WARM UP Three friends went to the gym to work out. None of the friends would tell how much he or she could leg-press, but each hinted at their friends’ leg-press amounts. Chen said that Juanita and Lou averaged 87 pounds Juanita said that Chen leg-pressed 6 pounds more than Lou. Lou said that eight times Juanita’s amount equals seven times Chen’s amount. Find out how much each friend could leg-press.

Organizing Information

OBJECTIVES Learn good ways to organize information Practice using dimensional analysis and unit conversion Solve logic problems Improve at working cooperatively Define and use direct variation

INTRODUCTION “If one and a half chickens lay one and a half eggs in one and a half hours, then how long does it take six monkeys to make nine omelets?” What sort of problem-solving strategy can you apply to the silly problem above? You could draw a picture or make a diagram. You could assign variables to all sorts of unknown quantities. However, do you have enough information to solve the problem? Sometimes the best strategy is to begin organizing what you know and what you want to know. With information organized, you may then find a way to get to the solution.

EXAMPLE A How many second are in a calendar year? SOLUTION: First identify what you know and what you need to know. Know Need to know 1 year Number of seconds It may seem like you don’t have enough information, but consider these commonly know facts: 1 year = 365 days (non leap year) 1 day = 24 hours 1 hour = 60 minutes 1 minute = 60 seconds

SOLUTION CONT. You can write each equality as a fraction and multiply the chain of fractions such that the units reduce to leave seconds. 31,536,000 seconds There are 31,536,000 seconds in a non-leap calendar year.

DIRECT VARIATION In the example you figured out there are 31,536,000 seconds in 1 year. If you want to know how many seconds there are in 5 years, you can just multiply this number by 5. When quantities are related like this, they are said to be in direct variation. Another example of direct variation is the relationship between the number of miles you travel and the time you spend traveling. In each case the relationship can be expressed in an equation of the form y = kx, where k is called the constant of variation.

EXAMPLE B To qualify for the Interlochen 470 auto race, each driver must complete two laps of the 5 mile track at an average speed of 100 miles per hour (mi/h). Due to some problems at the start, Naomi averages only 50 mi/h on her first lap. How fast must she go on the second lap to qualify for the race? SOLUTION: Sort the information into two categories: what you know and what you might need to know. Assign variables to the quantities that you don’t know. Use a table to organize the information, and include the units for each piece of information.

EXAMPLE B Know Might need to know Speed for the first lap: 50 mi/h Speed for second lap (in mi/h): s Average speed for both laps: 100 mi/h Time for first lap (in h): t1 Length of each lap (in mi): 5 Time for second lap (in h): t2 Use the units to help you find connections between the pieces of information. Speed is measured in miles per hour and therefore calculated by dividing distance by time. You might also remember the relationship distance = rate x time or d = rt. Because you know the distance and rate for the first lap, you can solve for the time: t = d/t.

SOLUTION CONT. It takes one-tenth of an hour, or 6 minutes, to do the first lap. You know the distance and speed for the first and second laps together, so solve for the time for both laps. Then you can subtract to find what you are looking for, the time for the second lap. Note that the time for both laps is the same as the time for the first lap. This means the time for the second lap, t2, must be zero. It’s not possible for Naomi to complete the second lap in no time, so she cannot qualify for the race. Would the solution be different if the laps were 10 miles long? d miles long?

CLASS INVESTIGATION WHO OWNS THE ZEBRA? There are five houses along one side of Birch Street, each of a different color. The home owners each drive a different car, and each has a different pet. The owners all read a different newspaper and plant only one thing in their garden.

CONCEPT PRACTICE Investigation: Who owns the zebra? CP 0.3 #1-5