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APPLICATIONS: SOLVING WORD PROBLEMS USING LINEAR EQUATIONS Introduction: Solving an algebraic problem consists of finding a number or set of numbers (the unknown), when other numbers related to the unknown (the data), are given. The relation between the data and the unknown will be the basis to set up an equation whose solution is the answer to the problem. Example: Certain number added to its half is 240. What is the number? Referring to the unknown as x, the equation would be x 1 x 240 2 The solution, x = 160, is the desired number. In fact, 160 + 160/2 = 240. Steps for Solving Word problems: Solving word problems through algebraic equations requires, in general, the following steps: 1. 2. 3. 4. 5. Identification of unknowns Selection of variables to represent the unknowns Set up the equation Solve the equation Interpretation and verification of results. Example: A mother is seven years older than twice the age of her son. If the mother is 47 years old, how old is her son? Unknown: Son's age Variable: x Equation: 2x + 7 = 47 Solution: 2x = 40 x = 20 Answer: The son is 20 years old Verification: 2(20)+7=40+7=47. Number Problems: Statements expressing relationships between numbers. The goal is to translate the words into algebraic expressions, set up the equation and solve it by known means. Key words: Increase, more, more than Decrease, less, less than Consecutive integers Consecutive odd or even integers Times, times more, of (after a fraction) Times less Is Meaning Addition Subtraction Differ by 1 Differ by 2 Multiplication Division Equal Examples: 1. One third of a number is 7 less than one-half of the number. Find the number. Let the number be x. The corresponding equation is: 1 1 x x7 3 2 Multiplying by 6 we get 2x 3x 42 or x 42 Therefore the number is 42. In fact, 42/3 = 14, and 42/2 - 7 = 21 - 7 = 14. 2. Find two consecutive even integers such that 4 times the larger is 8 less than 5 times the smaller. First even integer: x Second even integer: x+2 Equation: 4(x + 2) = 5x - 8 x = 16 The numbers are 16 and 18. In fact, 4(18) = 72 and 5(16) - 8 =80 - 8 = 72. 3. The sum of two numbers is 24. The smaller is three times less than the larger. Find the numbers. Larger number: x Smaller number: x/3 Equation: x + x/3 = 24 3x + x = 72 4x = 72 x = 18 The numbers are: 18 and 6 In fact, 18 + 6 = 24 and 18/3 = 6. Percent Problems: Sometimes the relation between two numbers is expressed as a percent. "Per-cent" means "per-hundred" or "over a hundred", and it is represented by the symbol %. Thus, x% = x/100. Examples: 1. What is 70% of 48? Unknown: x Equation: x = (70/100)(48) x = 33.6 2. 238 is what percent of 350? Unknown: x Equation: Solution: 238 = (x/100)(350) 238 = 3.5x x = 68 68% When we deposit money in an account, the amount deposited id called the principal (P), and the amount earned over a period of time is called the interest (I). The interest received at the end of one year at a rate r is: I=Pxr As the next example shows, this formula is used to solve problems involving capital gain. 3. Two sums of money totaling $20,000 are invested at 5% and 6% respectively. Find the two amounts if together in one year they earn $1,080. First amount : Second amount : Equation: Principal x 20000-x Rate 5% 6% Interest 0.05x 0.06(20000-x) 0.05x + 0.06(20000 - x) = 1080 5x + 6(20000 - x) = 108000 5x - 120000 - 6x = 108000 x = 12000 Amount invested at 5% : $12,000 Amount invested at 6% : $8,000. Notice that 12,000 + 8000 = 20,000, and also that 0.05(12000) + 0.06(8000) = 1080. Markup id the amount added to the cost of an item to determine its selling price. It is usually expressed as either, a percent of the cost or a percent of the selling price. Let us consider the following example. 4. The cost of a radio is $80. What is the selling price if the markup is 20% of the selling price? 5. Selling price : x Markup : 0.20x Equation: x = 80 + 0.20x 0.80x = 80 x = 100 The selling price is $100. Mixture Problems: We will study here three examples involving mixtures and alloys. 1. How many liters of water need to be added to 6 liters of an 8% solution of salt and water in order to obtain a 5% salted solution. Amount of water in liters : Concentration of salt : Amount of salt in liters : Equation : Original 6 8% 0.08(6) Added x 0% 0 Final x+6 5% 0.05(x + 6) 0.08(6) + 0 = 0.05(x + 6) 48 = 5(x + 6) 5x = 18 x = 3.6 The amount of water to be added is 3.6 liters. 2. If we mix 48 ounces of a 4% iodine solution with 40 ounces of a 15% iodine solution, what is the percentage of iodine in the mixture? First Amount of solution in ounces: 48 Concentration of iodine: 4% Amount of iodine in ounces: (4/100)(48) Second 40 15% (15/100)(40) Mixture 88 x% (x/100)(88) Equation: 0.04(48) + 0.15(40) = 0.88x 4(48) + 15(40) = 88x 192 + 600 = 88x x=9 The mixture is 9% iodine. 3. An engineer mixed a 48% aluminum alloy with a 72% aluminum alloy to make a 57% aluminum alloy. If there are 20 pounds more of the 48% alloy than the 72% alloy, what is the weight in pounds of the final alloy? Weight of alloy in pounds: Percent of aluminum: Pounds of aluminum: First x + 20 48% 0.48(x + 20) Second x 72% 0.72x Mixture 2x + 20 57% 0.57(2x + 20) Equation: 0.48(x + 20) + 0.72x = 0.57(2x + 20) 48(x + 20) + 72x = 57(2x + 20) 48x + 960 + 72x = 114x + 1140 6x = 180 x = 30 The weight of the final alloy is 2(30) + 20 = 80 pounds. Coin and Value problems: These problems involve relationships between different types of coins and the corresponding values of various items. The following examples are a typical sample of this category. 1. Maria has $4.45 in dimes and quarters. If she has 28 coins in all, how many are dimes and how many are quarters? Dimes Quarters Coin value in cents: 10 25 Number of coins: x 28 - x Total value in cents: 10x 25(28 - x) Equation: 10x + 25(28 - x) = 445 10x + 700 - 25x = 445 -15x = -225 x = 17 Maria has 17 dimes and 28 - 17 = 11 quarters. 2. George bought three books for his History, Chemistry, and Mathematics classes. The Chemistry book costs $5 more than the History book. The Mathematics book costs twice as much as the Chemistry book. If the total bill for all three books was $145, what is the price of each one? History Chemistry Mathematics Price of book in dollars: x x+5 2(x + 5) Equation: x + x+5 + 2(x + 5) = 145 4x = 130 x = 32.5 Price of History book $32.50 Price of Chemistry book $37.50 Price of mathematics book $75 Motion Problems: These problems are based on the relationship existing between the distance, the velocity, and the time. The relationship is given by the formula d=vxt where d = distance, v = velocity, and t = time. Two typical examples are shown next. 1. Two cars that are 375 miles apart from each other, and whose speeds differ by 5 miles per hour, are moving toward each other. Under these circumstances they will meet in 3 hours. What is the speed of each car? First car Second car Speed in miles per hour: x x+5 Time in hours: 3 3 Distance in miles: 3x 3(x + 5) Equation: 3x + 3(x + 5) = 375 6x = 360 x = 60 The speed of the first car is 60 mph The speed of the second car is 65 mph. 2. Michael drove 45 minutes at a certain speed. Then he increased the speed by 16 mph for the rest of the trip. If the total distance traveled was 114 miles, and it took him 2 hours and 15 minutes, how many miles did he drive at the higher speed? Speed in mph: Time in hours: Distance in miles: Equation: First part x 45/60 = 3/4 (3/4)x Second part x + 16 2 1/4 - 3/4 = 3/2 (3/2)(x + 16) (3/4)x + (3/2)(x + 16) = 114 3x + 6(x + 6) = 456 9x = 360 x = 40 The distance traveled at the higher speed was (3/2)(40 + 16) = 84 miles. Temperature Problems: Three common scales used to measure temperatures are the Fahrenheit, the Celsius or Centigrade, and the Kelvin or Absolute. The relationship between these scales is given by the equations: F = (9/5) + 32 and K = C + 273 Exercises: 1. At what temperature the Fahrenheit and the Centigrade scales coincide? If x is the common temperature, the equation relation both scales becomes: x = (9/5)x + 32 5x = 9x + 160 4x = -160 x = -40 Therefore the two scales coincide at -40 degrees. 2. Find the Kelvin or Absolute temperature corresponding to 86 Fahrenheit degrees. First we will find the corresponding Centigrade reading. The equation is: 86 = (9/5)C + 32 (9/5)C = 54 C = 30 Thus the Kelvin reading is, K = 30 + 273 = 303 degrees. Lever Problems: A uniform bar with negligible weight and balanced on a support (fulcrum) with weights on both sides of it is called a lever. The distance from a weight to the fulcrum is referred to as the arm of the weight. For a lever to be balanced the following equation must be safisfied, F1 x d1 = F2 x d2 Nutcrackers, scissors, and balances are simple machines to which the lever equation applies. Examples: 1. Maria and Francisco together weight 75 lb. They balance a 10 feet teeterbore when Francisco is 4 ft from the fulcrum. Find their respective weights. Weight: Arm of weight: Equation: Francisco x 4 Maria 75 - x 6 4x = 6(75 - x) 10x = 450 x = 45 Francisco's weight is 45 lb, and Maria's weight is 75 - 45 = 30 lb. 2. A lever is balanced when a 90 lb weight is placed on one side, 8 ft away from the fulcrum, and a 40 lb and 120 lb weights are placed 2 ft apart from each other on the side of the fulcrum, with the 40 lb weight closer to it. How far from the fulcrum is the 40 lb weight? Equation: 40x + 120(x + 2) = 90(8) 60x = 480 x=3 The 40 lb weight is 3 ft away from the fulcrum. Geometry Problems. Examples: 1. The length of a rectangle is 3 ft less than twice its width and its perimeter is 42 ft. Find the dimensions of the rectangle. Dimensions in feet: Equation: Width x Length 2x - 3 2x + 2(2x - 3) = 42 6x = 48 x=8 The width of the rectangle is 8 ft, and its length is 2(8) - 3 = 13 ft. 3. One of two complementary angles is 9o more than twice the other. Find the two angles. Measure in degrees: Equation: Angle x Complementary 2x + 9 x + 2x+9 = 90 3x = 81 x = 27 One angle measures 27o and its complement measures 90 - 27 = 63o. ************************************************************************* EXERCISES 1. Find three consecutive odd integers such that, the product of the first and the third minus the product of the first and the second, is 11 more than the third number. 2. The cost of a tape deck is $570. What is the selling price of the tape if the markup is 40% of the selling price? 3. The annual interest earned by $18,000 is $206 more than the interest earned by $16,000 invested at a rate 0.8% less than the one corresponding to the $18,000. What is the rate of interest for each amount? 4. Find the Celsius reading corresponding to 23 Fahrenheit degrees. 5. If one of two supplementary angles is four times the other, what is the measure of each angle? *************************************************************************