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Chapter 9.5--Equations Containing Fractions Chapter 9.5--Equations Containing Fractions Part 1: Solve Equations Containing Fractions If we have an equation containing fractions we usually multiply both sides of the equation with the common denominator of the fractions. Then we can solve the equation without fractions. Of course, there is a hazard to this technique! You may get what looks like a bona fide solution that really is not a solution at all. We call these extraneous solutions and they must be discarded. 4x x−4 Question 9: Solve Multiply both sides by 4x x−4 5 5x x−4 x−4 x − 4 5x − 4 5x x−4 x − 4 4x 5x − 20 5x Look Ma! No Fractions!!! 4x 20 x 5 Always check your answers. 20 1 Question 12: Solve Multiply by 5? ? 55 ? Yes 1 x x4 3− 4 x4 x4 x 3x 4 − 4 x 3x 12 − 4 −2x 8 x −4 Check your answer −4 −44 Notice we are trying to divide by zero! Ouch! So -4 is not a real solution. There is no solution. Question 16 Now you solve this one: 5 3n−8 n n2 Cross-multiply 5n 10 3n 2 − 8n Moving everything to the left: 3n 2 − 13n 10 0 3n − 10n − 1 0 n 10 or 1 3 W Clarke 1 12/3/2004 Chapter 9.5--Equations Containing Fractions Part 2: Solve Proportions A ratio is a fraction relating two quantities. A proportion is an equation that sets two ratios equal to each other. Example: Speed is a ratio. It is the fraction of distance over time. When we say miles per hour, we mean the number of miles divided by the number of hours. To solve proportions, all we have to do is multiply both sides by the product of the two denominators. In actual practice this means we multiply each fraction by the denominator of the other fraction. We call this cross-multiplication. 16 64x Question 27: Solve the proportion: 9 Multiply by 9x . This is equivalent to cross-multiplying. 16x 9 64 Now divide by 16 x 964 9 4 36 16 6 2 Question 36: Now you solve this proportion: x−1 2x1 Cross multiply 22x 1 6x − 1 4x 2 6x − 6 −2x −8 x 4 2 Question 43: A simple syrup is made by dissolving 2 c of sugar in 3 c of boiling water. At this rate, how many cups of sugar are required for 2 c of boiling water. sugar Use the ratio wat er 2 2/3 22 s s 2 Multiply both sides by 2 2 3 Multiply top and bottom by 3 12 2 s So we need to use 6 c of sugar. W Clarke 2 12/3/2004 Chapter 9.5--Equations Containing Fractions Question 48: Leonardo da Vinci measured various distances on the human body in order to make accurate drawings. He determined that in general, the ratio of the kneeling height 3 of a person to his or her standing height is 4 . Using this ratio determine the standing height of a person who has a kneeling height of 48 in. kneeling Use st anding 3 4 48 s Cross multiply 3s 48 4 s 484 64 3 The person is 64 inches tall. Part 3: Solving similar triangles. Triangles are similar if corresponding pairs of angles are equal and corresponding sides are in proportion. In the diagram below, Triangles ABC and DEF are similar. F A B n m j i k C E p D We get the following proportions: j j i k i k m n, m p, n p Notice that in writing these proportions, I always put sides from the first triangle on top and those from the second triangle on the bottom. Of course if you flipped all of the fractions you would still have valid proportions. W Clarke 3 12/3/2004 Chapter 9.5--Equations Containing Fractions Question 65: Find the perimeter of triangle ABC. D C 10 m 7.5 m A 4 in F B AC 7.5 4 5 4 7.5 5 BC 10 30 5 4 5 E 5m Now multiply both sides by 7.5 AC 6 in BC 8 in Question 71 Now you do this one: Given MP and NQ intersect at O, NO measures 25 ft, MO measures 20 ft, and PO measures 8 ft. Find the length of QO. Q 20 ft M O 8 ft P 25 ft N We assume the triangles are similar. QO 25 8 20 Multiply by 25 QO W Clarke 825 20 4 10 12/3/2004 Chapter 9.5--Equations Containing Fractions Question 83: Three people put their money together to buy lottery tickets. The first person put in $25, the second person put in $30, and the third person put in $35. One of their tickets was a winning ticket. If they won $4.5 million, what was the first person's share of the winning? First person’s input Use the ratio T ot al First person’s share T ot al W innings 25 253035 f 4.5 Multiply by 4.5 (million) 254.5 90 1. 25 f So the first person's portion is $1.25 million. W Clarke 5 12/3/2004