A state department of education, as a question on the public-release form of a statewide standardized test, released a question to the public in which eighth graders have to determine how many millilters (mL) are in 4.6 liters (L). (If anyone from this particular state department is reading this, the sample form LM has “millimeters” in the question, not “milliliters,” a minor typo that will not bother us too much.)
This is a straightforward question, since both units use the metric system. All we really have to know is the meaning of the prefixes. In the metric system, a complete table of prefixes can be found at
http://www.simetric.co.uk/siprefix.htm
Such a comprehensive table is probably overkill, but there it is for your reference. A very common mistake with the metric prefixes is that students think the prefix “milli-” is related to the English word “million.” It really is related to “thousandths.” Oddly enough, that mistake, which is quite common in classrooms, is not presented as one of the answer choices.
The problem asks how many milliliters (mL) are in 4.6 liters (L). An abbreviated table of Le Système International d’Unités (SI) prefixes is shown below, along with their multipliers:
| Prefix | Multiplier | Abbreviation |
| mega- | 1,000,000 | M |
| kilo- | 1,000 | k |
| deci- | 0.1 | d |
| centi- | 0.01 | c |
| milli- | 0.001 | m |
| micro- | 0.000 001 | µ |
The way to read this table is like so: Suppose you have 20 kilometers (km). You look at the table and see that the multiplier for “kilo-” is 1,000. You say: “20 kilometers (your problem) is 20 × 1,000 (the multiplier in the table) = 20,000 meters (the base unit in your problem).
If we have 4.6 L and we need to know how many mL that is, we need to multiply by the inverse of 0.001, which would be 1,000. We do this because the prefix “milli-” means the number is already multiplied by 0.001, so to reverse that, we multiply by the inverse.
Thus, 4.6 L = 4.6 × 1,000 = 4,600 mL.
What we did, since we started with the base unit (L) and had to convert it to the unit with the prefix (mL), we needed to multiply by the inverse of the number in the table. Since the number in the table for our prefix (milli-) was 0.001, we needed to multiply by 1,000 (the inverse of the multiplier in the table).
Common sense takes over
Another way to look at the conversion is that 1 millimeter (mm) is much smaller than 1 meter (m). As a matter of fact, 1 mm is only 1/1000th the size of 1 m. So when converting L to mL above, you should get a much larger number, since liters are much bigger (a thousand times bigger) than milliliters.
If you would like more practice on unit conversion problems, visit our online library at VoxLearn.org and search for “unit conversion.”
Using the English (or U.S.) system of units
Although converting from kilometers to meters in the metric system can be as easy as multiplying by the correct power of 10 or moving the decimal point over so many places in the correct direction, using the English system, which is not based on powers of 10, can be a little trickier.
The principle is the same: multiply by the correct value to convert from one unit to another (sometimes this number is called a “conversion factor). The problem is, the number is not likely to be a power of 10, so the decimal-point-moving trick won’t work in the English or U.S. system. You’re going to have to learn the conversion factors.
Many of these, students will have learned in earlier grades. For example, it is likely most fifth graders know how many quarts are in a gallon. We will not review those here. But the more advanced unit conversion problems, within the same measurement system, should be learned in middle school or so.
Measurement of Length
The basic unit of length in the English system is the inch. Conversion factors are as follows:
(1) How many miles are there in 13,200 yards?
To make the conversion, we have to apply the conversion factor from the table above that tells us how many yards there are in a mile. It is best to remember that dividing something by an equivalent quantity makes the quotient equal to 1, and that multiplying something by 1 gives you back that something. This is a very important rule to remember when doing unit conversion: You need to multiply your original quantity (including the units) by the right form of a quantity that equals 1, so that you get the units you need.
In this problem, we start with 13,200 yards, and we need to convert that to miles. We are given that 1 mile = 1,760 yards. This conversion factor means that
When you divide something by something that is equal to it (in this case, dividing 1,760 yards by one mile), you get 1.
Multiplying something by 1 gives you back that exact same thing.
We’re going to make use of this second rule now. Take our quantity, 13,200 yards. We multiply that by “1” (in the form of 1 mile / 1,760 yards) to get our answer in terms of miles. The yards “cancel” each other out, since x divided by x is 1, no matter what x is.
After you “cancel” the yards, you are left with “miles” in the numerator. That’s what you want, since your answer needs to be in terms of miles.
As for the numbers, you have 13,200 in the numerator and 1,760 in the denominator. This is just a fraction: 13,200 / 1,760. To get the real number in simplest form, divide the numerator by the denominator.
13,200 / 1,760 = 7.5, so 13,200 yards = 7.5 miles.
Measurement of Weight
It is difficult to know which units you will be responsible for in your state for the English units of weight. The basic unit is the pound, and we have the following conversion factors:
It is important not to confuse an ounce of weight with a fluid ounce, which is a measure of liquid volume. There is a nice description of this difference on mathforum (link), where you will learn that 1 fluid ounce of water weighs just a tiny fraction more than 1 ounce of weight, also known as an avoirdupois ounce.
Here, we are talking about an ounce of weight. There are 16 of those ounces in a pound of weight. And there are 2,000 pounds in a ton of weight. If something weighs a “ton,” it weighs 2,000 pounds by definition.
(2) If your weight, in pounds, is 120, what is your weight in ounces?
All of these unit conversion problems are going to be the same: we start with what we’re given and find the most useful conversion factor that can turn it into the units we want. Our starting point here is “pounds” and we want our answer in terms of “ounces.” That means it would be ideal if we could find a conversion factor that had ounces and pounds in it. Luckily, we have one of those:
Now we have 120 and 16 on the top, so we multiply them. The units canceled properly, leaving us with just “ounces” in the numerator, and since 120 × 16 = 1,920, we know the answer is 120 pounds = 1,920 ounces.
Are you getting the hang of it? Unit conversion problems like this require only a few steps. First, find a useful conversion factor—one that has the units of your original problem and the units you want the answer in.
Then, set it up so that your original units cancel, leaving only the units that you are interested in, by multiplying by some form of 1.
Measurement of Capacity
With volume, especially liquid volume, the English system gets real complicated. Below are just a few conversion factors for liquid measures in the U.S. System, which differs a little bit from the British Imperial System:
(3) If oil costs $40 per barrel (those were the days, huh?), how much does it cost per gallon?
We know the cost per barrel, and since 1 barrel, for oil anyway, = 42 gallons, we can set up our conversion as follows:
As you see, the technique is the same: take what we start with and find a conversion factor with the right units, which will be equivalent to 1, and multiply the original problem by that conversion factor. Cancel the “barrel” in denominator of our original value (remember that “per barrel” means to put “barrel” in the denominator) and in the numerator of the conversion factor. This will leave us with “$ / gallon” as the unit, which is what the problem asked for.
Then, do the arithmetic: $40 / 42 is a little higher than 95 cents per gallon of oil. Cheap—well, cheaper than it is now, anyway.
Measurement of Speed or Rate
Occasionally, you get a problem where you will have to use more than one conversion factor. An example might be converting 3 gallons into teaspoons. You have one conversion factor that works for gallons and ounces (1 gallon = 128 ounces) and another conversion factor that works for ounces and teaspoons (1 ounce = 6 teaspoons). What you don’t have is a conversion factor that works for gallons and teaspoons.
You will therefore have to string two (or possibly more) conversion factors together. This is OK, because, remember, each conversion factor, when written as a fraction, is equal to 1. You can multiply a quantity by 1 as much as you want (quantity × 1 × 1 × 1 × 1 … = quantity, no matter how many 1’s you multiply it by.
This is often the case with speed or rate problems because, for instance, the numerator is often in units of length and the denominator in terms of time, if you have a speed problem. For example, 65 mph would be written as 65 miles / hour.
(4) How fast is a car going, in feet per second (ft/sec), if it is traveling at 65 miles per hour (mph)?
To answer this, we need to convert miles into feet first. We happen to have a good conversion factor for this, shown above: 1 mile = 5,280 feet.
Next, we need to convert hours into seconds. We have not discussed time in this post, because it is usually learned long before eighth grade. The appropriate conversion factors are 1 hour = 60 minutes and 1 minute = 60 seconds.
Starting, then, with our problem of 65 miles / hour, we multiply by conversion factors, arranged so that the units we don’t want cancel out and the units we do want remain:
The units all cancel properly, leaving us with feet in the numerator and seconds in the denominator, which is what “feet per second” means. All we have left to do is the arithmetic.
65 × 5,280 ÷ 60 ÷ 60 = 95.333 … ft/sec
And cheetahs can run even faster than that!
How big is an acre?
The definition of an acre says that it is a rectangular plot of land, 1 furlong by 1 chain. That doesn’t help much, does it? A furlong is 220 yards, and a chain is 1/10 furlong, or 22 yards. Therefore an acre is 220 × 22 = 4,840 square yards in area.
We will not concern ourselves with whether or not it will have to be those exact dimensions on the sides, but just get the conversion factor in your head:
It’s kind of an easy number to remember, isn’t it?
(5) How many square feet is an acre?
For this, you need to remember how many square feet there are in a square yard. There are three feet in a yard (conversion factor: 1 yard = 3 feet), but what that means in terms of square feet and square yards, is that there are 9 square feet in 1 square yard. Take a look at this diagram:
That is, the entire outside square is 1 square yard, and each individual square inside is 1 square foot in area. There are 9 individual square-foot squares inside the one, big square-yard square. The conversion factor is therefore
And given our conversion factor above that 1 acre = 4,840 square yards, we have the following conversion of 1 acre to square feet:
This means an acre is 43,560 square feet in area.