3.5: Solving Multistep Conversion Problems
- Page ID
- 283825
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- Convert a measurement into a different unit in cases where an equality between the two units is not available; i.e. perform conversions that take two or more steps.
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 inches. 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 miles which is in the same measurement system as inches. We know that 0.6214 mi =1 km.
Then we convert miles to feet, remembering that \(1\; \rm{mi}\) = \( 5280\; \rm{ft}\). Finally, we convert feet to inches using 1 ft. = 12 in.
Concept Map
kilometers → miles → feet → inches
Calculation
\[ \begin{align} 54.7 \; \cancel{\rm{km}} \times \dfrac{0.6214 \; \cancel{\rm{mi}}}{1\; \cancel{\rm{km}}} \times \dfrac{5280\; \cancel{\rm{ft.}}}{\cancel{1 \rm{mi}}} \times \dfrac{12\; \cancel{\rm{in.}}}{\cancel{1 \rm{ft.}}} & = 2,153,643 \; \rm{in.} \\ &= 2.15 \times 10^6\; \rm{in.} \end{align}\]
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
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 |
\[1 \: \text{d} \times \frac{24 \: \text{hr}}{1 \: \text{d}}\times \frac{60 \: \text{min}}{1 \: \text{hr}} \times \frac{60 \: \text{s}}{1 \: \text{min}} = 86,400 \: \text{s} \nonumber\] |
An IV bag contains 250 mL of solution. How many quarts (qt.) of solution are in the bag? Note: 1 L = 1.06 qt.
Solution
When using the unit analysis method, if you are solving for a simple unit such as quarts you should start with the measured or given number that has a simple unit. A simple unit has only one part such as mL. A compound unit has two parts such as g/mL (grams per milliliter). Do not start with an equality like 1 L = 1.06 qt. Equalities will be used to make conversions later in the problem.
\[250\:mL\]
When you make a conversion factor put the unit from the top of the previous step on the bottom of the fraction.
\[250\:mL \times \frac{}{\: mL}\]
Consider whether you have the information needed to convert directly to the unit for the answer (an equality that relates the unit that you have to the unit that you want). In this case we do not have that information. However, we can use the metric system to convert from mL to L and then use the note provided in the problem to convert from L to qt.
\[250\:mL \times \frac{\: L}{\: mL} \times \frac{\: qt.}{\: L}\]
It is helpful to plan the units out first (either one step at a time or for all of them at once) and then fill in the numbers. Making sure that the units are in the right place will help you to know whether to multiply or divide by the numbers in the equalities.
Fill in the numbers. Make sure that each conversion factor (fraction) is made out of two values that are equal. Using the metric system, 1000 mL = 1 L. Using the information provided in the problem, 1 L = 1.06 qt. Match the numbers up with the correct units.
\[250\:mL \times \frac{1\:L}{1000\:mL} \times \frac{1.06\:qt.}{1\:L}\]
To solve, start with the given number, multiply it by values in numerators (tops of fractions) and divide by values in denominators (bottoms of fractions). It is not necessary to multiply or divide by 1 in the calculator since that does not change the value of a number. For this problem you could type: 250 \(\div\) 1000 x 1.06
\[250\:mL \times \frac{1\:L}{1000\:mL} \times \frac{1.06\:qt.}{1\:L} = 0.265\:qt.\]
The given value, 250 mL has only two significant figures. The metric system equality, 1000 mL = 1 L, is exact because those values are in the same measurement system. The other equality, 1 L = 1.06 qt., is not exact because L and qt. are from two different systems. Based on 1.06 that relationship has three significant figures. The value with the least significant figures was the 250 mL with two significant figures, so the answer should be rounded to two significant figures: 0.27 qt.
Exercise \(\PageIndex{1}\)
Perform each conversion in a multistep calculation.
- 65.0 hours to milliseconds (note: the metric system can be used to convert seconds to milliseconds)
- 15 inches to millimeters
- 1200 feet to kilometers
- Answer a:
- \(65.0\:h \times \frac{60\:min}{1\:h} \times \frac{60\:s}{1\:min} \times \frac{1000\:ms}{1\:s} = 2.34 \times 10^{8}\:ms \)
- Answer b:
- 380 mm
- Answer c:
- 0.37 km
Career Focus: Pharmacist
A pharmacist dispenses drugs that have been prescribed by a doctor. Although that may sound straightforward, pharmacists in the United States must hold a doctorate in pharmacy and be licensed by the state in which they work. Most pharmacy programs require four years of education in a specialty pharmacy school. Pharmacists must know a lot of chemistry and biology so they can understand the effects that drugs (which are chemicals, after all) have on the body. Pharmacists can advise physicians on the selection, dosage, interactions, and side effects of drugs. They can also advise patients on the proper use of their medications, including when and how to take specific drugs properly. Pharmacists can be found in drugstores, hospitals, and other medical facilities. Curiously, an outdated name for pharmacist is chemist, which was used when pharmacists formerly did a lot of drug preparation, or compounding. In modern times, pharmacists rarely compound their own drugs, but their knowledge of the sciences, including chemistry, helps them provide valuable services in support of everyone’s health.
A woman consulting with a pharmacist. This image was released by the National Cancer Institute, an agency part of the National Institutes of Health. Image used with permission (Public Domain; Rhoda Baer (Photographer) via NIH).
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.
Contributors
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