Guess and Check Tables, AlgebraDate: 10/12/2002 at 00:45:58 From: B.b. Subject: About Guess and Check Tables... Hey Dr. Math! Can you explain this question for me? I don't understand it. In science class, Hector decides to measure the mass of his math, science, and Spanish textbooks. He discovers that the mass of his Spanish text is five more than four times the mass of his math book, while the mass of his science text is 110 grams less than his Spanish book. Combined, the mass of the texts is 1,178 grams. Find the mass of each book. A friend said to subtract 105 grams from the mass of the Spanish book. I didn't get it because the problem says '110 grams less.' She calculated and finally, her answers were: The math book is 142 grams, the Spanish book is 573 grams and the science book is 463 grams. Add them all together and get 1,178 grams. Is she right? Date: 10/12/2002 at 15:22:55 From: Doctor Ian Subject: Re: About Guess and Check Tables... Hi, We can check her answer by comparing it to the original problem description. First, the weights _do_ add up to 1178 grams. So the answer is plausible. If the math book is 142 grams, then the Spanish book should be 5 more than 4 times that, or 5 + 4(142) = 5 + 4(100 + 40 + 2) = 5 + 400 + 160 + 8 = 573 grams So it's looking pretty good. And the science book should be 110 grams less than this, or 473 grams. So it looks like she's right. On the other hand, her instructions to you don't make sense to me, either. To solve this with a table, you would make places to guess the masses of the various books: Math Spanish Science Total ---- ------- ------- ----- Suppose you guess that the math book is 50 grams. Math Spanish Science Total ---- ------- ------- ----- 50 Once you've chosen that, the mass of the Spanish book is fixed: Math Spanish Science Total ---- ---------- ------- ----- 50 4*50 + 5 ^ | 5 more than 4 times the mass of the math book And if the mass of the Spanish book is known, the mass of the science book is also fixed: Math Spanish Science Total ---- ---------- ----------------- ----- 50 4*50 + 5 (4*50 + 5) - 110 ^ | 110 grams less than the Spanish book Now, suppose we add these all up. We could go ahead and do the arithmetic, to get 50 + 4*50 + 5 + 4*50 + 5 - 110 = 50 + 200 + 5 + 200 + 5 - 110 = 410 - 110 = 300 Which is way too low. But we can do something clever, which is to do only _some_ of the arithmetic: 50 + 4*50 + 5 + 4*50 + 5 - 110 = 50 + 4*50 + 4*50 + 5 + 5 - 110 = 50*(1 + 4 + 4) + 10 - 110 = 50*9 - 100 Now, this is interesting, because we know where the 50 came from. It was our original guess! So we could just use _this_ expression to check our other guesses: Math = 100, total = 100*9 - 100 = 800 (still too low) Math = 200, total = 200*9 - 100 = 1700 (too high) and so on. Later on, when you get into algebra, you'll see how you can avoid guessing altogether by using a name instead of a number to work backward from the answer: guess + 4*guess + 5 + (4*guess + 5 - 110) = 1178 guess*9 - 100 = 1178 guess*9 = 1278 guess = 1278/9 = 142 Don't worry if you couldn't follow what I just did there. It's sort of like a preview of a movie that you won't get to see until later. The main thing to remember now is that although you've normally been taught to combine numbers whenever you see them, sometimes there are good reasons to hold off doing that, especially if you're looking for some kind of pattern that you can use to generate smarter guesses. I hope this helps. Write back if you'd like to talk more about this, or anything else. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ Date: 10/12/2002 at 16:40:59 From: B.b. Subject: Thank you (About Guess and Check Tables...) Hello, Dr. Ian! It's B.b. asking you about 'Guess and Check Tables'. I reread it over and over again and got it! Thanks a lot! :) You guys are a HUGE help to me! |
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