6 MODULE 5. PERCENTS 5b Solving Equtions Mening of Proportion In this skill we review equtions tht involve percents. review the mening of proportion. Our first tsk is to Proportions. A proportion is sttement tht equtes two rtios or rtes. Extremes nd Mens One concept tht is needed is the ide of extremes nd mens. Extremes nd Mens. The first nd fourth terms re clled the extremes of the proportion. The second nd third terms re clled the mens of the proportion. In the proportion, the terms nd d re the extremes; the terms b nd c re the mens. extremes mens If we multiply both sides of the proportion by the common denomintor, ( ) ( c bd = bd b d) then cncel, we get the following result. ( ) ( c bd = b d b d) d = bc This leds to the following observtion. Product of Extremes nd Mens. In the proportion the product of the mens equls the product of the extremes. Tht is, d = bc.
5B. SOLVING EQUATIONS 7 We get n equivlent result using technique clled cross multipliction. Product of mens = bc Product of extremes = d EXAMPLE 1. Solve the proportion for x: Solution. Cross multiply, then solve the resulting 4 = x 12 Originl proportion. 4 x = 12 Products of mens nd extremes re equl. 4x =6 4x 4 = 6 4 x =9 Divide both sides by 4. Check. Substitute 9 for x into the originl proportion nd check. 4 = x 12. Solve the proportion for n: 2 = n 9 4 = x 12 4 = 9 12 Originl proportion. Substitute 9 for x. Cross multiply. 4 = 9 12 Product of mens = 6 Product of extremes = 6 Thus, the solution 9 checks. Answer: 6 EXAMPLE 2. Solve the proportion for x: 2x +1 15 = 1. Solve the proportion for y: 6+2y 18 = 8 9
8 MODULE 5. PERCENTS Answer: 5 Solution. Cross multiply, then solve the resulting 2x +1 = 1 15 Originl proportion. (2x + 1)= 15(1) Products of mens nd extremes re equl. 6x + = 15 6x + =15 6x =12 6x 6 = 12 6 x = 2 On the left, distribute. On the right, multiply. Subtrct from both sides. Divide both sides by 6. Simplify both sides. Check. We ll leve it to our reders to check this solution. Solving Percent Problems There re three bsic types of percent problems: 1. Find given percent of given number. For exmple, find 25% of 640. 2. Find percent given two numbers. For exmple, 15 is wht percent of 50?. Find number tht is given percent of nother number. For exmple, 10% of wht number is 12? Let s begin with the first of these types: Find given percent of given number. Wht number is 6% of 120? EXAMPLE. Wht number is 25% of 640? Solution. Let x represent the unknown number. Trnslte the words into n Wht number is 25% of 640 x = 25% 640 Now, solve the eqution for x. x = 25% 640 Originl x =0.25 640 Chnge 25% to deciml: 25% = 0.25. x =160 Multiply: 0.25 640 = 160.
5B. SOLVING EQUATIONS 9 Thus, 25% of 640 is 160. Now we ll ddress our second item on the list: Find percent given two numbers. EXAMPLE 4. 15 is wht percent of 50? 14 is wht percent of 25? Solution. Let x represent the unknown percent. Trnslte the words into n 15 is wht percent of 50 15 = x 50 The commuttive property of multipliction llows us to chnge the order of multipliction on the right-hnd side of this Now we cn solve our eqution for x. 15 = 50x. 15 = 50x Originl 15 50 = 50x 50 Divide both sides by 50. 15 = x 50 Simplify right-hnd side. x =0.0 Divide: 15/50 = 0.0. But we must express our nswer s percent. To do this, move the deciml two plces to the right nd ppend percent symbol. Thus, 15 is 0% of 50. 0.0 = 0 0.% =0% Alterntive Conversion. At the third step of the eqution solution, we hd x = 15 50. We cn convert this to n equivlent frction with denomintor of. x = 15 2 50 2 = 0 Thus, 15/50 = 0/ = 0%. Answer: 56%
10 MODULE 5. PERCENTS The next exmple illustrtes the third type of percent problem: Find number tht is given percent of nother number. 20% of wht number is 45? EXAMPLE 5. 10% of wht number is 12? Solution. Let x represent the unknown number. Trnslte the words into n 10% of wht number is 12 10% x = 12 Chnge 10% to frction: 10% = 10/ = 1/10. 1 10 x =12 Now we cn solve our eqution for x. ( ) 1 10 10 x = 10(12) Multiply both sides by 10. Thus, 10% of 120 is 12. x =120 Let s do some ddition exmples with mixed number percentges. Wht number is 105 4 %of 222? EXAMPLE 6. Wht number is 105 1 4 % of 18.2? Solution. Let x represent the unknown number. Trnslte the words into n Wht number is 105 1 4 % of 18.2 x = 105 1 4 % 18.2 In this cse, the frction termintes s 1/4 = 0.25, so Now we cn solve our eqution for x. 105 1 %=105.25% = 1.0525. 4 x =105 1 4 % 18.2 Originl x =1.0525 18.2 5 1 4 %=1.0525. x =19.1555 Multiply.
5B. SOLVING EQUATIONS 11 Thus, 105 1 4 % of 18.2 is 19.1555. Answer: 24.765 EXAMPLE 7. 11 1 9 % of wht number is 20? 12 Solution. Let x represent the unknown number. Trnslte the words into n 2 % of wht number is 760? Chnge 11 1 9 % to frction. 11 1 9 % of wht number is 20 11 1 9 % x = 20 11 1 1 9 %=11 9 = 9 = 9 1 = 9 1 = 1 9 Percent: Prts per hundred. Mixed to improper: 11 1 9 =/9. Invert nd multiply. Cncel. Replce 11 1 9 % with 1/9 in the eqution nd solve for x. 1 9 x =20 111 9 %=1/9. ( ) 1 9 9 x = 9(20) Multiply both sides by 9. x =180 Thus, 11 1 9 % of 180 is 20. Answer: 6,000