SECTION A.8 Problem Solving: Interest, Mixture, Uniform Motion, Constant Rate Job Applications A67 Translating Verbal Descriptions into Mathematical Expressions (a) For uniform motion, the average speed of an object equals the distance traveled divided by the time required. Translation: If r is the speed, d the distance, and t the time, then = r d t . (b) Let x denote a number. The number 5 times as large as x is x5 . The number 3 less than x is −x 3. The number that exceeds x by 4 is +x 4. The number that, when added to x , gives 5 is −x 5 . EXAMPLE 1 1 Translate Verbal Descriptions into Mathematical Expressions Steps for Solving Applied Problems Step 1 Read the problem carefully, perhaps two or three times. Pay particular attention to the question being asked in order to identify what you are looking for. Identify any relevant formulas you may need ( π = = d rt A r , 2 , etc.). If you can, determine realistic possibilities for the answer. Step2 Assign a letter (variable) to represent what you are looking for, and if necessary, express any remaining unknown quantities in terms of this variable. Step 3 Make a list of all the known facts, and translate them into mathematical expressions.These may take the form of an equation or an inequality involving the variable. If possible, draw an appropriately labeled diagram to assist you. Sometimes, creating a table or chart helps. Step 4 Solve for the variable, and then answer the question. Step 5 Check the answer with the facts in the problem. If it agrees, congratulations! If it does not agree, review your work and try again. Now Work problem 9 Always check the units used to measure the variables of an applied problem. In Example 1(a), if r is measured in miles per hour, then the distance d must be expressed in miles, and the time t must be expressed in hours. It is a good practice to check units to be sure that they are consistent and make sense. 2 Solve Interest Problems Interest is money paid for the use of money. The total amount borrowed (whether by an individual from a bank in the form of a loan, or by a bank from an individual in the form of a savings account) is called the principal . The rate of interest , expressed as a percent, is the amount charged for the use of the principal for a given period of time, usually on a yearly (that is, per annum) basis. Simple Interest Formula If a principal of P dollars is borrowed for a period of t years at a per annum interest rate r , expressed as a decimal, the interest I charged is = I Prt (1)
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