Subtracting Mixed Numbers with BorrowingDate: 03/31/2009 at 22:29:05 From: Bejoy Subject: Simplify 6 (1/8) - 3 (4/12) For 6(1/8)-3(4/12) my teacher got the answer 2(19/24) but i'm not getting that answer. i'm getting 3(5/24) First i multiplied the denominator 8 by 3 then i multiplied the second denominator 12 by 2. And i got the common denominator 24. Now my answer is 6(3/24)-3(8/24). After this i subtracted those fractions and got 3(5/24) but my teacher got a totally different answer. i have no idea how did she get that answer. Can you please help me to get the answer that my teacher got? Date: 04/01/2009 at 11:19:41 From: Doctor Ian Subject: Re: Simplify 6 (1/8) - 3 (4/12) Hi Bejoy - When we subtract two mixed numbers, we subtract the whole number parts, and we subtract the fractional parts, and put them together to get an answer, right? You have 6 3/24 - 3 5/24 ---------- Now, you can subtract 3 from 6, 6 3/24 - 3 5/24 ---------- 3 ? But can you subtract 5/24 from 3/24? No, because 5/24 is larger! Let's think about money for a moment. Suppose we invent a coin that is worth 1/24 of a dollar. (It would be worth a little bit less than a nickel.) I have 6 dollars and 3 of those coins. I need to give you 3 dollars and 5 of those coins. Can I do it? No, I don't have enough coins. What I _can_ do is find someone to "make change" for a dollar, giving me 24 of those coins. Then I have 5 dollars and 27 coins. So I can give you 3 of the dollars, leaving me with 2; and I can give you 5 of the coins, leaving me with 22. In terms of your problem, this looks like 6 3/24 trade 1 for 24/24 5 27/24 - 3 5/24 -----------------> - 3 5/24 ---------- ---------- 2 22/24 or 2 11/12 Does it make more sense now? Note that this is what you already learned to do when you subtract whole numbers, and have to "borrow" or "regroup". For example, if I want to subtract 543 - 287 ----- I can think of subtracting the various groups separately, 500 40 3 5 hundreds 4 tens 3 ones - 200 80 7 or - 2 hundreds 8 tens 7 ones ------------ ---------------------------- I can't subtract 7 from 3, so I have to "borrow" 10 from the second column: 500 30 13 - 200 80 7 ------------- 6 I can't subtract 80 from 30, so I have to "borrow" 100 from the third column: 400 130 13 - 200 80 7 ------------- 50 6 And now I can finish up: 400 130 13 - 200 80 7 ------------- 200 50 6 -> 256 It's just the same idea, except sometimes you're using whole numbers, and sometimes you're using fractions. The same idea shows up when using units like gallons and quarts, 5 gal 1 qt 4 gal 5 qt - 2 gal 3 qt -> - 2 gal 3 qt ------------- ------------- 2 gal 2 qt and when using time, 5 hr 15 min 4 hr 75 min - 1 hr 45 min -> - 1 hr 45 min -------------- -------------- This shouldn't be surprising, since we _could_ write those last two problems in terms of mixed numbers! A quart is 1/4 of a gallon, and 1 minute is 1/60 of an hour: 5 1/4 gal 4 5/4 gal - 2 3/4 gal -> - 2 3/4 gal ------------ ------------- 2 2/4 gal 5 15/60 hr 4 75/60 hr - 1 45/60 hr -> - 1 45/60 hr ------------- ------------- 3 30/60 hr The only thing that differs among these cases is the group size. With the numbers you've learned about, each group (thousands, hundreds, tens, ones) is always 10 times larger than the previous one. But in other systems, we have other group sizes: 1 gal = 4 qt 1 qt = 2 pt 1 pt = 2 cup 1 hr = 60 min 1 min = 60 sec and so on. One final thing you might find helpful is that you can avoid this kind of borrowing altogether. How? Well, suppose I have 10 dollars, and you have 6 dollars. I have 4 dollars more than you, right? Now, suppose someone gives us EACH 4 more dollars. Now I have 14, and you have 10. But I still have 4 dollars more than you. That hasn't changed. Same as if someone gives each of us 329 dollars, or a million dollars. Adding the same amount to each number doesn't change the difference between them. Do you see why? How can I use this? Well, suppose I have something like 6 3/24 - 3 5/24 ---------- I can add 19/24 to each of those numbers, and the numbers change, but the difference doesn't: 6 3/24 6 22/24 6 22/24 - 3 5/24 -> - 3 24/24 -> - 4 ---------- ---------- ---------- And now I have an easy subtraction! Again, I can do this with something as basic as 504 505 555 - 249 -----> - 250 ------> - 300 ----- add 1 ----- add 50 ----- So really, I _never_ have to borrow to do a subtraction, if I don't want to. I can just keep adding until the number I'm subtracting is a nice, round number--then I can do the subtraction easily. Sometimes one way is easier, sometimes the other way. It's good to know about both, so you have a choice. And, if you solve a problem in two different ways, getting the same answer both ways can give you confidence that you haven't made a mistake. Anyway, does this help clear things up? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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