EARLAND'S CLASS RESOURCES



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To convert between units, you're usually given one measure and asked to convert to another measure. In these simple scenarios, all you have to do to convert is remember a fairly simple rule:

• going to smaller units mean going to bigger numbers,

• going to bigger units mean going to smaller numbers,

Here's how it works:

• Convert 3 gallons to quarts. Quarts are smaller than gallons; every gallon has four quarts. Since I'm converting from a larger unit (gallons) to a smaller unit (quarts), my answer needs to be a bigger number. So I multiply:

Answer:

• Convert 7920 yards to miles. Miles are bigger than yards; there are 1760 yards in every mile. Since I'm converting from a smaller unit (yards) to a bigger unit (miles), my answer needs to be a smaller number. So I divide:

Answer:

The above are examples of one-step conversions: But sometimes conversions are more complicated, or you're not sure which unit is "bigger". This applies especially in the case of conversions between English and metric units. For instance, which is "bigger", decaliters or Imperial gallons? Or consider rates: which is "bigger", 80 miles an hour or 40 meters per second? It's hard to see how the term "bigger" would apply here.

Cancelling Units!!!!

For these sorts of conversion, we use as many conversion factors as we need, setting up a long multiplication so the units we don't want cancel out. Note: I'm not talking here about numbers "cancelling out", like when you're multiplying fractions. Instead, I'm talking about treating the units ("feet", "miles", "seconds", etc) as though they were numbers, and cancelling them.

• Which is faster, going 80 miles an hour or going 40 meters per second?

Okay, I need to convert from "miles" to "meters" and from "hours" to "seconds". Looking in the back of my textbook (which is frequently a handy resource), I find the following conversion factors among the many listed:

Depending on the source and my predilection, I could have chosen other conversion factors. But these provide connections, one way or another, between "seconds" and "hours" and between "miles" and "meters", so they'll do.

To compare these two rates of speed, I need them to be in the same units. Flipping a coin, I decide that I'll convert the "80 miles per hour" to "meters per second". I need to set things up so the units will cancel:

Why did I put "1 hour" on top and "60 mins" underneath? Because I started with "80 miles per hour", so "hours" started out underneath. I want "hours" to cancel off, so the conversion factor for hours and minutes needed to have "hours" on top. That meant that "60 mins" had to be underneath.And that dictated the orientation of the next factor:.....Putting it all together, we get the following long string:

Now I cancel off the units:

Since the units cancel, leaving me with the "meters per second" that I need (circled above), I know the numbers must be in the right places. So to get my answer, all I have to do is grab a calculator and simplify the fraction multiplication:

This is a powerful technique. Cancelling units (also known as "unit analysis" or "dimensional analysis") is based on the principal that multiplying something by "1" doesn't change the value, and that any value divided by the same value equals "1".

• Suppose an object is moving at 66 ft/sec. How fast would you have to drive a car to keep pace with this object?

A car's speedometer doesn't measure feet per second, so you'll have to convert to some other measurement. You choose miles per hour. You know the following conversions: 1 minute = 60 seconds, 60 minutes = 1 hour, and 5280 feet = 1 mile. If 1 minute equals 60 seconds (and it does), then

The fact that the conversion can be stated in terms of "1", and that the conversion ratio equals "1" no matter which value is on top, is crucial to the process of cancelling units.

We have a measurment in terms of feet per second; we need a measurement in terms of miles per hour. To convert, we start with the given value with its units (in this case, "feet over seconds") and set up our conversion ratios so that all undesired units are cancelled out, leaving us in the end with only the units we want. Here's what it looks like:

Why did we set it up like this? Because, just like we can cancel duplicated factors when we multiply fractions, we can also cancel duplicated units:

I would have to drive at 45 miles per hour.

How did I know which way to put the ratios? How did I know which units went on top and which went underneath? I didn't. Instead, I started with the given measurement, wrote it down complete with its units, and then put one conversion ratio after another in line, so that whichever units I didn't want were eventually canceled out. If the units cancel correctly, then the numbers will take care of themselves.

If, on the other hand, I had done something like, say, the following:

...then nothing would have cancelled, and I would not have gotten the correct answer. By making sure that the units cancelled correctly, I made sure that the numbers were set up correctly too, and I got the right answer. This "setting up so the units cancel" is a crucial aspect of this process.

• You are mixing some concrete for a home project, and you've calculated according to the directions that you need six gallons of water for your mix. But your bucket isn't calibrated, so you don't know how much it holds. On the other hand, you just finished a two-liter bottle of soda. If you use the bottle to measure your water, how many times will you need to fill it?

• You find out that the average household in Mesa, Arizona, uses about 0.86 acre-feet of water every year. You get your drinking water home-delivered in those big five-gallon bottles for the water dispenser. How many of these water bottles would have to be stacked in your driveway to equal 0.86 acre-feet of water?

• You've been watching a highway construction project that you pass on the way home from work. They've been moving an incredible amount of dirt. You call up the information line, and find out that, when all eighty trucks are running with full crews, the project moves about nine thousand cubic yards of dirt each day. You think back to the allegedly "good old days" when work was all done manually, and wonder how many wheelbarrowsful of dirt would be equivalent to nine thousand cubic yards of dirt. You go to your garage, and see that your wheelbarrow is labeled on its side as holding six cubic feet. Since people wouldn't want to overfill their barrows, spill their load, and then have to start over, you assume that this stated capacity is a good measurement. How many wheelbarrow loads would it take to move the same amount of dirt as those eighty trucks?

Now Read Appendix B: p. 546( 547 in Your Text Books on SI UNITS

Copy the tables below

|Quantity Name |Unit Name |Unit Symbol |

| | | |

| | | |

| | | |

| | | |

| | | |

| | | |

Table 3:

|Prefix |Symbol |Factor |*** Conversion Factor |

| | |109 | |

| | |106 | |

| | |103 | |

| | |102 | |

| | |101 | |

| | |100 | |

| | |10-1 | |

| | |10-2 | |

| | |10-3 | |

| | |10-6 | |

| | |10-9 | |

Name: Date: Measurement Conversions [Metric to Metric]

1. 3.68 kg = ?g

2. 568 cm = ? m

3. 8700 ml = ? l

4. 25 mg = ?g

5. 0.101 cm = ? mm

6. 250 ml = ? l

7. 600 g = ?kg

8. 8900 mm = ? m

9. 0.000004 m = ? mm

10. 0.250 kg = ? mg

Now Read Appendix B: p. 548 ( 549 in Your Text Books on Scientific Notation

Now Read Appendix B: p. 549 ( 550 in Your Text Books on Significant Digits

Depends on the number of significant digits in the given data, as discussed in the rules below.

• Non-zero digits are always significant, so 22 has two significant digits, and 22.3 has three significant digits.

• With zeroes, the situation is more complicated:

a. Zeroes placed before other digits are not significant; 0.046 has two significant digits.

b. Zeroes placed between other digits are always significant; 4009 kg has four significant digits.

c. Zeroes placed after other digits but behind a decimal point are significant; 7.90 has three significant digits.

d. Zeroes at the end of a number are significant only if they are behind a decimal point Otherwise, it is impossible to tell if they are significant. To avoid uncertainty, use scientific notation to place significant zeroes behind a decimal point:

Ex. 200 has only one significant digit so if you mean 3, you write 2.00 x 10^2 to show that all three are significant

Ex. 8.200 x 103 has four significant digits , 8.20 x 103 has three significant digits , 8.2 x 103 has two significant digits

ADD/SUBTRACT – fewest number of decimals VS MULTIPLE/DIVIDE – fewest number of Sig. Fig.

Keep One Extra Digit in Intermediate Answers !!!

1. How many significant digits are in the following measurements?

a) 23.30 cm b) 1843.02 g c) 3.66 kg d) 0.0037 m e) 705000 s

2. Complete these addition problems.

a) 6.201 cm + 7.4 cm + 0.68 cm + 12.0 cm =

b) 1884 kg + 0.94 kg + 1.0 kg + 9.778 kg =

c) 16. 156 g + 28.2 g + 0.0058 g + 9.44 g =

3. Complete these subtraction problems.

a) 10.8 g – 8.264 g =

b) 2104.1 m – 463.09 m =

c) 16.50 mL – 8.0 mL =

4. Complete these multiplication problems.

a) 10.19 m x 0.013 m =

b) 3.2145 km x 4.23 km =

c) (7.50 x 106 m)(2.2 x 10-3 m) =

5. Complete these division problems.

a) 80.23 m

2.4 s

b) 4.301 kg

1.9 cm3

c) 6.6 x 108 m

2.31 x10-2

-----------------------

60 seconds : 1 minute

60 minutes : 1 hour

1 mile : 5280 feet

1 foot : 12 inches

2.54 centimeters : 1 inch

100 centimeters : 1 meter

1. The Introduction asks which is faster, 60 miles an hour or 60 feet a second. Well? (1 mi = 5280 ft; 1 hr = 3600 sec.)

2. How much does a 2-liter bottle of soda pop weigh in pounds? (Assume that the pop has the density of water, namely 1 kg/liter, and that the weight of the bottle itself is negligible.) (1 kg = 2.2 lb.)

3. An Englishman returning home from Norway has 860 kroner of pocket money that he never spent. How much is that in pounds? (Assume an exchange of NOK 12.32 = £ 1.00.)

4. In book IV of The Lord of the Rings, Frodo and Sam rappel down a cliff, using a rope 30 ells long. How high was the cliff in m, if the rope nearly reached the bottom? (1 ell = 45 in; 12 in = 1 ft; 1 ft = 39.37 cm.)

5. A Canadian vacationing in the States pays $1.689 a gallon for gasoline. What would be the equivalent price at home? (Gasoline is sold by the liter in Canada, as in most countries.) (1 US gal = 3.785 liter; assume a conversion rate of C$ 1.60 to the US dollar.)

6. You buy a 750 ml bottle of rum. How many rum-and-Cokes can you make, using an ounce and a half of rum in each drink? (1 US fluid ounce = 29.57 ml.)

7. What is 65 degrees in radian measure? (pi radians = 180°.)

Challenge Yourself:

8. How many cubic meters are there in a cubic mile? (1 mi = 1609.344 m.)

9. My 1967 Encyclopedia Britannica says that Lake Erie has a surface area of 9930 square miles and an average depth of 58 feet. How much water does it hold, in cubic miles? in liters? (1 mi = 5280 ft; 1 liter = 0.001 m³, and use the answer to the previous problem.)

10. Lake Erie has a surface area of 9930 square miles. If an inch of rain falls on the lake one day, how many gallons have been added to its volume? How many liters? (1 mi = 5280 ft; 1 ft = 12 in; 1 US gal = 231 in³; 1 US gal = 3.785 liters.)

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