We've all measured something at some point in our lives. Most of us have probably used the English System of measurement, as that is the measurement system the United States has used for a very long time. However, we are about the only one's who don't use the Metric System.
Without even really knowing it we've all used a method called dimensional analysis at some point in our lives to convert our measurements. Think about how many times have you needed to convert feet to inches or vice-versa. In dimensional analysis we set up a series of ratios using like rates to change our units to what we are looking for. To do this we use something known as a conversion factor(s).
| 1 Mile(mi) | = | 1760 Yards(yds) |
| 1 Mile(mi) | = | 5280 Feet(ft) |
| 1 Yard(yd) | = | 3 Feet(ft) |
| 1 Foot(ft or ') | = | 12 Inches(in or ") |
Your car can drive 300 miles on a tank of gas. How many yards can it drive?
First we need to find a conversion factor that matches miles to yards. We can see form the box above that 1 miles is equal to 1760 yards. The process behind dimensional analysis is to set up the conversion factor such that the unit you're trying to eliminate is opposite of where it currently is. Looking at our initial ratio:
$$\dfrac{300\hspace{0.25cm}\text{miles}}{1\hspace{0.25cm}\text{tank of gas}}$$
We can see here that we are comparing miles to gas and we want to eliminate the miles and compare yards to gas. To do that we'd set up our ratios such that:
$$\dfrac{300\hspace{0.25cm}\text{miles}}{1\hspace{0.25cm}\text{tank of gas}}\centerdot\dfrac{1760\hspace{0.25cm}\text{yards}}{1\hspace{0.25cm}\text{mile}}=\dfrac{300(1760)\color{red}{\cancel{\text{miles}}}\centerdot \color{black}\text{yards}}{1(1)\text{tank}\centerdot\color{red}{\cancel{\text{mile}}}}$$
$$\dfrac{528,000\hspace{0.25cm} \text{yds}}{\text{tank}}$$
Notice that since there was a unit of miles on top and a unit of miles on bottom we are able to reduce out the common unit, allowing us to have just the units we want.
A bicycle is traveling at 15 miles per hour. How many feet will it cover in 20 seconds?
To answer this question, we need to convert 15 miles per hour into feet per second. This will require multiple conversions factors. To get started we may just want to consider doing them one at a time and save doing it at the same time for later in the chapter. To start converting miles to feet we find our conversion factor of 1 mile is equal to 5280 feet.
$$\dfrac{15\hspace{0.25cm}\text{miles}}{1\hspace{0.25cm}\text{hour}}\centerdot \dfrac{5280\hspace{0.25cm}\text{feet}}{1\hspace{0.25cm}\text{mile}}=\dfrac{15(5280)\hspace{0.25cm}\text{feet}}{1\hspace{0.25cm}\text{hour}}=\dfrac{79,200\hspace{0.25cm}\text{feet}}{1\hspace{0.25cm}\text{hour}}$$
Now we need to convert the hours to seconds. This will require multiple conversion factors because there are 60 minutes in one hour and 60 seconds in one minute so we need both conversion factors being sure to always set it up that one unit is on top and the other on the bottom:
$$\dfrac{79,200\hspace{0.25cm}\text{feet}}{1\hspace{0.25cm}\text{hour}}\centerdot\dfrac{1\hspace{0.25cm}\text{hour}}{60\hspace{0.25cm}\text{minutes}} \centerdot \dfrac{1\hspace{0.25cm}\text{minute}}{60\hspace{0.25cm}\text{seconds}}=\dfrac{79,200(1)(1)\hspace{0.25cm}\text{feet}}{1(60)(60)\hspace{0.25cm}\text{seconds}} =\dfrac{79,200\hspace{0.25cm}\text{feet}}{3600\hspace{0.25cm}\text{seconds}}$$
$$=22\hspace{0.25cm} \text{feet/sec}$$
Which means that in 20 seconds the bicycle will have traveled $$(20\hspace{0.25cm}\text{sec})(22\hspace{0.25cm}\text{feet/sec})=440\hspace{0.25cm}\text{feet.}$$
For those of us who grew up in the United States, we see no issue with the English System of measurement. However, all over the world everyone else is using a standard unit of measure known as the Metric System. The Metric System works off a standard base unit and then each different unit is a multiple or fraction of ten.
For the sake of measuring length the Metric System uses something known as a Meter(m) which is just slightly longer than one yard. As previously mentioned the Metric System works on a base ten system. You may have not noticed this on a ruler but if you count the number of lines between 1cm and 2cm there are nine little lines. Each of those lines are known as millimeters. Technically speaking 10mm = 1cm. Which if you think about it, it is a lot easier to convert than 12in to 1ft.
| 1 Kilometer (km) | = | 1000 Meters |
| 1 Hectometer (hm) | = | 100 Meters |
| 1 Deckameter (dam) | = | 10 Meters |
| 1 Meter (m) | = | 1 Meter |
| 1 Decimeter (dm) | = | $\frac{1}{10}$ Meter |
| 1 Centimeter (cm) | = | $\frac{1}{100}$ Meter |
| 1 Millimeter (mm) | = | $\frac{1}{1000}$ Meter |
When converting in the Metric system it may be just as easy to think about multiplying by $10^n$ where $n$ is the number of times you have to move down the hierarchy or dividing by $10^n$ where $n$ is the number of times you have to move up the hierarchy. Think of it kind of like a ladder and for each step you go up you divide by ten and for each step you go down you multiply by ten.
Convert the following:
First to move from kilometers to meters would require us to move down the "ladder" from km $\rightarrow$ hm $\rightarrow$ dam $\rightarrow$ m, that means we'd move down the ladder $n=3$ steps. Thus we would multiply 4.5 time $10^3$
$$4.5(10^3)=4.5(1000)=4,500\text{m}$$
To move from centimeters to meters would require us to move up the "ladder" from cm $\rightarrow$ dm $\rightarrow$ m, which means we'd go up $n=2$ steps. Thus, we'd divide 0.0257 by $10^2$
$$\dfrac{0.0257}{10^2}=\dfrac{0.0257}{100}=0.000257\text{m}$$
One snack size bag of cheese Ritz Bits contains 200mg of sodium. According to Healthline.com the average adult should consume at most 2.3 grams per day. What percentage of your daily sodium intake would this serving be?
First we need to convert the 200mg to grams. Grams is a form of measurement that weighs things. Just pay attention to the general form. That means we would need to go from Milli to our base unit of grams. To do that we'd need to go up the ladder from mg $\rightarrow$ cg $\rightarrow$ dg $\rightarrow$ g, which is three steps, and since it it up we divide:
$$\dfrac{200}{10^3}=\dfrac{200}{1000}=0.2\text{g}$$
Next we need to calculate the percentage of daily total which is part divided by whole times $100\%$ like we learned in Chapter 1.
$$\dfrac{0.2}{2.3}\centerdot 100\%=0.08696\centerdot100\%=8.696\%$$
There may come a time in your life when you need to be able to convert between a Metric measurement and an English measurement. That is when the information below comes in handy.
| 1 mile | = | 1.61 kilometers | 1 kilometers | = | 0.62 miles |
| 1 yard | = | 0.914 meters | 1 meter | = | 3.28 feet |
| 1 foot | = | 0.305 meters | 1 meter | = | 1.09 yards |
| 1 inch | = | 2.54 centimeters | 1 centimeter | = | 0.394 inches |
Note: Depending on how you choose your conversion factors your answers can vary ever so slightly based on rounding. Many of these decimal values are rounded for ease of use.
Your friend in Spain wants to know how far you can throw a football. You know that you can throw a football 75 yards. How many meters would that be?
As we can see, there are two options of conversion factors when it comes to converting from yards to meters. Either 1 yard is 0.914 meters or 1 meter is 1.09 yards. We will do both conversions just to show you the difference. But in reality, you'd just choose one factor and use your dimensional analysis to convert to the Metric System.
Option 1: 1 yard = 0.914 meters
\[ \dfrac{75\text{ yards}}{1} \cdot \dfrac{0.914\text{ meters}}{1\text{ yd}} = 68.55\text{ m} \]
Option 2: 1 meter = 1.09 yards
\[ \dfrac{75\text{ yards}}{1} \cdot \dfrac{1\text{ meter}}{1.09\text{ yds}} = 68.81\text{ m} \]
As you can see, there is a slight difference between the two. This is just because of the rounding error that occurs in the conversion factor.