Elapsed Time and the Grouping FactorDate: 03/08/2003 at 15:02:46 From: Marylyn Wills Subject: Elapsed time How do I explain why you can't just subtract the two times given in a word problem and get the time elapsed? For instance... Carl studies from 5:25 to 5:45. How long does he study? The answer is simply 20 minutes, after subtracting the 2. BUT if he studies from 3:48 p.m. to 7:05 p.m., if you subtract the two, I get 3:57. I know you have to convert something to 60 minutes, but WHY? Thank you! Date: 03/09/2003 at 03:46:34 From: Doctor Ian Subject: Re: Elapsed time Hi Marylyn, The easiest way to find elapsed times is to add, not subtract. If Carl studies from 3:48 to 7:05, then he studies for 12 minutes from 3:48 to 4:00 he studies 3 hours from 4:00 to 7:00 he studies 5 minutes from 7:00 to 7:05 for a total of 12 minutes + 3 hours + 5 minutes = 3 hours + 17 minutes This is how I'd do it. It's very much like making change. But if you want to use subtraction, you have to be careful about borrowing. Let's look at a typical subtraction problem: 45 - 17 ---- Here, we can't subtract 7 from 5, so we have to borrow from the next column over. So we end up with 3 | 15 - 1 | 7 ----|--- 1 | 8 and then we're okay. But what happened here? We made use of the fact that a digit in the tens column is worth 10 times as much as the same digit in the ones column. That is, we traded one 10 for ten 1's. Now let's look at a subtraction of times, e.g., 7:05 - 3:48 ------ Now the 'normal' rules of borrowing no longer work. A 1 in the leftmost column is worth 1 minute; a 1 in the second column is worth 10 minutes; but a 1 in the third column is NOT worth 100 minutes. It's only worth 60 minutes. So borrowing from hours would look like 7:05 6:65 - 3:48 ---> - 3:48 ------ ------ To get a better feel for what's going on, suppose we want to do the following subtraction: 3 gallons, 2 quarts, 4 ounces - 1 gallon, 3 quarts, 11 ounces -------------------------------- We start with the smallest units, and find that we can't subtract 11 from 4, so we have to borrow some ounces by converting a quart. A quart is 32 ounces, so now we have 3 gallons, 1 quart, 36 ounces - 1 gallon, 3 quarts, 11 ounces -------------------------------- Now we have a similar problem with quarts. A gallon is 4 quarts, so we have 2 gallons, 5 quart, 36 ounces - 1 gallon, 3 quarts, 11 ounces -------------------------------- and now we can proceed smoothly: 2 gallons, 5 quart, 36 ounces - 1 gallon, 3 quarts, 11 ounces -------------------------------- 1 gallon, 2 quarts, 25 ounces The point here is that in the general case, the way borrowing works depends on how things are grouped, because borrowing is the inverse of grouping. In the decimal system, the grouping factor is always the same - you group 10 of something into the next largest thing, 10 ones -> 1 ten 10 tens -> 1 hundred 10 hundreds -> 1 thousand and so on. But in other systems, the grouping might be just about anything at all: 8 ounces -> 1 cup 2 cups -> 1 pint 2 pints -> 1 quart 4 quarts -> 1 gallon 60 seconds -> 1 minute 60 minutes -> 1 hour 24 hours -> 1 day 365 days -> 1 year 12 inches -> 1 foot 3 feet -> 1 yard 1760 yards -> 1 mile So the moral of the story is that when you borrow, it pays to keep track of what kind of thing you're borrowing. Does this make sense? I hope this helps. Write back if you'd like to talk more about this, or anything else. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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