 in to ft
12 mi to km
Answer a:
\(4.3007 x 10^{-11} kg \)
Answer b:
83.75 ft
Answer c:
19 km
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Summary
In multistep conversion problems, the previous unit is canceled for each step and the next unit in the sequence is produced, each successive unit canceling out until only the unit needed in the answer is left.