Flipping and Switching FractionsDate: 01/18/2002 at 11:39:19 From: Russell Marks Subject: Solving n in a fraction I need the equations for solving the following problems: n/2 = 5/10 1/n = 5/10 1/2 = n/10 1/2 = 5/n Thanks for your help. Date: 01/18/2002 at 12:47:17 From: Doctor Ian Subject: Re: Solving n in a fraction Hi Russell, The first thing to notice is that there are really only TWO cases that you have to worry about: ? b 1) - = - We don't know the numerator on the left. a c a b 2) - = - We don't know the denominator on the left. ? c Why? Because if you have the variable on the other side of the equation, you can always just switch sides. That is, if something = something_else then it's also true that something_else = something The second thing to notice is that there is really only ONE case that you have to worry about, because if something = something_else then it is also true that 1/something = 1/something_else (so long as neither of them is zero). So if a/? = b/c is true, then so is 1/(a/?) = 1/(b/c) 1*(?/a) = 1*(c/b) Remember, to divide by a fraction, you multiply by its inverse. ?/a = c/b Let's see how we can apply that to your equations: 1) n/2 = 5/10 We already like this one. We don't need to change it. 2) 1/n = 5/10 Flip both sides to get n/1 = 10/5. 3) 1/2 = n/10 Switch sides to get n/10 = 1/2. 4) 1/2 = 5/n Switch sides to get 5/n = 1/2. Then flip to get n/5 = 2/1. Okay, so now we know that we can always end up with something like ?/a = b/c What can we do with that? Well, we can think back to when we first learned about division. The way division is _defined_ is that whenever K = MN, it's also true that K/M = N and K/N = M (again, so long as neither M nor N is zero). Don't just take my word for this - check out a few examples: 12 = 3 * 4 so 12/3 = 4 and 12/4 = 3 20 = 4 * 5 so 20/4 = 5 and 20/5 = 4 15 = 2 * 7.5 so 15/2 = 7.5 and 15/7.5 = 2 and so on. Well, we have something that looks like K/M = N right? Let's match up the pieces: K | ?/a = b/c | \_/ | | M N This means that, by definition, K M | | ? = a(b/c) \_/ | N So once you have something like n/3 = 10/6 you can use the definition of division to write n = 3*(10/6) Let's look at an example of the toughest case: 12/4 = 18/? 18/? = 12/4 Switch sides. ?/18 = 4/12 Flip. ? = 18(4/12) The answer. Does this help? Write back if you'd like to talk about this some more, or if you have any other questions. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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