Dimensional
analysis practice
Rules for rounding numbers: If the number to be dropped is 5 or greater, then round up the preceding number. If the number to be dropped is less than 5, then leave the preceding number as is. Examples: (a) 34.6251 rounded to 2 decimal places becomes 34.63. (b) 0.1499 rounded to one decimal place becomes 0.1. (c) 0.1499 rounded to 3 decimal places becomes 0.150. (d) 19.501 rounded to the nearest integer becomes 20.
1. Convert 25 feet per second into kilometers per hour. Show one
decimal place in the answer.
25 ft. |
X |
12 in. |
X |
1 m |
X |
1 km |
X |
60 sec. |
X |
60 min. |
= |
27.4 km |
You see that each conversion factor is written such that you can cancel one unit from numerator or denominator in each multiplication step: feet in the first multiplication step, inches in the second, and so on.
If you have trouble seeing how such problems are done when written in this chain fashion, you can do them one step at a time, of course. However, if you calculate the numbers one step at a time, don't round until you get to the end, the final answer. Rounding always makes the answer somewhat less accurate, i.e. introduces some error. And more rounding steps are likely to introduce more error.
2. Convert 25 grams per liter into pounds per gallon. Round the answer to two decimal places. Note that the
problem requires moving from the metric system (grams and liters) to the
foot-pound system (pounds and gallons).
Therefore, you need to use two of those "special" three
conversion factors that you were told to
memorize for moving from system to system.
25 g |
X |
1 lb |
X |
1 L |
X |
4 qt. |
= |
0.21 lb. |
3. Most doctors would be concerned if they saw a patient's blood
cholesterol concentration greater than 220 milligrams of cholesterol per
deciliter of blood. Is a cholesterol reading of 1500 micrograms per milliliter
a cause for concern? Round the answer to the nearest integer.
(Note that it is common to see concentrations of solutes in blood expressed in
terms of deciliters.)
1500 μg |
X |
1 g |
X |
1 mg |
X |
1000 mL |
X |
1 L |
= |
150 mg |
Here are two steps involving scientific notation (powers-of-10). Note that 1 g/10^{6} μg might be written as 10^{-6} g/μg, so that the power-of-10 expression might be in the numerator with a negative exponent. Likewise, 1 mg/10^{-3} g might be written as 10^{3} mg/g or as 1000 mg/g or as 1 mg/0.001 g; all four of these are the same, but some people will have serious difficulty with such expressions. What are the rules for multiplying and dividing when exponents are involved? This is basic algebra again. Now is the time to get this stuff straight, rather than stumbling through another course, losing points on tests because this was never clear and still isn't. If you need help with this, don't be afraid or embarrassed to ask. Consult your TA first; come see Dr. Rawn if you still need help.
4. On your first visit to Canada suppose you're cruising down the
road at a leisurely 50 miles per hour. A Mountie
pulls you over (Canadian horses are fast) and writes a summons, charging you
with exceeding the speed limit. You remember that the last sign you saw read
"SPEED LIMIT 50 km/hr", and you argue that "50 is 50, right?". The Mountie is not amused.
So what's the deal here? Round the answer to the nearest
integer.
50 mile |
X |
5280 ft. |
X |
12 in. |
X |
1 m |
X |
1 km |
= |
80 km |
Note again that you must move from foot-pound (miles) to metric (km); so, you'll need that conversion factor for interconverting inches and meters that you are required to use: 39.4 inch/meter. The steps before that here are simply the ones needed to get to the point where you can use that conversion factor to cross over.
You probably can see that there's a shorter way to work many such problems if you know more conversion factors. For instance, this last problem would be shorter if you used a conversion factor that relates kilometers to miles directly. Of course, there is one, but remember that in this course we will always use a particular conversion factor to cross from one system to the other…no shortcuts, please…we're trying to learn a method. It's too easy to store a bunch of these factors in a calculator and then push a button without understanding how the method works.
5. The following two conversions are done incorrectly. What is wrong with each one? These are examples of mistakes that are commonly made by students, mistakes that do cost points on exams for instance.
(a) Convert 25 milligrams to grams.
25
mg |
X |
1
g |
= |
25
X 10^{3} g |
The units (mg) cancel OK here. But the conversion factor, itself, is wrong. It says that 1 g = 0.001 mg. A moment of thought should tell us that it should be 10^{-3} g / mg. Sometimes that signals a "careless" mistake; sometimes it just shows that the student doesn't understand scientific notation. Here's an example of a case where thinking about the answer for a moment may help one to catch such a mistake. If we know that milligrams are smaller than grams, then how could 25 mg be equal to 25 thousand grams. Yikes!
(b) Convert 25 micrograms to grams.
25
μg |
X |
1
μg |
= |
25
X 10^{6} g |
Here the conversion factor is correct. It's true that 1 microgram = 10^{-6}
gram (same as 10^{6} microgram = 1 gram). But the conversion factor is
written incorrectly for this problem. How do we know? We need to replace
microgram units with gram units, but we can't cancel the μg
with the conversion factor written as it is. It needs to be inverted. Remember:
the units must come out right, and this problem asks for grams in the answer.
The problem here, setup as shown, would give squared micrograms in the
numerator and grams in the denominator. Though the wrongness of that may
seem obvious now, the same kind of mistake commonly occurs when students fail
to pay attention to units or carelessly neglect to write the units in every
step when solving a problem. Paying attention to the units in working a
problem can help avoid getting the wrong answer.