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Units of Measurement
Units of Measurement. Most numbers in geometry problems are without units, but for a few real-world problems, the test does include units. So this is relatively rare, but it does happen sometimes in geometry problems. I already discussed the issue of units and unit conversions in the lesson Introduction to Motion Questions in the Words Problem module.
So if you haven’t seen that lesson, I would strongly advise watching that lesson first. Because it will make everything in this lesson make much more sense. Okay. There are several common units of length, inches, feet, yards, centimeters, meters, kilometers, and miles.
Ordinary lengths are measured in terms of a unit of length. Area is measured in terms of a unit of length squared. For example, square inches or centimeters squared. Volume is measured in terms of a length of unit cubed so, cubic inches or cubic centimeters, something along those lines. If the entire problem contains just one unit, then it poses no challenge.
The problem is trickier and more test like, if it introduces more than one unit. For example, it gives information in terms of one unit and asks for an answer in terms of another unit. To change from one unit to another, we need to use unit conversions. Some unit conversions might be familiar. So for Americans certainly they would know 1 foot equals 12 inches.
Folks from other countries might be more familiar with 1 meter equals 100 centimeters. Because different units of length are used in different parts of the world the test almost always gives these. So it’s very common that this would actually appear on the test. Once you’re given one of these equations, you change it into a fraction.
When we write them as a fraction, it’s a fraction equal to one. And that means we can multiply and divide anything by that fraction, and it wouldn’t change its fundamental value. To change from one unit to another we simply multiply by the unit conversion fraction. That has the given unit in the denominator and the desired unit in the numerator.
For example, if we want to change feet to inches, we need a unit conversion that has feet on the bottom and inches on the top. So 12 inches over 1 foot. The we multiple, the feet cancel and we get 6 times 12, 72. 72 inches. Changing units gets trickier when areas and volumes are involved.
Suppose we have to change an area of 5 square meters to centimeters. To cancel the factors of meters, we have to multiply by the unit conversion a 100 centimeters over 1 meter. We have to multiply that by that twice. That is to say, we need to multiply by the unit conversion squared. So we multiply 5 squared meters by 100 centimeters over 1 meter.
That conversion factor squared. And of course, a 100 squared is 10,000. And 5 times 10,000 is 50,000. So 5 square meters is 50,000 square centimeters. Think about a square, 1 foot by 1 foot. Since there are 12 inches in a foot, this square would be 12 inches by 12 inches.
And we could visualize it like this. So obviously there’s much more than 12 square inches in that square foot, 12 square inches would just be one tiny column, or one tiny row. The whole square foot is 12 times 12, or 144 square inches. One square foot equals 144 square inches. Here’s a practice problem.
Pause the video and then we’ll talk about this. Okay. A bathroom floor is rectangular, 2 meters by 3 meters. It is to be tiled with square titles, 4 centimeters on a side. How many tiles will it take to fill the floor? So this is very typical of a test problem.
We’re given two different kinds of units and we have to figure out how to make them work with each other. So first of all, the area of that floor, that’s 2 times 3, or 6 square meters. Now let’s change that to square centimeters. So we multiply that by 100 centimeters over 1 meter, that squared, 100 squared is 10,000, 6 times 10,000 is 60,000.
So 60,000 square centimeters, that’s the area of the whole floor. And the area of the tile, it’s 4 by 4, so we wanna divide that by 4 centimeters squared. And what I’m gonna suggest is, to make things easier, let’s divide by 4 once and to then divide by 4 again. So 60 divided by 4 that’s easy, that’s 15.
Now we have 15,000 we want to divide that by 4. Well here’s one way to think about it, 15,000 is 16,000 minus 1. Well 16,000 divided by 4, that’s clearly just 4,000 and 1,000 divided by 4, that’s 250. So 4,000 minus 250. That’s 3750, so that’s a very easy way to get to the answer without even touching a calculator.
Similarly, if we have to convert a volume, we need to multiply by the cube of the scale factor. When this rare problem occurs in the test, it almost always involves metric units because powers of ten are very easy to cube. So, one cubic meter, if we wanna change that to centimeters, we’d have to multiply by 100 centimeters over 1 meter, that factor cubed.
And of course 100 cubed is 1 million. So 1 cubic meter equals 1 million cubic centimeters. Here’s another practice problem. Pause the video and then we’ll talk about this. So they give us the dimensions of the pool in meters. And then, we want to fill it with sand which is given in cubic millimeters.
So first of all, we can figure out the volume of the pool. The volume of the pool 2 times 5 times 8. Well, that’s 10 times 8, which is 80. 80 cubic meters. And then, the question is, how many cubic millimeters are there in 80 meters. Well, we’re gonna take that 80 meters and multiply it by the conversion factor, 1000 millimeters over 1 meter.
And that conversion factor is gonna be cubed. And so here, it makes sense to do things in scientific notation. We get 10 to the 3rd cubed is 10 to the 9th. So, we have 8 times 10 to 9th. Incidentally that would be 80 billion. Using the American definition of billion.
That would be 80 billion. And we could rewrite that as 80 times 10 to the 10th. In summary, change units with unit conversions, which are almost always given in the text of the problem. For areas, we have to square the unit-conversion fraction. For volumes, we have to cube the unit-conversion fraction.
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