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How to Set Up the Different Types of Word Problems DISTANCE Word Problems à Rate • Time = Distance Key Words: Distance, Rate, Time Be sure to make sure your units of time all agree. Often these problems will mix minutes and hours and see if the student notices. Before you do any algebra all the units must agree, regardless of whether they are units of time or distance. Rate Car #1 Car #2 • • • Time = = = Distance Standard Linear Equation Set–Ups: Plug in numbers and variables for rate and time, then multiply across to get algebraic expressions for the individual distances traveled. Shaded areas must be filled in first. Rate Car #1 Car #2 • • • Time A. When the distances are equal, the resulting equation will be: distance = = = first car B. When the total distance is given, the resulting equation will be: distance Distance = distance first car C. When the difference in distances is given, the resulting equation will be: distance second car + distance longer second car = total distance – distance shorter = difference D. If one distance is less than the other, add the difference to the shorter distance when forming the equation. distance longer = distance shorter + extra distance Rational (Fractional) Equation Set–Ups: Here we will be plugging in information about the distances and either the rate or time, but not both. Then we need to use “division” to get rational expressions for the missing quantity which will be time or rate, respectively, so if r gt = D , then t = Rate Car #1 Car #2 D D and r = . r t • • • Time = = = Distance A. When the rates are equal, plug in for distance and time, and get algebraic expressions for the rates. The resulting equation will be: rate first car = rate second car B. When the times are equal, plug in for distance and rate, and get algebraic expressions for the times. The resulting equation will be: time first car = time second car C. When the total time is given, plug in for distance and rate, and get algebraic expressions for the times. The resulting equation will be: time first car + time second car = total time. D. For vehicle or wind/water current, the respective rates are: (speed in still water/wind) ± (speed of the current). Usually these problems involve a boat/canoe traveling up stream/down stream, or with current/against current, or an airplane traveling into a head wind or with a tail wind. How to Set Up Different Types of Word Problems 1 of 4 LIQUID MIXTURE Word Problems Key Words: Percent, Mixture, Gallons (or Liters), Mixed Together, Added To, Combined Remember that percents must go in as decimals. Also, the shaded areas must be filled in first. Be sure to use the fact that the first row forms an equation, where the left–hand side has to equal the right–hand side. This will help you to fill in all six shaded boxes. Solution 1 Volume of Solution + + Solution 2 = = Mixture % Active Ingredient (goes in as a decimal) Volume of Active Ingredient + = A. The sum of the first two columns, gives the third. (Only used for the first and third rows.) B. We multiply down each column to get the volume of active ingredient. C. The last row represents the equation to be solved. (The equation to be solved stands for the volume of active ingredient, which remains unchanged in this mixture process.) SOLID MIXTURE Word Problems Key Words: Pounds (lbs.), Ounces (oz.), Price per Pound ($ / lb), Total Cost The shaded areas must be filled in first. Be sure to use the fact that the first row forms an equation, where the left–hand side has to equal the right–hand side. This will help you to fill in all six shaded boxes. Item 1 Weight of Each Item being Mixed Together + + Item 2 = = Mixture Price per Unit of Weight Total Cost of Each Item + = A. The sum of the first two columns, gives the third. (Only used for the first and third rows.) B. We multiply down each column to get the total cost of each item. C. The last row represents the equation to be solved. (The equation to be solved indicates that the cost of the final mix is determined by adding the costs of the individual ingredients being mixed in.) How to Set Up Different Types of Word Problems 2 of 4 SIMPLE INTEREST Word Problems (Principal • Rate • Time = Interest) Key Words: Investment, Principal, Rate, Time, Interest The shaded areas must be filled in first. The time for these problems is almost always measured in years, where the time set to 1 year for ease of computation. If the problem does not explicitly mention a time period of “1 year” it may say “annual” instead, this means for a period of “1 year”. Principal • First Investment Second Investment • • • = = • • = Rate Time A. If the interest is the same on both investments then your equation will be B. If the total amount of interest if given, add the two interests together: C. If one interest is more than the other interest, use Interest I1 I2 I1 = I2 I1 + I2 = total interest I1 + difference = I2 MONEY Word Problems (like Buying Tickets & Employment) Key Words: Price per Item (Dollars per Hour), Total, Total Cost (Total Salary) This grid can be used for problems that involve buying tickets (adult and child, or, evening and matinee) and also for working two jobs, with two different salaries, and different times spent at each job. Take special note that there are only five boxes that need to be filled in, prior to multiplying down the first two columns. The box shaded in black is not used, since the concept of total does not apply in these physical situations. First Kind of Item Number of Items Second Kind + + of Item = = Totals Price per Item (goes in as a decimal) Total Cost of Each Item + = A. The sum of the first two columns, gives the third. (Only used for the first and third rows.) B. We multiply down each of the first two columns to get the total cost of each item. C. The last row represents the equation to be solved. (The equation to be solved indicates that the cost of both items together is determined by adding the costs of the individual items.) How to Set Up Different Types of Word Problems 3 of 4 WORK Word Problems (these problems always have fractions in their set–ups) Key Words: Rate, Work Together, Completed, Job, Total Time Here we need to make sure that the units of time agree with the units of the rate. It doesn’t matter what these units are, and they will vary from problem to problem, but each problem must be internally consistent. The shaded areas are filled in first, then we multiply across to get rational expressions for the portion of the work completed by each person. Time Alone à Rate of Work Rate is (1 / time alone) • Time Worked = Fraction of Task Completed (P = Part) 1st Person • = P1 2nd Person • = P2 Most common form of the equation looks like: Total Time 1 1 + 1424 3 1424 3 Fraction of Work Done by First Person Fraction of Work Done by Second Person = 1 { 1 Job Completed OR 1 1 Total Time + Total Time = 1 First Time Second Time A. If the task is completed, the sum of the parts must equal 1, because one job is completed. P1 + P2 = 1 B. If the task is not completed, then the sum is equal to that part of the task. C. If the two objects or people are working against each other (forces in opposition), then the parts are subtracted instead of added. This situation usually occurs when you have one pipe filling a tank, and another pipe emptying a tank. They keywords are usually FILL and EMPTY. The filling portion goes to the left of the subtraction, and the emptying portion to the right. 1 1 Total Time − Total Time = 1 FILL Time EMPTY Time How to Set Up Different Types of Word Problems 4 of 4