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## 1.3.1: Problem Solving and Unit Conversions

## Learning Objectives

## Converting Between Units with Conversion Factors

Of course, there are other ratios which are not listed in Table \(\PageIndex{1}\). They may include:

- Ratios embedded in the text of the problem (using words such as per or in each , or using symbols such as / or %).
- Conversions in the metric system, as covered earlier in this chapter.
- Common knowledge ratios (such as 60 seconds \(=\) 1 minute).

If you learned the SI units and prefixes described, then you know that 1 cm is 1/100th of a meter.

\[ 1\; \rm{cm} = \dfrac{1}{100} \; \rm{m} = 10^{-2}\rm{m} \nonumber \]

\[100\; \rm{cm} = 1\; \rm{m} \nonumber \]

Suppose we divide both sides of the equation by \(1 \text{m}\) (both the number and the unit):

\[\mathrm{\dfrac{100\:cm}{1\:m}=\dfrac{1\:m}{1\:m}} \nonumber \]

## Performing Dimensional Analysis

\[ 3.55 \; \rm{m} \times \dfrac{100 \; \rm{cm}}{1\; \rm{m}} \nonumber \]

The final step is to perform the calculation that remains once the units have been canceled:

\[ \dfrac{3.55}{1} \times \dfrac{100 \; \rm{cm}}{1} = 355 \; \rm{cm} \nonumber \]

quantity (in old units) × conversion factor = quantity (in new units)

## General Steps in Performing Dimensional Analysis

- Identify the " given " information in the problem. Look for a number with units to start this problem with.
- What is the problem asking you to " find "? In other words, what unit will your answer have?
- Use ratios and conversion factors to cancel out the units that aren't part of your answer, and leave you with units that are part of your answer.
- When your units cancel out correctly, you are ready to do the math . You are multiplying fractions, so you multiply the top numbers and divide by the bottom numbers in the fractions.

## Significant Figures in Conversions

How do conversion factors affect the determination of significant figures?

- Numbers in conversion factors based on prefix changes, such as kilograms to grams, are not considered in the determination of significant figures in a calculation because the numbers in such conversion factors are exact.
- Exact numbers are defined or counted numbers, not measured numbers, and can be considered as having an infinite number of significant figures. (In other words, 1 kg is exactly 1,000 g, by the definition of kilo-.)
- Counted numbers are also exact. If there are 16 students in a classroom, the number 16 is exact.
- In contrast, conversion factors that come from measurements (such as density, as we will see shortly) or that are approximations have a limited number of significant figures and should be considered in determining the significant figures of the final answer.

## Example \(\PageIndex{1}\)

- 101,000 ns to seconds
- 32.08 kg to grams
- 1.53 grams to cg
- Conversion factors are used to convert one unit of measurement into another.
- Dimensional analysis (unit conversions) involves the use of conversion factors that will cancel unwanted units and produce the appropriate units.

## Course: CHEM 333D. Problem Solving in Organic Chemistry I (1)

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