Multiplying Fractions, Dividing Common Factors: Why Does It Work? Date: 06/10/2010 at 14:10:42
From: Jim Subject: Multiplying Fractions A student\’s former teacher taught her students that if the numerator of one fraction and the denominator of the other fraction had a common factor, then they could divide each by that factor before multiplying. For example, say we have 4/6 x 8/9 Since the 6 and 8 are multiples of 2, you could divide both by two, creating the following result: 4/3 x 4/9. When multiplied, 4/6 x 8/9 has the same result as 4/3 x 4/9. My question is. why does this work? What is the underlying math principle? My kids and I would love to know! Thanks for your help. Date: 06/10/2010 at 15:37:00 From: Doctor Ian Subject: Re: Multiplying Fractions Hi Jim, Let\’s start with a particularly simple example, where one numerator is the same as the other denominator: 2 3 – * – 3 5 The first thing to notice is that when we follow the rule for multiplying fractions, we get 2 3 2 * 3 – * – = —– 3 5 3 * 5 Now, because multiplication is commutative, we can change the order of the operands, so we could rewrite that as 2 3 2 * 3 3 * 2 – * – = —– = —– 3 5 3 * 5 3 * 5 And we can un-multiply that to get 2 3 2 * 3 3 * 2 3 2 – * – = —– = —– = – * – 3 5 3 * 5 3 * 5 3 5 But 3/3 is the same as 1, and multiplying by 1 is the same as doing nothing: 2 3 2 * 3 3 * 2 3 2 2 2 – * – = —– = —– = – * – = 1 * – = – 3 5 3 * 5 3 * 5 3 5 5 5 So here, the numerator and denominator just cancel out. Does this make sense? The important idea here is that when multiplying fractions, we can group factors in the numerator and denominator any way we want. And if we have something that appears in both the numerator and denominator, we can separate that out to get n/n, or 1. That\’s really what we\’re _doing_ when we cancel. We\’re separating out n/n pairs, which turn into 1, and thus disappear.
This really comes into its own when we break things into prime factors before multiplying. For example, 4 35 — * — =? 15 72 We can break the numerators and denominators up into prime factors: 4 35 2*2 5*7 — * — = — * ——— 15 72 3*5 2*2*2*3*3 Those all just get multiplied together. 4 35 2*2*5*7 — * — = ————- 15 72 2*2*2*3*3*3*5. and if we align them, we can spot pairs that can be canceled: 4 35 2 * 2 * 5 * 7 — * — = —————————— 15 72 2 * 2 * 2 * 3 * 3 * 3 * 5 We could go to the trouble of separating them. 4 35 2 2 5 7 — * — = – * – * – * ————- 15 72 2 2 5 2 * 3 * 3 * 3. but in practice, we don\’t bother. We would just cross them out, and multiply what\’s left: x x x 4 35 2 * 2 * 5 * 7 7 7 — * — = —————————— = ——- = — 15 72 2 * 2 * 2 * 3 * 3 * 3 * 5 2*3*3*3 54 x x x This is a lot easier than doing the multiplication, then reducing it by finding common factors. Still making sense? If so, we\’re almost done! Let\’s look at your example: 4 8 – * – =? 6 9 When we note that 6 and 8 have a common factor. 4 2*4 — * — =? 2*3 9. we recognize that we could separate those out, but we just toss the common factor, and keep the rest: 4 4 — * — =? 3 9 It\’s the same result as writing everything out, and moving everything around — but a lot quicker. Does this help? – Doctor Ian, The Math Forum http://mathforum. org/dr. math/ Date: 06/11/2010 at 08:08:27 From: Jim Subject: Thank you (Multiplying Fractions) After hearing me drone on and on about the importance of the commutative property, my 7th graders really connected with this explanation. They are now making up their own examples trying to stump each other. This is also giving them invaluable practice working with factors. Thanks so much! Multiply two or more fractions.
Multiply a fraction by a whole number. Multiply two or more mixed numbers. Solve application problems that require multiplication of fractions or mixed numbers. Just as you add, subtract, multiply, and divide when working with whole numbers, you also use these operations when working with fractions. There are many times when it is necessary to multiply fractions and. For example, this recipe will make 4 crumb piecrusts: 5 cups graham crackersPPPPPPP 8 T. sugar Ptsp. vanilla Suppose you only want to make 2 crumb piecrusts. You can multiply all the ingredients by, since only half of the number of piecrusts are needed. After learning how to multiply a fraction by another fraction, a whole number or a mixed number, you should be able to calculate the ingredients needed for 2 piecrusts. When you multiply a fraction by a fraction, you are finding a fraction of a fraction. Suppose you have PBy dividing each fourth in half, you can divide the candy bar into eighths. Then, choose half of those to get. In both of the above cases, to find the answer, you can multiply the numerators together and the denominators together. If the resulting needs to be simplified to lowest terms, divide the numerator and denominator by common factors. You can also simplify the problem before multiplying, by dividing common factors. You do not have to use the simplify first shortcut, but it could make your work easier because it keeps the numbers in the numerator and denominator smaller while you are working with them. When working with both fractions and whole numbers, it is useful to write the whole number as an (a fraction where the numerator is greater than or equal to the denominator). All whole numbers can be written with a 1 in the denominator. For example:, and. Remember that the denominator tells how many parts there are in one whole, and the numerator tells how many parts you have.
Often when multiplying a whole number and a fraction the resulting product will be an improper fraction. It is often desirable to write improper fractions as a mixed number for the final answer. You can simplify the fraction before or after rewriting as a mixed number. See the examples below. If you want to multiply two mixed numbers, or a fraction and a mixed number, you can again rewrite any mixed number as an improper fraction. So, to multiply two mixed numbers, rewrite each as an improper fraction and then multiply as usual. Multiply numerators and multiply denominators and simplify. And, as before, when simplifying, if the answer comes out as an improper fraction, then convert the answer to a mixed number. As you saw earlier, sometimes its helpful to look for common factors in the numerator and denominator before you simplify the products. In the last example, the same answer would be found if you multiplied numerators and multiplied denominators without removing the common factor. However, you would get, and then you would need to simplify more to get your final answer. Now that you know how to multiply a fraction by another fraction, by a whole number, or by a mixed number, you can use this knowledge to solve problems that involve multiplication and fractional amounts. For example, you can now calculate the ingredients needed for the 2 crumb piecrusts. Often, a problem indicates that multiplication by a fraction is needed by using phrases like half of, a third of, or Pof. You multiply two fractions by multiplying the numerators and multiplying the denominators. Often the resulting product will not be in lowest terms, so you must also simplify. If one or both fractions are whole numbers or mixed numbers, first rewrite each as an improper fraction. Then multiply as usual, and simplify.