# 1.10: Solving Multistep Conversion Problems

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## 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 Convert kilometers to meters to millimeters: use conversion factors 1000 meters per 1 kilometer and 1 millimeter per 0.001 meter

#### 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. Convert milliseconds to seconds to microseconds: use conversion factors 0.001 second per millisecond and 1 microsecond per 1 million seconds
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. Convert day to hour to minute to second: use conversion factors 24 hours per day, 60 minutes per hour, and 60 seconds per minute
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
$$4.3007 \times 10^{-11} kg$$
$$83.75\, ft$$
$$19\, km$$
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## Summary

In multi-step 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.

1.10: Solving Multistep Conversion Problems is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.