4.2 Multiplying Fractions Multiplying Fractions If a, b, c, and d are numbers with (b and d not zero), then we rewrite the multiplication as a single fraction: a c a c b d b d Then use methods of section 4. to reduce the fraction to lowest terms. Example : Find the product. 9 0 Figure out the sign of the product first! The product of two numbers with different signs is negative. Write the negative sign at the beginning of your work. Rewrite each factor using prime factorization. 9 0 9 0 2 2 2 6 You Try It : Find the product. 0 b) 7 2 4 7 Carreon
Example 2: Find the product. 4 4 Figure out the sign of the product first! The product of two numbers with the same sign is positive. Rewrite each factor using prime factorization. You Try It 2: Find the product. 4 4 44 2 2 2 2 2 2 2 2 22 b) 6 77 24 Carreon 2
CAUTION: In examples and 2, you will notice that when we rewrote the product as a single fraction, we did not actually find the product of the numerators nor the product of the denominators before we prime factored. Why can t we just multiply the numerators and multiply the denominators at the beginning? To answer this question, we will work the following problem completing the multiplication first: 9 24 6 2 If we multiply the numerators first, we get 9 24 9 24 6 2 62 26 400 Notice that we have to spend time finding 9 24 and 6 2, which would involve using vertical multiplication: 24 9 and 26 6 2 80 400 We still have to rewrite each factor using prime factorization. 26 222 400 222 2 Notice that this involves undoing the vertical multiplication you just completed! This is why we don t actually find the product at the beginning of the problem. 2 2 2 2 2 2 2 2 27 0 With more experience multiplying fractions, you may find that you are able to mentally divide out common factors. This technique is similar to using common factors to reduce in section 4.. In example, we show to use this technique to simplify the problems in example and 2. Carreon
Example : Find the product by using common factors. b) 9 0 4 4 Figure out the sign of the product first! The product of two numbers with different signs is negative. Write the negative sign at the beginning of your work. 9 0 9 0 Look at each factor in the numerator and determine if they share any common factors with the denominator. If so, find the greatest common factor. and 0 have a greatest common factor of. and 9 have a greatest common factor of. Mentally divide each factor by the corresponding greatest common factor. 0 2 and 9 Make sure you use a slash to show the original factor cancels. Write the result of the mental division above the factors in the numerator and below the factors in the denominator. 9 0 2 2 6 Carreon 4
b) Figure out the sign of the product first! The product of two numbers with the same sign is positive. 4 4 44 Look at each factor in the numerator and determine if they share any common factors with the denominator. If so, find the greatest common factor. 4 and have a greatest common factor of 4. 4 and have a greatest common factor of. Mentally divide each factor by the corresponding greatest common factor. 4 4 4 and 4 Make sure you use a slash to show the original factor cancels. Write the result of the mental division above the factors in the numerator and below the factors in the denominator. 4 4 Notice that some of the remaining factors still have a common factor. and have a greatest common factor of. Mentally divide out the greatest common factor. Carreon
Note that the last three steps can be condensed: 4 4 However, this takes a great deal of keeping track of the numbers and you may be more likely to make a careless error. You Try It : Find the product by using common factors. 22 b) 6 77 24 So which is better? Using prime factorization or common factors to multiply? This is up to you! Both techniques (if carried out correctly) lead to the same answer. Some problems you will find that using common factors is easier; some problems you will find it easier to use prime factorization. We suggest starting off by using prime factorization because it leads to less errors by allowing you to see exactly which factors cancel out. (It may also lead to more partial credit on exam, should you make a minor error.) Once you get more comfortable with multiplying fractions, you may want to try using common factors or a combination of both techniques. Just make sure you carefully track your steps to avoid making a minor errors. Carreon 6
Example 4: Find the product. 6x y 4 y 0x Figure out the sign of the product first! The product of two numbers with different signs is negative. Write the negative sign at the beginning of your work. Rewrite each factor using prime factorization. Don t forget to write the variables in expanded notation. 6 x y 6 x y y 0x y 0x 4 4 2 2 x 7 y y y y 2x x x x 2 2 x 7 y y y y 2 x x x x 27 y y x x x 2 4y x You Try It 4: Find the product. 6 x y b) 9y 4x a c 2c 4a 6 6 6 2 Carreon 7
Example : Find 4 of 84. When the key word of follows a fraction, it indicates we are supposed to multiply. Replace of with a multiplication symbol. 84 Write 84 as a fraction 84. Rewrite each factor using prime factorization. of 84 4 84 4 84 4 84 4 227 22 2 2 7 2 2 7 6 6 You Try It : Find 2 of 96. Carreon 8
Example 6: Some financial advisors suggest that you should spend no more than 7 of your pre-tax income on your monthly mortgage payment. If your pre-tax income is $600, what is the highest monthly mortgage payment you can afford? The problem is asking for the highest monthly mortgage payment you can afford. The problem states that you are only able to spend 7 of your pre-tax income on your monthly mortgage payment. Translate 7 of your pre-tax income into mathematical expression. 7 $600 of follows a fraction so of means multiply Your pre-tax income is $600. 7 600 7600 Rewriting multiplication as a single fraction 260 Using prime factorization or dividing out common factors The highest monthly mortgage payment you can afford is $260. You Try It 6: Some financial advisors suggest that you should spend no more than 7 of your pre-tax income on your monthly mortgage payment. If your pre-tax income is $600, what is the highest monthly mortgage payment you can afford? Carreon 9