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1.8: Problem-Solving Strategies: Make a Table and Look for a Pattern

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Learning Objectives

  • Read and understand given problem situations.
  • Develop and use the strategy: make a table.
  • Develop and use the strategy: look for a pattern.
  • Plan and compare alternative approaches to solving the problem.
  • Solve real-world problems using selected strategies as part of a plan.

Introduction

In this section, we will apply the problem-solving plan you learned about in the last section to solve several real-world problems. You will learn how to develop and use the methods make a table and look for a pattern. Let’s review our problem-solving plan.

Step 1

Understand the problem

Read the problem carefully. Once the problem is read, list all the components and data that are involved. This is where you will be assigning your variables.

Step 2

Devise a plan – Translate

Come up with a way to solve the problem. Set up an equation, draw a diagram, make a chart or construct a table as a start to solving your problem.

Step 3

Carry out the plan – Solve

This is where you solve the equation you developed in Step 2.

Step 4

Look – Check and Interpret

Check to see if you used all your information. Then look to see if the answer makes sense.

Read and Understand Given Problem Situations

The most difficult parts of problem-solving are most often the first two steps in our problem-solving plan. You need to read the problem and make sure you understand what you are being asked. If you do not understand the question, then you can not solve the problem. Once you understand the problem, you can devise a strategy that uses the information you have been given to arrive at a result.

Let’s apply the first two steps to the following problem.

Example 1:

Six friends are buying pizza together and they are planning to split the check equally. After the pizza was ordered, one of the friends had to leave suddenly, before the pizza arrived. Everyone left had to pay $1 extra as a result. How much was the total bill?

Step 1

Understand

We want to find how much the pizza cost.

We know that five people had to pay an extra $1 each when one of the original six friends had to leave.

Step 2

Strategy

We can start by making a list of possible amounts for the total bill.

We divide the amount by six and then by five. The total divided by five should equal $1 more than the total divided by six.

Look for any patterns in the numbers that might lead you to the correct answer.

In the rest of this section you will learn how to make a table or look for a pattern to figure out a solution for this type of problem. After you finish reading the rest of the section, you can finish solving this problem for homework.

Develop and Use the Strategy: Make a Table

The method “Make a Table” is helpful when solving problems involving numerical relationships. When data is organized in a table, it is easier to recognize patterns and relationships between numbers. Let’s apply this strategy to the following example.

Example 2

Josie takes up jogging. On the first week she jogs for 10 minutes per day, on the second week she jogs for 12 minutes per day. Each week, she wants to increase her jogging time by 2 minutes per day. If she jogs six days per week each week, what will be her total jogging time on the sixth week?

Solution

Step 1

Understand

We know in the first week Josie jogs 10 minutes per day for six days.

We know in the second week Josie jogs 12 minutes per day for six days.

Each week, she increases her jogging time by 2 minutes per day and she jogs 6 days per week.

We want to find her total jogging time in week six.

Step 2

Strategy

A good strategy is to list the data we have been given in a table and use the information we have been given to find new information. We can make a table with the following headings.

Week Minutes per Day Minutes per Week

We are told that Josie jogs 10 minutes per day for six days in the first week and 12 minutes per day for six days in the second week. We can enter this information in our table:

Week Minutes per Day Minutes per Week
1 10 60
2 12 72

You are told that each week Josie increases her jogging time by 2 minutes per day and jogs 6 times per week. We can use this information to continue filling in the table until we get to week six.

Week Minutes per Day Minutes per Week
1 10 60
2 12 72
3 14 84
4 16 96
5 18 108
6 20 120

Step 3

Apply strategy/solve

To get the answer we read the entry for week six.

Answer In week six Josie jogs a total of 120 minutes.

Step 4

Check

Josie increases her jogging time by two minutes per day. She jogs six days per week.

This means that she increases her jogging time by 12 minutes per week.

Josie starts at 60 minutes per week and she increases by 12 minutes per week for five weeks.

That means the total jogging time = 60 + 12 \times 5 = 120 \ minutes

The answer checks out.

You can see that by making a table we were able to organize and clarify the information we were given. It also helped guide us in the next steps of the problem. This problem was solved solely by making a table. In many situations, this strategy would be used together with others to arrive at the solution.

Develop and Use the Strategy: Look for a Pattern

Look for a pattern is a strategy that you can use to look for patterns in the data in order to solve problems. The goal is to look for items or numbers that are repeated or a series of events that repeat. The following problem can be solved by finding a pattern.

Example 3

You arrange tennis balls in triangular shapes as shown. How many balls will there be in a triangle that has 8 layers?

Solution

Step 1

Understand

We know that we arrange tennis balls in triangles as shown.

We want to know how many balls there are in a triangle that has 8 layers.

Step 2

Strategy

A good strategy is to make a table and list how many balls are in triangles of different layers.

One layer It is simple to see that a triangle with one layer has only one ball.

Two layers For a triangle with two layers we add the balls from the top layer to the balls of the bottom layer. It is useful to make a sketch of the different layers in the triangle.

Three layers we add the balls from the top triangle to the balls from the bottom layer.

We can fill the first three rows of the table.

Number of Layers Number of Balls
1 1
2 3
3 6

We can see a pattern.

To create the next triangle, we add a new bottom row to the existing triangle.

The new bottom row has the same number of balls as there are layers.

- A triangle with 3 layers has 3 balls in the bottom layer.

To get the total balls for the new triangle, we add the number of balls in the old triangle to the number of rows in the new bottom layer.

Step 3

Apply strategy/solve:

We can complete the table by following the pattern we discovered.

Number of balls = number of balls in previous triangle + number of layers in the new triangle

Number of Layers Number of Balls
1 1
2 3
3 6
4 6 + 4 = 10
5 10 + 5 = 15
6 15 + 6 = 21
7 21 + 7 = 28
8 28 + 8 = 36

Answer There are 36 balls in a triangle arrangement with 8 layers.

Step 4

Check

Each layer of the triangle has one more ball than the previous one. In a triangle with 8 layers,

layer 1 has 1 ball, layer 2 has 2 balls, layer 3 has 3 balls, layer 4 has 4 balls, layer 5 has 5 balls, layer 6 has 6 balls, layer 7 has 7 balls, layer 8 has 8 balls.

When we add these we get: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36 balls

The answer checks out.

Notice that in this example we made tables and drew diagrams to help us organize our information and find a pattern. Using several methods together is a very common practice and is very useful in solving word problems.

Plan and Compare Alternative Approaches to Solving Problems

In this section, we will compare the methods of “Making a Table” and “Looking for a Pattern” by using each method in turn to solve a problem.

Example 4

Andrew cashes a $180 check and wants the money in $10 and $20 bills. The bank teller gives him 12 bills. How many of each kind of bill does he receive?

Solution

Method 1: Making a Table

Step 1

Understand

Andrew gives the bank teller a $180 check.

The bank teller gives Andrew 12 bills. These bills are mixed $10 bills and $20 bills.

We want to know how many of each kind of bill Andrew receives.

Step 2

Strategy

Let’s start by making a table of the different ways Andrew can have twelve $10 bills and $20 bills.

Andrew could have twelve $10 bills and zero $20 bills or eleven $10 bills and one $20 bills, so on.

We can calculate the total amount of money for each case.

Step 3

Apply strategy/solve

$10 bills $20 bills Total amount
12 0 \$10(12) + \$20(0) = \$120
11 1  \$10(11) + \$20(1) = \$130
10 2 \$10(10) + \$20(2) = \$140
9 3  \$10(9) + \$20(3) = \$150
8 4 \$10(8) + \$20(4) = \$160
7 5 \$10(7) + \$20(5) = \$170
6 6 \$10(6) + \$20(6) = \$180
5 7 \$10(5) + \$20(7) = \$190
4 8 \$10(4) + \$20(8) = \$200
3 9 \$10(3) + \$20(9) = \$210
2 10 \$10(2) + \$20(10) = \$220
1 11 \$10(1) + \$20(11) = \$230
0 12 \$10(0) + \$20(12) = \$240

In the table we listed all the possible ways you can get twelve $10 bills and $20 bills and the total amount of money for each possibility. The correct amount is given when Andrew has six $10 bills and six $20 bills.

Answer: Andrew gets six $10 bills and six $20 bills.

Step 4

Check

Six $10 bills and six $20 bills = 6(\$10) + 6(\$20) = \$60 + \$120 = \$180.

The answer checks out.

Let’s solve the same problem using the method “Look for a Pattern.”

Method 2: Looking for a Pattern

Step 1

Understand

Andrew gives the bank teller a $180 check.

The bank teller gives Andrew 12 bills. These bills are mixed $10 bills and $20 bills.

We want to know how many of each kind of bill Andrew receives.

Step 2

Strategy

Let’s start by making a table of the different ways Andrew can have twelve $10 bills and $20 bills.

Andrew could have twelve $10 bills and zero $20 bills or eleven $10 bills and one $20 bill, so on.

We can calculate the total amount of money for each case.

Look for patterns appearing in the table that can be used to find the solution.

Step 3

Apply strategy/solve

Let’s fill the rows of the table until we see a pattern.

$10 bills $20 bills Total amount
12 0 \$10(12) + \$20(0) = \$120
11 1 \$10(11) + \$20(1) = \$130
10 2 \$10(10) + \$20(2) = \$140

We see that every time we reduce the number of $10 bills by one and increase the number of $20 bills by one, the total amount increased by $10. The last entry in the table gives a total amount of $140 so we have $40 to go until we reach our goal. This means that we should reduce the number of $10 bills by four and increase the number of $20 bills by four. We have

Six $10 bills and six $20 bills

6(\$10) + 6(\$20) = \$180

Answer: Andrew gets six $10 bills and six $20 bills

Step 4

Check

Six $10 bills and six $20 bills = 6(\$10) + 6(\$20) = \$60 + \$120 = \$180.

The answer checks out.

You can see that the second method we used for solving the problem was less tedious. In the first method, we listed all the possible options and found the answer we were seeking. In the second method, we started with listing the options but we looked for a pattern that helped us find the solution faster. The methods of “Making a Table” and “Look for a Pattern” are both more powerful if used alongside other problem-solving methods.

Solve Real-World Problems Using Selected Strategies as Part of a Plan

Example 5:

Anne is making a box without a lid. She starts with a 20 \ in \times 20 \ in square piece of cardboard and cuts out four equal squares from each corner of the cardboard as shown. She then folds the sides of the box and glues the edges together. How big does she need to cut the corner squares in order to make the box with the biggest volume?

Solution

Step 1

Understand

Anne makes a box out a 20 \ in \times \ 20 \ in piece of cardboard.

She cuts out four equal squares from the corners of the cardboard.

She folds the sides and glues them to make a box.

How big should the cut out squares be to make the box with the biggest volume?

Step 2

Strategy

We need to remember the formula for the volume of a box.

Volume = Area of base \times height

Volume = width \times length \times height

Make a table of values by picking different values for the side of the squares that we are cutting out and calculate the volume.

Step 3

Apply strategy/solve

Let’s “make” a box by cutting out four corner squares with sides equal to 1 inch. The diagram will look like this:

You see that when we fold the sides over to make the box, the height becomes 1 inch, the width becomes 18 inches and the length becomes 18 inches.

Volume = width \times length \times height

Volume = 18 \times 18 \times 1 =324 \ in^3

Let’s make a table that shows the value of the box for different square sizes:

Side of Square Box Height Box Width Box Length Volume
1 1 18 18 18 \times 18 \times 1 = 324
2 2 16 16 16 \times 16 \times 2 = 512
3 3 14 14 14 \times 14 \times 3 = 588
4 4 12 12 12 \times 12 \times 4 = 576
5 5 10 10 10 \times 10 \times 5 = 500
6 6 8 8 8 \times 8 \times 6 = 384
7 7 6 6 6 \times 6 \times 7 = 252
8 8 4 4 4 \times 4 \times 8 = 128
9 9 2 2 2 \times 2 \times 9 = 36
10 10 0 0  0 \times 0 \times 10 = 0

We stop at a square of 10 inches because at this point we have cut out all of the cardboard and we cannot make a box anymore. From the table we see that we can make the biggest box if we cut out squares with a side length of three inches. This gives us a volume of 588 \ in^3.

Answer The box of greatest volume is made if we cut out squares with a side length of three inches.

Step 4 Check

We see that 588 \ in^3 is the largest volume appearing in the table. We picked integer values for the sides of the squares that we are cut out. Is it possible to get a larger value for the volume if we pick non-integer values? Since we get the largest volume for the side length equal to three inches, let’s make another table with values close to three inches that is split into smaller increments:

Side of Square Box Height Box Width Box Length Volume
2.5 2.5 15 15 15 \times 15 \times 2.5 = 562.5
2.6 2.6 14.8 14.8 14.8 \times 14.8 \times 2.6 = 569.5
2.7 2.7 14.6 14.6 14.6 \times 14.6 \times 2.7 = 575.5
2.8 2.8 14.4 14.4 14.4 \times 14.4 \times 2.8 = 580.6
2.9 2.9 14.2 14.2 14.2 \times 14.2 \times 2.9 = 584.8
3 3 14 14 14 \times 14 \times 3 = 588
3.1 3.1 13.8 13.8 13.8 \times 13.8 \times 3.1 = 590.4
3.2 3.2 13.6 13.6 13.6 \times 13.6 \times 3.2 = 591.9
3.3 3.3 13.4 13.4 13.4 \times 13.4 \times 3.3 = 592.5
3.4 3.4 13.2 13.2 13.2 \times 13.2 \times 3.4 = 592.4
3.5 3.5 13 13 13 \times 13 \times 3.5 = 591.5

Notice that the largest volume is not when the side of the square is three inches, but rather when the side of the square is 3.3 inches.

Our original answer was not incorrect but it was obviously not as accurate as it could be. You can get an even more accurate answer if we take even smaller increments of the side length of the square. We can choose measurements that are smaller and larger than 3.3 inches.

The answer checks out if we want it rounded to zero decimal places but, A more accurate answer is 3.3 inches.

Review Questions

  1. Go back and find the solution to the problem in Example 1.
  2. Britt has $2.25 in nickels and dimes. If she has 40 coins in total how many of each coin does she have?
  3. A pattern of squares is out together as shown. How many squares are in the 12^{th} diagram?
  4. Oswald is trying to cut down on drinking coffee. His goal is to cut down to 6 cups per week. If he starts 24 cups the first week, cuts down to 21 cups the second week, and 18 cups the third week, how many weeks will it take him to reach his goal?
  5. Taylor checked out a book from the library and it is now 5 days late. The late fee is 10 cents per day. How much is the fine?
  6. How many hours will a car traveling at 75 miles per hour take to catch up to a car traveling at 55 miles per hour if the slower car starts two hours before the faster car?
  7. Grace starts biking at 12 miles per hour. One hour later, Dan starts biking at 15 miles per hour, following the same route. How long would it take him to catch up with Grace?
  8. Lemuel wants to enclose a rectangular plot of land with a fence. He has 24 feet of fencing. What is the largest possible area that he could enclose with the fence?

Review Answers

  1. $30
  2. 5 dimes and 35 nickels
  3. 23 squares
  4. 7 weeks
  5. 50 cents
  6. 5.5 hours
  7. 5 hours
  8. 3 \ ft \ in^3

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