Explaining Proportions and Unit PricesDate: 09/21/2006 at 11:47:01 From: Ta Subject: 5 lbs. divided by 10 lbs. = .5 lbs. How?? You need 5 lbs of flour to make 10 lbs of batter. How many pounds of flour do you need for 1 lb of batter? I know how to do percents, and I know how to get these answers. But I can't grasp this concept: Why does dividing (or getting the %) the 5 lbs by the whole (10 lbs) give you the amount for a SINGLE pound? Date: 09/21/2006 at 16:00:32 From: Doctor Ian Subject: Re: 5 lbs. divided by 10 lbs. = .5 lbs. How?? Hi Ta, Really, what's going on here is a proportion. It takes 5 lbs of flour to make 10 lbs of batter: 5 lbs flour ------------- 10 lbs batter Now, the main thing about a proportion is, you can scale it by n/n, where n is any number. For example, if we want to double the recipe, we'd have 5 lbs flour 2 10 lbs flour ------------- * - = -------------- 10 lbs batter 2 20 lbs batter If we wanted 40 lbs of batter, we'd have to quadruple it: 5 lbs flour 4 20 lbs flour ------------- * - = -------------- 10 lbs batter 4 40 lbs batter Does this make sense? If so, we can ask: By what would we have to scale 10 lbs of batter to end up with 1 lb of batter? That would be 1/10, right? So 5 lbs flour 1/10 5/10 lbs flour ------------- * ---- = -------------- 10 lbs batter 1/10 1 lbs batter The principle doesn't change just because the scale factor isn't an integer. Note that this is all that's going on when we compute unit fractions, e.g., 189 cents 1/12 (189 * 1/12) cents ---------- * ---- = ------------------ 12 ounces 1/12 1 ounce or speeds, 242 miles 1/1.5 (242 * 1/1.5) miles --------- * ----- = ------------------- 1.5 hours 1/1.5 1 hour or even percentages--except in this case, we want a "unit" of 100, since a percentage is a fraction with 100 in the denominator: 36 1/129 100 (36 / 129) * 100 --- * ----- * --- = ---------------- = [(36 / 129) * 100]% 129 1/129 100 100 In the other cases, once we end up with 1 in the denominator, we just express it using "per": 26.3 miles ---------- = 26.3 miles per hour 1 hour So you see, it's really just the same idea--proportions--used again and again, in slightly different ways. So once you've played around with proportions enough to really understand them, everything else will click into place, too. Does that make sense? Let me know if you need more help. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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