Make a Table

Example A motorcycle and a car start from the same point and travel in the same direction along the same road. The motorcycle starts first and travels 20 mi. before the car starts. The motorcycle averages 53 mi./hr.

Example The car averages 58 mi./hr. How many hours will it take the car to catch up with the motorcycle?

Example Think: What are you trying to find? We are finding the time it takes for the car to catch up with the motorcycle.

Example Think: What formula have you seen in the past that relates rate, time, and distance? Use the basic formula for distance, d = rt, where d = distance traveled, r = rate (or speed of vehicle), and t = time spent traveling.

Example Think: How could you organize the information to find out when the two vehicles have traveled the same distance? How far ahead is the motorcycle when the car begins to travel?

Example Since you are trying to find the time that the car travels, let the time (t ) start when the car begins to travel. The motorcycle has a 20 mi. head start, so at time 0 the motorcycle has traveled 20 mi.

Example The calculation for d after one hour for the motorcycle is 20 + 53(1) = 73 mi. The distance that the car has traveled after one hour is 58(1) = 58 mi.

Example Make a table to track these distances as the hours increase. Continue the table until both vehicles have traveled the same distance.

Motorcycle Car r t d = 20 + rt d = rt 53 1 20 + 53(1) = 20 + 53 = 73 58 58(1) = 58 53 2 20 + 53(2) = 20 + 106 = 126 58 58(2) = 116 53 3 20 + 53(3) = 20 + 159 = 179 58 58(3) = 174 53 4 20 + 53(4) = 20 + 212 = 232 58 58(4) = 232

Example According to the table, the car traveled the same distance as the motorcycle after 4 hr. Therefore, it will take the car 4 hr. to catch up with the motorcycle.

Example Think: Does this answer seem reasonable? Is there any other way this problem could be solved? This is a reasonable answer, and when checked it is correct.

Example How many ways can you make $0.70 in change if pennies are not used?

Quarters Dimes Nickels 2 2 1 2 4 1 4 1 3 1 2 5 1 7 1 9 7 6 2

Quarters Dimes Nickels 5 4 4 6 3 8 2 10 1 12 14

For each of the following exercises, devise a plan for finding the solution. Explain the steps of your plan, and use the plan to solve the problem.

Exercise Two cars start from the same point and travel in the same direction along the same road. One car starts first and travels 35 mi. before the other car starts.

Exercise The first car travels at an average rate of 55 mi./hr., and the second car travels at an average rate of 62 mi./hr. How many hours will it take the second car to catch up to the first?

Miles Traveled Hours Car 1 Car 2 35 1 90 62 2 145 124 3 200 186 4 255 35 1 90 62 2 145 124 3 200 186 4 255 248 5 310

Exercise One car travels north at a constant rate of 59 mi./hr. A second car starts at the same time at the same point and travels south at a constant rate of 66 mi./hr. How many hours will it be before they are 625 mi. apart?

Hours Distance Between 1 59 + 66 = 125 2 2(125) = 250 3 3(125) = 375 4 4(125) = 500 5 5(125) = 625

Exercise List all of the different three-digit numbers that you can write using the digits 2, 3, and 4, once each.

Beginning with 2 3 4 234 324 423 243 342 432

Exercise A baseball coach decided that his first four batters would be Ames, Bailey, Cox, and Davis. The coach does not want Ames to bat first or second. How many different batting orders can the coach make?

B C D A B C A D B D C A B D A C C B D A C B A D C D B A C D A B D B C A D B A C D C A B D C B A

Exercise A bag contains tickets numbered 9, 2, 1, 4, and 0. Suppose you draw three tickets at random without replacing those previously drawn and add the numbers together. How many different totals are possible?

Numbers Chosen Sum 0, 1, 2 3 0, 1, 4 5 0, 1, 9 10 0, 2, 4 6 0, 2, 9 11 0, 4, 9 13 1, 2, 4 7 1, 2, 9 12 1, 4, 9 14 2, 4, 9 15

Exercise Amy, Beth, Carol, and Dottie are to sit in one row of desks, one behind the other. Their teacher does not want Beth and Carol to be seated one behind the other. How many different ways can the four students be arranged?

A B D C A C D B B A C D B A D C B D A C B D C A C A B D C A D B C D A B C D B A D B A C D C A B

Exercise Josie wants to make change for a half dollar. How many different ways can she do so if pennies cannot be used in the change?

Quarters Dimes Nickels 2 1 2 1 3 1 5 5 4 2 3 4 2 6 1 8 10

Exercise Two cars leave towns that are 459 mi. apart at the same time and travel toward each other along the same road. The first car travels at an average rate of 48 mi./hr., and the second car travels at an average rate of 54 mi./hr.

Exercise How long will it take until the two cars pass each other?

Hours Distance Between 1 459 − 102 = 357 2 357 − 102 = 255 3 255 − 102 = 153 4 153 − 102 = 51