Strategies for Solving Word Problems with Fractions and Units
Date: 08/02/2007 at 09:21:19 From: Jamie Subject: word problems with fractions On a road map with a scale of 1/4 inch per 10 miles, the highway from Waukee to Winterset is 1 3/8 inches long. How many miles long is this highway? I can't figure out if I need to multiply, divide, add or subtract. For example: 1 3/8 * 1/4 = 11/8 * 1/4 = 11/8 * 2/8 = 22/8 = 2 6/8 I get stuck because the answer is a whole number with no fraction. I tried subtracting and adding as well but I cannot get a whole number answer.
Date: 08/02/2007 at 10:00:54 From: Doctor Ian Subject: Re: word problems with fractions Hi Jamie, When you don't know what to do, sometimes it's helpful to use simpler numbers to help you see what's going on. For example, we might ask: On the map, the highway from A to B is 1/4 of an inch long. How long is the highway? It's easy to see that, since 1/4 of an inch corresponds to 10 miles, the answer must be: 10 miles. Okay, how about this one? On the map, the highway from A to B is 2/4 of an inch long. How long is the highway? It's got to be twice as long as the last one, so it's 2*10 = 20 miles. If we do this enough times, we can see that if we can write the distance as "something over 4", we can just multiply the something by 10 to get the distance we want. Does that make sense? Now, suppose we have something where the numbers don't work out so nicely, like On the map, the highway from A to B is 2/5 of an inch long. How long is the highway? We'd like to write 2/5 as "something over 4". To do that, we set up a proportion, 2 ? - = - 5 4 and we can solve that to find that 2 * 4 ----- = ? 5 8 - = ? 5 1.6 = ? (If you haven't learned how to solve a proportion like that, see Flipping and Switching Fractions http://mathforum.org/library/drmath/view/58193.html and let me know if it's not clear.) That is, 2 1.6 - = --- 5 4 So the highway must be 1.6*10 = 16 miles long. We could also do this with a fraction: 2 8/5 - = --- 5 4 so the highway is 8/5*10, or 16 miles long. Does this make sense? If so, let's try it with your numbers: 1 3/8 is the same as 11/8, so we have 11 ? -- = - 8 4 and we can find that 11 5 1/2 -- = ----- Just divide both the numerator and denominator 8 4 by 2, to get an equivalent fraction. So the length of the highway is 5 1/2 * 10 = 55 miles long. So far, so good? Note that including units in your calculations can also help you see when you're going astray. You wrote 1 3/8 * 1/4 = 11/8 * 1/4 = 11/8 * 2/8 = 22/8 = 2 6/8 But what are the units on those figures? 1 3/8 is the number of map inches, and 1/4 is the number of map inches corresponding to 10 miles. So we really have 1 3/8 map inches * 1/4 map inches per 10 miles = ... which gives you units of (map inches) squared per 10 miles... which isn't what you want. You'd like to end up with miles. Now, suppose we divide, instead: 1 3/8 map inches ----------------------------- 1/4 map inches per 10 miles That looks pretty messy, but it's the same sort of thing we do when we divide a distance by a rate to get a time: 90 miles ----------------- = 3 hours 30 miles per hour So we end up with 1 3/8 ----- (units of 10 miles) 1/4 Now, do you see how important the units are? If we leave them out, it would be easy to think that 1 3/8 divided by 1/4 is the number of MILES in the highway. But it's not! It's the number of UNITS OF 10 MILES, which is a very, very different thing. So we get 1 3/8 ----- (units of 10 miles) 1/4 = 1 3/8 * 4/1 (units of 10 miles) = 11/8 * 4/1 (units of 10 miles) = 11/2 (units of 10 miles) = 5 1/2 (units of 10 miles) = 5 1/2 * 10 miles = 55 miles which is what we got doing it the other way. So there are two lessons here, that you want to remember long after you've forgotten about this particular problem: 1) When in doubt, try simpler numbers to see if you can spot a pattern you can use for any numbers. 2) Include units in your calculations as a way of helping you spot nonsense calculations. Does this help? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
Date: 08/04/2007 at 09:51:42 From: Jamie Subject: Thank you (word problems with fractions) Thank you very much! I get it now.
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