she says.
The Arithmetic of Measures
Our favorite teacher, Ms Barnes, strode into our office
the other day. She was not amused.
"It's all very well," she began, "to explain the math of Galileo's telescope, and Leibniz's calculus and Einstein's relativity and Newton's gravity. I'm sure my students have learned a lot, and enjoyed your stories, even though some of them may have had a little difficulty in following the mathematics. And the Media issue, which had a math test for reporters, was good.
"But," she continued, "I've been waiting for you to get into the math my students need every day."
We wondered what math that might be.
She looked at us as if we belonged in the corner with a dunce cap on our head.
"Kids buy things all the time," she began, "and should know what's a bargain and what's not. They should be able to estimate how long it will take them to walk to the mall, and they should know something about density. They need to be able to compute what they should tip at a cafe. They should know how many feet there are in a mile, how many ounces in a cup, and how many pounds in a ton. They should be able to compute how far away a bolt of lightning struck. They must be able to change miles to kilometers, kilograms to pounds, and liters to fluid ounces."
That sounded sensible to us, and we asked if she would give us a hand. She agreed.
Calculating Time from Speed and Distance
"Suppose one of my students is off to the Mall," she began. He knows that it's 20 blocks from his home. If he walks, how long will it take him to get there?
"To find a time, you need to know distance and speed. Because speed=distance/time, and so time=distance/speed. We know the distance is 20 blocks. So how fast can we walk? An adult walking fast travels maybe 3 miles per hour (mph). Let's suppose a young person makes 2 1/2 mph. So time is 20 blocks/2 1/2 mph. What's wrong with that?"
We said distance has to be in miles, so when we divide miles by miles-per-hour we'll get hours.
"Correct," said Ms Barnes. "So how do we convert blocks to miles? A block in one part of town may be longer or shorter than one in another part. But a good guess is that there are ten blocks to a mile. So 20 blocks might be 2 miles.
"So the time is 2 miles/2 1/2 mph, or 2 miles/(5/2 mph), which is 2x2/5 hours = 4/5 hours. And 4/5 of 60 minutes is 48 minutes.
Time for a Bike Ride
"Of course, our student could have ridden his bicycle. Then he might have gone 12 miles per hour, and would get to the mall in 2 miles/(12 mph), which is 1/6 hour or 10 minutes. (But bikes can travel faster than 12 mph. A cyclist once went 512 miles in 24 hours, which is 512/24 = 21 mph. At the famous Tour de France race, bikers maintain 30mph average during a single day.)
How Long Before we can Buy a Game Boy?
"So now he's at the Mall. He goes to the Game Store, and finds that a Gameboy console with a 30-game card costs $285. He's got $47 in the bank. But he
gets a $7 allowance every week, and can save $4 of that. How long will it be before he can buy the
Gameboy?
"He'll need $285-$47, or $238. If he saves $4/week, it'll take $238/($4/week), or approximately $240/4=60 weeks. Since a year is 52 weeks, he'll be able to buy the game in 13 months or so.
What's the Best Candy Buy?
"Our student goes 'round the mall and finds a booth that sells candy. His favorite chocolate bar is the Gnarly, and he can buy one bar for $1.50, or a pack of seven bars for $9.00. Sounds like the pack is a good deal, but what would he save? Seven bars at $1.50 would be a total of $10.50, so he'd save $1.50.
When Lightning Strikes
"You're out for a walk and there's thunder and lightning. How far away did that lightning hit? Well, light travels very fast -- 186,000 miles every second. So there's practically no time between when the bolt hits its target and you see it. However, sound travels much slower: about 1100 feet per second. So what you must do is count the seconds between when you see the flash and when you hear the thunder. You multiply that time by 1100, and you'll get the distance in feet. For example, if there's a 6-second delay, the bolt hit 6 seconds times 1100 feet/second, or 6600 feet away.
Tipping the Waiter or Waitress
"You eat lunch at a cafe, and your bill is $6.75. How big a tip should you leave? An average tip is 15%, so you need to find 15% or $6.75. Ten percent would of course be $0.675 -- call it 68 cents. Fifteen percent is 10% plus half of 10%. Half of 68 cents is 34 cents, so the 15% tip would be 68+34=$1.02 -- call it a dollar.
All have the same volume: a cubic foot--1 foot high, 1 foot wide, and 1 foot deep. Would you expect
them all to weigh the same?"
We guessed the lead cube weighed more than the gold one,
which was heavier than the one filled with water.
"You'd be wrong," Ms Barnes replied. "Gold is heavier than lead. The density of gold is 1205 pounds per cubic foot, of lead is 708 lb./cu.ft., and of water 62.4 lb./cu.ft. Since each of the blocks in our figure has a volume of one cubic foot, their densities tell us what they weigh.
"These days," she continued, "gold sells for about $1000 per ounce. Suppose we wanted to know how big an ounce is. To figure that out, we have to know something about pounds and ounces."
She showed us a table.
"Here are some numbers everyone should know -- measures of weight, volume, and length. Note the ounce -- abbreviated oz. -- is used to measure both volume and weight. The volume measure is called a 'fluid ounce'.
Converting Between Measuring Units
"With these numbers we can find how big an ounce of gold is. But first, let's learn how to convert from one kind of measuring unit to another. The useful thing to learn is that units -- pounds or feet or quarts or miles -- can be 'canceled out' just like numbers can."
We asked how numbers can be cancelled.
"Surely you know that," Ms Barnes replied. "Suppose we want to divide 294 by 78. Looks complicated? The thing to do is to find the factors of those numbers, by dividing. Divide 294 by 2 and get 147, so 294=2x147. But 147 is divisible by 3, because 1+4+7=12, which is divisible by 3. 147/3=49. And 49=7x7. So 294=2x3x7x7. Now how about 78? It's divisible by 2, so 78=2x39. But 39=3x13, so 78=2x3x13. And to divide 294 by 78, we have 2x3x7x7/2x3x13. The two's and the three's 'cancel out', so 294/78=49/13.
The same thing works with units. Here's one example: How many ounces in 7 pounds? We know there are 16 ounces in a pound. So 7 pounds=7 pounds x 16 ounces/pound. The pounds 'cancel' out, and we're left with 7x16=112 ounces. Or suppose we want to know how many yards there are in 5280 feet -- which is a mile. We do this: 5280 feet=5280 feet/(3 feet/yard)= 5280/3 yards -- the feet 'cancel out'. So a mile is 5280/3=1760 yards.
"Now let's apply this to density." She showed us some calculations. "We want the cubic inches in an ounce of gold. Here in blue you see the calculation. We multiply 1205 pounds per cubic feet of gold by 16 ounces in any pound. That gives us 19,280 ounces of gold in a cubic foot. So we can compute what our block of gold is worth. Since gold is worth $1000 per ounce, the total cube is worth $1000 times 19,280, or $19,280,000! Almost $20 million.
"Now we divide by 1728 cubic inches in a cubic foot."
We wondered where the 1728 came from.
"A cubic foot measures 12 inches by 12 inches by 12 inches. And 12 cubed is 1728. So you see why the calculation works? Just look at the units:
(pounds/cubic foot) times (ounces/pound) = ounces/cubic foot. The pound units 'cancel out'. And if we divide 19280 oz./cu. ft. by 1728 cu. in./cu. ft., we get 11.15 oz./ cu. in. -- the cubic feet 'cancel out'.
"So there are 11.15 ounces of gold in a cubic inch. That means an ounce of gold fits in 1/(11.15) = .09 cubic inches. And a cube with that volume has a side the cube root of 0.09, or about .45 inches -- roughly half an inch.
"The red figures show how to find the size of a glass that'll hold a cup of water. A cup is 8 fluid ounces. We divide 231 cubic inches per gallon by 128 fluid ounces per gallon, and get 1.8 cubic inches per fluid ounce. Eight times that is a cup. And remembering that the most efficient cup has a height equal to twice its radius. So twice the radius times pi times the radius squared must be 14.4 cubic inches. And we find the radius of the cup is about 1 and 1/3 inches.
not Civilized?"
M. Enerve Complains
About Feet and Pounds
This has been a week full of irritated visitors. The day after Ms Barnes came complaining
about our seeming lack of interest in the basics of math, an excitable gentleman stormed into our office and accused us of being
behind the times. We tried to calm him down, but during his entire visit he waved his arms to emphasize his points. (Upon
discovering that he is French, we realized he would favor us with a lot of arm-waving.)
"Why do you use these ancient measures?" he asked. "I weigh myself on the machine and am told I weigh 165 pounds. The speed limit sign says I can't drive over 35 miles per hour. On the highway I find the next town is 48 miles away. The sign on the bank says the temperature is 67 degrees. In the grocery store I must buy a quart of milk. What are these pounds and miles and quarts? If the temperature is 67 degrees I must be on fire.
"Long ago we French realized that the English miles and pounds and quarts and degrees of temperature were ridiculous. There was a much better way of dealing with length and weight and volume temperature."
We told him we thought miles and so on were fine.
"Bah!" was his reply. "What is an inch, exactly? Why does a foot contain twelve of them? And a yard 36? Why are there 5280 feet in a mile? Why are there 2000 pounds in a ton? We live in a decimal world, with ten fingers."
We guessed he was going to tout the metric system.
"But of course," he replied. "A beautiful system. And with tens everywhere." He showed us a Scaling Table. "You have 5280 feet in a mile, we have 1000 meters in a kilometer. You have 16 ounces in a pound, we have 1000 grams in a kilogram. You have 4 quarts in a gallon, we have 10 deciliters in a liter."
We confessed we admired all the tens, but told him that we seem to agree on time We have 1000 milliseconds in a second, and a million in a microsecond. And we wondered whether his metric system has 60 minutes in an hour, 24 hours in a day, and 365 days in a year. Would it maybe be better to have 100 minutes in an hour, ten hours in a day, 10 days in a week, and 100 weeks in a year.
M Enerve waved his arms again and reminded us that this would make a year equal to 1000 days, but that the earth goes around the sun in only 365.
"All you foolish Americans should use grams, meters, liters, and Centigrade degrees for our measures," he argued. And while you're at it, you must learn how to convert from English to metric units."
He showed us another table.
"You must learn that a kilogram is 2.2 pounds, for example. And you should be able to calculate that my 165 pounds is really 165 pounds divided by 2.2 pounds per kilogram, or 75 kilograms. A speed of 35 miles per hour is really 35 mph times 1.6 kilometers per mile, or 56 kilometers per hour. The town 48 miles away is 48 miles times 1.6 kilometers per mile, or 77 kilometers from here. A quart of milk is 1 quart times 0.95 liters per quart, or 0.95 liters. And a temperature of 67 degrees Fahrenheit is 67-32=35 degrees F. above freezing. Centigrade freezing is 0 degrees, and there are .56 degrees centigrade in each degree Fahrenheit, so the temperature is really 20 degrees Centigrade.
"If it were 67 degrees Centigrade, that would mean it's 1.8 times 67 equals 121 degrees above freezing Fahrenheit, or 121+32=153 degrees Fahrenheit. Which is why I said I must be on fire.
We thanked M Enerve for his kindness in helping us, and told him we'd try to educate the readers of the Gnarly Gnews.
(Continued from left column)
"But of course that's not the bike's volume. It's the
volume of the space occupied by a bike. And most of that volume is air.
"So how can we find the volume of a bicycle? It's easy. Fill a big tub with water -- right to the top. Then lower the bike into the tub, and collect that water that overflows from the tub. The volume of the water equals the volume of the bike! You just divide that volume into the bike's weight, and you get its real density. Back in about 270 BC Archimedes used this trick to prove that a crown bought by King Hieron was not solid gold. He measured the volume of the crown by using the overflowing tub, and then showed that a chunk of gold with that volume weighed more than the crown did.
"The goldsmith who made the crown had pretended it was pure gold, and he found himself without a head after Archimedes' experiment.
"Of course, you may wonder what your own body density is. You could do Archimedes' trick in a bath tub, but that's a bit awkward. Suppose we pretend your body is made of cylinders and a box shape. If you're 5 foot 7 inches (67 inches) tall and weigh 140 pounds your volume may be sort of like the green figure on the right. Your head is a cylinder 8 inches in diameter and 11 inches tall. Your two arms and two legs are also cylinders, with the dimensions as shown. And your body is the shape of a box 28 inches high, 16 inches wide and 5 inches deep.
"The calculations are below on the left. We find the volume of a cylinder by squaring the radius, multiplying by the height and then by pi. That is:
"Cylinder volume = pi r2 h.
"The volume of a box is just the product of its three dimension.
"Doing the calculations and adding the volumes we get a total volume of 3976 cubic inches. Dividing that into your 140-pound weight gives a density of 0.035 pounds per cubic inch. And we can change pounds per cubic inch into pounds per cubic foot by multiplying by 1728 cubic foot per cubic inch. So we find the density of our green person is 60.5 pounds per cubic foot. Of course those green figures may not be a good approximation to a real human, so we should better say the density is somewhere in the neighborhood of 60 pounds per cubic foot -- say, between 50 and 70.
"If you look at the Density table above, you'll see the density of water is 62.4 pounds per cubic foot -- not far from the density of our strange teen- ager. And that's not surprising, for our bodies contain a whole lot of water.
The gram, centigrade, and the liter.
The foot, quart, and pound
Lost a whole lot of ground
To a system which was a lot neater.
Having been scolded by both Ms Barnes and the excitable Frenchmen, we sought out
our young friend Chloe, hoping she'd brighten our day and give us her view of measurements. As we expected, she was both
cheerful and full of history. We told her we were trying to learn about measurements.
some stones? A small one, and then one twice that size? And three times as big? And so on. And with their balance they could
find that a rabbit weighed the same as four stones. And a loaf of bread two stones. Who knows how they decided the size? But even today the English weigh themselves in what they call stones which are equal to 14 of what we now call pounds? Me, I weight about
eight stones.
"So in 1790, the French government set up, you know, a commitee. To create a new
measurement system? They decided Mouton was right? They'd base the unit of length on the size of the earth. They called it a meter? And 1000 meters would be a kilometer? And the distance from the north pole to the equator would be 10,000 kilometers. That's how the
kilometer was defined? With, like, 1000 meters to the kilometer. So the distance around the earth would be 40,000 km?
Sam'll Answer 
