Borrowing in SubtractionDate: 12/01/98 at 23:00:14 From: Christy Subject: Subtraction with three numbers When I work on the problem: 100 - 99 ---- I don't know why the zero in the tens place becomes a 9 when I borrow from it. I know that I am taking the 1 from the hundreds position to borrow, but why does the zero in the tens become a nine? Date: 12/02/98 at 12:03:30 From: Doctor Peterson Subject: Re: Subtraction with three numbers Hi, Christy. I like to teach "borrowing" using money. If you want, get out some play money ($100's, $10's, and $1's) and try acting out what I say. It's fun, even if you don't need it. When we write 100, it means 1 hundred, 0 tens, and 0 ones. So lay out three piles (two of them will be empty!): $100's $10's $1's ------ ----- ---- 1 0 0 I like to have fun with this and pretend these piles represent three people, "Mr. Hundreds," "Mr. Tens," and "Mr. Ones." Now suppose we come along and want to get $99 from them (9 tens from Mr. Tens and 9 ones from Mr. Ones). We first go to Mr. Ones and ask for the 9 ones, but he doesn't have any ones today, so he says "I'll be right back," and goes next door to Mr. Tens to ask for a ten dollar bill that he can change into 10 ones. Mr. Tens doesn't have any, but he goes next door to Mr. Hundreds and asks for a hundred dollar bill that he can change into 10 tens. He's in luck! He takes it, goes to the bank, and now he has 10 tens: $100's $10's $1's ------ ----- ---- 0 10 0 Now he can give one of his tens to Mr. Ones, who changes it to 10 ones: $100's $10's $1's ------ ----- ---- 0 9 10 Now Mr. Ones can give us 9 of his 10 ones: $100's $10's $1's ------ ----- ---- 0 9 10 -9 -- 1 Then we can go to Mr. Tens and ask for all of his 9 tens: $100's $10's $1's ------ ----- ---- 0 9 1 -9 -- 0 So they have nothing left but a single one. Now let's write all this down the way we usually do: 1 0 0 - 9 9 ------ We can't subtract 9 from 0, so we "go next door" to borrow a ten. Since there are no tens to borrow, before we can do that we have to go yet another place over to borrow a hundred, which turns into 10 tens: 0 10 / / 1 0 0 - 9 9 ------ Now we can take one from the tens (leaving 9 tens), which turns into 10 ones, from which we can subtract 9: 0 9 10 / / / 1 0 0 - 9 9 ------ 1 Finally, we can subtract the 9 tens (and no hundreds): 0 9 10 / / / 1 0 0 - 9 9 ------ 0 0 1 So where did that 9 come from in the tens place? It's the 10 tens we borrowed (which was one hundred), minus the 1 ten that was borrowed from it by the ones. Try adding: 99 + 1 ---- and you'll see a double carry just like the double borrow in your subtraction. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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