USING RECTANGLES TO MULTIPLY 5.1.1 through 5.1. Two ways to find the area of a rectangle are: as a product of the (height)! (base) or as the sum of the areas of individual pieces of the rectangle. For a given rectangle these two areas must be the same, so Area as a product = Area as a sum. Algebra tiles and, later, generic rectangles, provide area models to help multiply epressions in a visual, concrete manner. See the Math Notes bo on page 218. Eample 1 Using Tiles The algebra tile pieces 2 + 6 + 8 are arranged into a rectangle inside the cornerpiece as shown at right. The area of the rectangle can be written as the product of its base and height or as the sum of its parts. ( 12 + 4) ( 12 + 2)!!!!!!!=!!!!!!!2 14+ 26 4+ 8 base height area as a product area area as a sum height cornerpiece 2 base Eample 2 Using Generic Rectangles A generic rectangle allows us to organize the problem in the same way as the first eample without needing to draw the individual tiles. It does not have to be drawn accurately or to scale. Multiply (! )(2 + 1). - 2 +1 -!6!!!! 2 2 2 +1 ( )( 2 + 1) = 2 2! 5! area as a product area as a sum 0
Write a statement showing: area as a product equals area as a sum. 1. 2.. 4. 5. 6. 5 2!5!15 + 6 18y!12 y!2 y 2 y!2 12y!8 + 4 Multiply. 7. ( + 2) ( 2 + 7) 8. ( 2! 1) ( + 1) 9. ( 2) (! 1) 10. ( 2y! 1) ( 4y + 7) 11. ( y! 4) ( y + 4) 12. ( y) (! 1) 1. (! 1) ( + 2) 14. ( 2y! 5) ( y + 4) 15. ( y) (! y) 16. (! 5) ( + 5) 17. ( 4 + 1) 2 18. ( + y) ( + 2) 19. ( 2y! ) 2 20. (! 1) ( + y + 1) 21. ( + 2) ( + y! 2) Answers 1. ( + 1) ( + ) = 2 + 4 + 2. ( + 2) ( 2 + 1) = 2 2 + 5 + 2. ( + 2) ( 2 + ) = 2 2 + 7 + 6 4. (! 5) ( + ) = 2! 2! 15 5. 6( y! 2) = 18y! 12 6. ( + 4) ( y! 2) = y! 2 + 12y! 8 7. 6 2 + 25 + 14 8. 6 2!! 1 9. 2 2! 2 10. 8y 2 + 10y! 7 11. y 2! 16 12. y! y 1. 2 + 5! 2 14. 2y 2 + y! 20 15. y! y 2 16. 9 2! 25 17. 16 2 + 8 + 1 18. 2 + 2 + y + 2y 19. 4y 2! 12y + 9 20. 2 + y! y! 1 21. 2 + y + 2y! 4 Chapter 5: Multiplication and Proportions 1
SOLVING EQUATIONS WITH MULTIPLICATION 5.1.4 To solve an equation with multiplication, first use the Distributive Property or a generic rectangle to rewrite the equation without parentheses, then solve in the usual way. See the Math Notes bo on page 198. Eample 1 Solve 6( + 2) = ( 5 + 1) Use the Distributive Property Subtract 6 Subtract Divide by 9 6 + 12 = 15 + 12 = 9 + 9 = 9 1 = Eample 2 Solve ( 2! 4) = ( 2 + 1) ( + 5) Rewrite using generic rectangles Subtract 2 2 Subtract 11 Divide by!15 2 2! 4 = 2 2 + 11 + 5!4 = 11 + 5!15 = 5 = 5!15 =! 1 2
Solve each equation. 1. ( c + 4) = 5c + 14 2.! 4 = 5( + 2). 7( + 7) = 49! 4. 8(! 2) = 2( 2! ) 5. 5! 4(! ) = 8 6. 4y! 2( 6! y) = 6 7. 2 + 2(2! 4) = 244 8. ( 2! 4) = ( 2 + 1) (! 2) 9. (! 1) ( + 7) = ( + 1) (! ) 10. ( + ) ( + 4) = ( + 1) ( + 2) 11. 2! 5( + 4) =!2( + ) 12. ( + 2) ( + ) = 2 + 5 + 6 1. (! ) ( + 5) = 2! 7! 15 14. ( + 2) (! 2) = ( + ) (! ) 1 15. ( + 2) = ( 1 + 2) (! ) 2 2 Answers 1.!1 2.!.5. 0 4. 2 5.!4 6. 7. 42 8. 2 9. 0.5 10.!2.5 11.!14 12. all numbers 1. 0 14. no solution 15.!12 Chapter 5: Multiplication and Proportions
SOLVING MULTI-VARIABLE EQUATIONS 5.1.5 Solving equations with more than one variable uses the same process as solving an equation with one variable. The only difference is that instead of the answer always being a number, it may be an epression that includes numbers and variables. The usual steps may include: removing parentheses, simplifying by combining like terms, removing the same thing from both sides of the equation, moving the desired variables to one side of the equation and the rest of the variables to the other side, and possibly division or multiplication. Eample 1 Eample 2 Solve for y Subtract Divide by!2 Simplify! 2y = 6!2y =! + 6 y =!+6!2 y = 2! Solve for y Subtract 7 Distribute the 2 Subtract 2 Divide by 2 Simplify 7 + 2( + y) = 11 2( + y) = 4 2 + 2y = 4 2y =!2 + 4 y =!2+4 2 y =! + 2 Eample Eample 4 Solve for y =! 4 Add 4 y + 4 = Divide by y + 4 = Solve each equation for the specified variable. Solve for t Divide by pr I = prt I pr = t 1. y in 5 + y = 15 2. in 5 + y = 15. w in 2l + 2w = P 4. m in 4n = m! 1 5. a in 2a + b = c 6. a in b! 2a = c 7. p in 6! 2(q! p) = 4 p 8. in y = 1 4 + 1 9. r in 4(r! s) = r! 5s Answers (Other equivalent forms are possible.) 1. y =! 5 + 5 2. =! 5 y +. w =!l + P 2 4. m = 4n+1 5. a = c!b 2 6. a = c!b!or! b!c!2 2 7. p = q! 8. = 4y! 4 9. r = 7s 4
SETTING UP PROPORTIONS 5.2.1 and 5.2.2 A convenient way to set up proportions is to arrange the information in a table. Once the information is placed into a labeled table, the proportion can be written directly from the table. See the Math Notes bo on page 211. Eample 1 A tree casts a 4-foot shadow. At the same time, a 4 1 2 foot tall boy casts a 10-foot shadow. Determine the height of the tree. Eliminate the denominators. Divide by 10. height (ft) shadow (ft) 4 4.5 10 4.5 = 4 10! $! 45 4.5% & = 45 4 $ 10 % & 10 = 19.5 = 19.5 Eample 2 Kim noticed that 100 vitamins cost $1.89. At this rate, how much should 50 vitamins cost? Eliminate the denominators. Divide by 100. Vitamins () Cost ($) 100 1.89 50 100 50 = 1.89! 50 100 $! 50% & = 50 1.89 $ % & 100 = 661.5 = $6.62 Chapter 5: Multiplication and Proportions 5
Solve each problem by writing a proportion and solving it. 1. Joe came to bat 464 times in 11 games. At this rate, how many times should he epect to bat in a full season of 162 games? 2. Mario s car needs 12 gallons to go 20 miles. At the same rate, how far can he travel with 10 gallons of gas?. If 50 empty soda cans weigh 1 2 pounds, how much would 70 empty soda cans weigh? 4. There is a $4 ta on an $800 motor scooter. How much ta would there be on a $1000 motor scooter? 5. A dozen donuts cost $.50. How much should three donuts cost? 6. In 5 minutes, Suki s car goes 25 miles. If she continues at the same rate, how long will it take her to drive 90 miles? 7. In a city of three million people,,472 were surveyed. Of those surveyed, 28 of them watched the last Lonely Alien movie. If the survey represents the city s TV viewing habits, about how many people in the city watched the movie? 8. Julio runs 10 mile in 1 1 2 minutes. At that rate how long will it take him to run a mile? 9. It is now 7:51 p.m. The movie that you have waited three weeks to see starts at 8:15 p.m. While standing in line for the movie, you count 146 people ahead of you. Nine people buy their tickets in 70 seconds. Will you be able to buy your ticket before the movie starts? 10. A biscuit recipe uses 1 teaspoon of baking powder for 2 4 powder is needed for three cups of flour? cup of flour. How much baking 11. The germination rate for zinnia seeds is 78%. This means that 78 out of every 100 seeds will sprout and grow. If Jim wants 60 plants for his yard, how many seeds should he plant? 12. L.J. s car has a gas tank which holds 19 gallons of fuel. If the gas tank was full and he used eight gallons to drive 200 miles, does the car have enough gas to go another 250 miles? Answers 1. about 574 times 2. about 267 miles. 4.9 pounds 4. $42.50 5. $0.88 6. 126 minutes 7. about 24,194 people 8. 5 minutes 9. yes, in about 19 min. 10. 2 teaspoons 11. about 77 seeds 12. yes, 275 miles 6
SOLVING PROPORTIONS 5.2.2 To solve proportions (equal ratio equations) remove the denominators by multiplying both fractions (ratios) by the common denominator. If you cannot determine the common denominator, multiply the denominators and then multiply both fractions (ratios) by the result. Then solve in the usual manner. Another approach is to begin by multiplying to remove the denominator of the fraction (ratio) with the variable. In the first eample below, multiply by 12; in the second eample, multiply by. When there is a variable in both fractions, multiply by the product of both denominators. Eample 1 Eample 2 Solve for 12 = 7 10 Solve for p 11 5 = p + 2! Multiply by 60 60 $! 12% & = 60 7 $ 10% & Simplify 5 = 42 Divide by 5 Eample = 42 5 = 8.4! Multiply by 15 15 11 $! 5 % & = 15 p + 2 $ % & Simplify = 5( p + 2) Distribute the 5 = 5 p + 10 Subtract 10 Divide by 5 2 = 5 p p = 2 5 p = 4.6 Solve for Multiply by 12 Simplify Subtract 4 and 9 + 4 =! 1 12 + % $ 4 & ' = 12! 1 % $ & ' ( + ) = 4(! 1) + 9 = 4! 4! =!1 = 1 Chapter 5: Multiplication and Proportions 7
Solve each proportion. 1. 20 = 5 2.! 4 12 = 2. 100 40 = 5 2 4. 6 5 =! 10 5. 8 7 = 1 2 6.! 4 =! 7 10 7. 6 11 = 5 4 8. 6 + 2 = 4! 8 9. 9! 6 = 24 2 10. 10 = 24 9 11. 4 5 =! 2 7 12. 6 7 = 42 1. 4 7 = 81 15 14. 100 = 5 20 15. 100 = 7 5 Answers 1. 12 2. 12. 1 4.! 0.27 5.!5.69 6.! 4.14 7.! 16.04 8.!4 9.!6 10.! 8.89 11.! 0.4 12. 16. 1. 9.45 14. 25 15. 20 8