3/4 of _____ Is 90 in Three Ways
Date: 03/24/2010 at 22:36:23 From: Rachel Subject: 3/4 of _____ is 90 Another one was 5/6 of 120 is ______ How do I find the answer and explain it to my son? Not sure what to be dividing by -- I think you divide -- but then how to explain it simply to my son in the 4th grade? 4 divided by 90 then multiply by 3? I know that is wrong, but doesn't it have something to do with division using the demoninator?
Date: 03/25/2010 at 09:17:17 From: Doctor Ian Subject: Re: 3/4 of _____ is 90 Hi Rachel, The most basic thing to know here is that a multiplication and a division are two different ways of expressing the same relationship. For example, these equalities ... 3 * 4 = 12 3 = 12 / 4 4 = 12 / 3 ... are just three ways of saying the same thing. Sometimes, we know two factors, and need a product: 1/2 * 6 = _____ But sometimes, we have a product and a factor, and need the other factor: 3/4 * _____ = 90 In this case, we can either (1) use guess-and-improve to identify the missing factor, or (2) rewrite the multiplication as a division, and evaluate that. The first approach might look like this: Could _____ be 12? 3/4 * 12 = (3 * 12)/4 = 36/4 = 9 So 12 is too small! Could _____ be 400? 3/4 * 400 = (3 * 400)/4 = 1200/4 = 300 So 400 is too large. The value we're looking for must be somewhere between 12 and 400. How about 200? 3/4 * 200 = (3 * 200)/4 = 600/4 = 150 Still too large. So we know it's between 12 and 150. How about 120? 3/4 * 120 = (3 * 120)/4 = 360/4 = 90 So now we know what the answer is. Note how I used each guess to improve my choice of the next guess. You can often solve problems pretty quickly this way, if you pay attention to what you're doing. But let's look at the second approach. We can rewrite the multiplication ... 3/4 * _____ = 90 ... as the equivalent division ... _____ = 90 / (3/4) This is good if you know the rule for dividing by a fraction, i.e., that to divide by a/b, you multiply by b/a: _____ = 90 / (3/4) = 90 * (4/3) = (90 * 4)/3 = 360/3 = 120 So we get the answer a little more quickly this way. There is a third way to think about it, which is this. If I have an equation that is true, like ... 2 + 3 = 5 ... and I multiply both sides by the same amount, e.g., ... 4 * (2 + 3) = 4 * (5) ... then the equation must still be true. (Does that make sense? If not, write back and we can talk about this, since it's one of the basic ideas that makes everything else work.) It's useful to know that the product of a fraction and its reciprocal is 1. For example, 3/4 * 4/3 = (3*4)/(4*3) = 12/12 = 1 So given something like ... 3/4 * _____ = 90 ... we can multiply both sides by the reciprocal of 3/4, 4/3 * 3/4 * _____ = 4/3 * 90 1 * _____ = 4/3 * 90 _____ = 4/3 * 90 Remember, if the first equation was true, then this last one must also be true! And note that it ends up at the same place as doing the equivalent division: That is, we multiply 90 by 4/3, which is the same as dividing it by 3/4. I know this seems like a lot, but the key thing to focus on here is that even if you end up doing a division, you can always start by writing the equivalent multiplication, which is often easier to understand. Once you have it, you have all kinds of options available to you. Does this help? Let me know if this helps. If it doesn't, we can look for some other way to approach it that will work better. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
Date: 03/25/2010 at 18:07:12 From: Rachel Subject: Thank you (3/4 of _____ is 90) THANK YOU!!!!!!! I look to Mr. Math website a lot to help my son with his homework, and then be able to help him understand. Your team has never let me down! The website is fantastic. Thank you again for your time! God bless, Rachel
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