1.8: Problem-Solving Strategies: Make a Table and Look for a Pattern
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 a problem.
- Solve real-world problems using the above 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.
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. Once you understand the problem, you can devise a strategy to solve it.
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?
Solution
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.
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 each week, what will be her total jogging time on the sixth week?
Solution
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.
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 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 a 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 |
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.
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 is \begin{align*}60 + 12 \times 5 = 120 \ minutes\end{align*}.
The answer checks out.
You can see that making a table helped us organize and clarify the information we were given, and helped guide us in the next steps of the problem. We solved this problem solely by making a table; in many situations, we would combine this strategy with others to get a solution.
Develop and Use the Strategy: Look for a Pattern
Looking for a pattern is another strategy that you can use 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 rows?
Solution
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 rows.
Strategy
A good strategy is to make a table and list how many balls are in triangles of different rows.
One row: It is simple to see that a triangle with one row has only one ball.
Two rows: For a triangle with two rows, we add the balls from the top row to the balls from the bottom row. It is useful to make a sketch of the separate rows in the triangle.
\begin{align*}3 = 1 + 2\end{align*}
Three rows: We add the balls from the top triangle to the balls from the bottom row.
\begin{align*}6 = 3 + 3\end{align*}
Now we can fill in the first three rows of a table.
Number of Rows | 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 rows. (For example, a triangle with 3 rows has 3 balls in the bottom row.)
To get the total number of balls for the new triangle, we add the number of balls in the old triangle to the number of balls in the new bottom row.
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 rows in the new triangle
Number of Rows | Number of Balls |
---|---|
1 | 1 |
2 | 3 |
3 | 6 |
4 | \begin{align*}6 + 4 = 10\end{align*} |
5 | \begin{align*}10 + 5 = 15\end{align*} |
6 | \begin{align*}15 + 6 = 21\end{align*} |
7 | \begin{align*}21 + 7 = 28\end{align*} |
8 | \begin{align*}28 + 8 = 36 \end{align*} |
Answer There are 36 balls in a triangle arrangement with 8 rows.
Check
Each row of the triangle has one more ball than the previous one. In a triangle with 8 rows,
row 1 has 1 ball, row 2 has 2 balls, row 3 has 3 balls, row 4 has 4 balls, row 5 has 5 balls, row 6 has 6 balls, row 7 has 7 balls, row 8 has 8 balls.
When we add these we get: \begin{align*}1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36 \ balls\end{align*}
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
Understand
Andrew gives the bank teller a $180 check.
The bank teller gives Andrew 12 bills. These bills are a mix of $10 bills and $20 bills.
We want to know how many of each kind of bill Andrew receives.
Strategy
Let’s start by making a table of the different ways Andrew can have twelve bills in tens and twenties.
Andrew could have twelve $10 bills and zero $20 bills, or eleven $10 bills and one $20 bill, and so on.
We can calculate the total amount of money for each case.
Apply strategy/solve
$10 bills | $ 20 bills | Total amount |
---|---|---|
12 | 0 | \begin{align*}\$10(12) + \$20(0) = \$120\end{align*} |
11 | 1 | \begin{align*}\$10(11) + \$20(1) = \$130\end{align*} |
10 | 2 | \begin{align*}\$10(10) + \$20(2) = \$140\end{align*} |
9 | 3 | \begin{align*}\$10(9) + \$20(3) = \$150\end{align*} |
8 | 4 | \begin{align*}\$10(8) + \$20(4) = \$160\end{align*} |
7 | 5 | \begin{align*}\$10(7) + \$20(5) = \$170\end{align*} |
6 | 6 | \begin{align*}\$10(6) + \$20(6) = \$180\end{align*} |
5 | 7 | \begin{align*}\$10(5) + \$20(7) = \$190\end{align*} |
4 | 8 | \begin{align*}\$10(4) + \$20(8) = \$200\end{align*} |
3 | 9 | \begin{align*}\$10(3) + \$20(9) = \$210\end{align*} |
2 | 10 | \begin{align*}\$10(2) + \$20(10) = \$220\end{align*} |
1 | 11 | \begin{align*}\$10(1) + \$20(11) = \$230 \end{align*} |
0 | 12 | \begin{align*}\$10(0) + \$20(12) = \$240\end{align*} |
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.
Check
Six $10 bills and six $20 bills \begin{align*}\rightarrow 6(\$10) + 6(\$20) = \$60 + \$120 = \$180\end{align*}
The answer checks out.
Let’s solve the same problem using the method “Look for a Pattern.”
Method 2: Looking for a Pattern
Understand
Andrew gives the bank teller a $180 check.
The bank teller gives Andrew 12 bills. These bills are a mix of $10 bills and $20 bills.
We want to know how many of each kind of bill Andrew receives.
Strategy
Let’s start by making a table just as we did above. However, this time we will look for patterns in the table that can be used to find the solution.
Apply strategy/solve
Let’s fill in the rows of the table until we see a pattern.
$10 bills | $20 bills | Total amount |
---|---|---|
12 | 0 | \begin{align*}\$10(12) + \$20(0) = \$120\end{align*} |
11 | 1 | \begin{align*}\$10(11) + \$20(1) = \$130\end{align*} |
10 | 2 | \begin{align*}\$10(10) + \$20(2) = \$140 \end{align*} |
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 increases 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. That would give us six $10 bills and six $20 bills.
\begin{align*}6(\$10) + 6(\$20) = \$60 + 120 = \$180\end{align*}
Answer: Andrew gets six $10 bills and six $20 bills.
Check
Six $10 bills and six $20 bills \begin{align*}\rightarrow 6(\$10) + 6(\$20) = \$60 + 120 = \$180\end{align*}
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 by listing the options, but we found a pattern that helped us find the solution faster. The methods of “Making a Table” and “Looking 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. 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 of a \begin{align*}20 \ in \times 20 \ in\end{align*} 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.
\begin{align*}\text{Volume} = \text{Area of base} \times \text{height}\end{align*}
\begin{align*}\text{Volume} = \text{width} \times \text{length} \times \text{height}\end{align*}
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.
\begin{align*}\text{Volume} = \text{width} \times \text{length} \times \text{height}\end{align*}
\begin{align*}\text{Volume} = 18 \times 18 \times 1 = 324 \ in^3\end{align*}
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 | \begin{align*}18 \times 18 \times 1 = 324\end{align*} |
2 | 2 | 16 | 16 | \begin{align*}16 \times 16 \times 2 = 512\end{align*} |
3 | 3 | 14 | 14 | \begin{align*}14 \times 14 \times 3 = 588\end{align*} |
4 | 4 | 12 | 12 | \begin{align*}12 \times 12 \times 4 = 576\end{align*} |
5 | 5 | 10 | 10 | \begin{align*}10 \times 10 \times 5 = 500\end{align*} |
6 | 6 | 8 | 8 | \begin{align*}8 \times 8 \times 6 = 384\end{align*} |
7 | 7 | 6 | 6 | \begin{align*}6 \times 6 \times 7 = 252\end{align*} |
8 | 8 | 4 | 4 | \begin{align*}4 \times 4 \times 8 = 128\end{align*} |
9 | 9 | 2 | 2 | \begin{align*}2 \times 2 \times 9 = 36\end{align*} |
10 | 10 | 0 | 0 | \begin{align*}0 \times 0 \times 10 = 0 \end{align*} |
We stop at a square of 10 inches because at this point we have cut out all of the cardboard and we can’t make a box any more. 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 \begin{align*}588 \ in^3\end{align*}.
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 \begin{align*}588 \ in^3\end{align*} 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 | \begin{align*}15 \ \times \ 15 \ \times \ 2.5 = 562.5\end{align*} |
2.6 | 2.6 | 14.8 | 14.8 | \begin{align*}14.8 \times 14.8 \times 2.6 = 569.5\end{align*} |
2.7 | 2.7 | 14.6 | 14.6 | \begin{align*}14.6 \times 14.6 \times 2.7 = 575.5\end{align*} |
2.8 | 2.8 | 14.4 | 14.4 | \begin{align*}14.4 \times 14.4 \times 2.8 = 580.6\end{align*} |
2.9 | 2.9 | 14.2 | 14.2 | \begin{align*}14.2 \times 14.2 \times 2.9 = 584.8\end{align*} |
3 | 3 | 14 | 14 | \begin{align*}14 \times 14 \times 3 = 588\end{align*} |
3.1 | 3.1 | 13.8 | 13.8 | \begin{align*}13.8 \times 13.8 \times 3.1 = 590.4\end{align*} |
3.2 | 3.2 | 13.6 | 13.6 | \begin{align*}13.6 \times13.6 \times 3.2 = 591.9\end{align*} |
3.3 | 3.3 | 13.4 | 13.4 | \begin{align*}13.4 \times 13.4 \times 3.3 = 592.5\end{align*} |
3.4 | 3.4 | 13.2 | 13.2 | \begin{align*}13.2 \times 13.2 \times 3.4 = 592.4\end{align*} |
3.5 | 3.5 | 13 | 13 | \begin{align*}13 \ \times \ 13 \ \times \ 3.5 = 591.5 \end{align*} |
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 not as accurate as it could be. We can get an even more accurate answer if we take even smaller increments of the side length of the square. To do that, we would choose smaller measurements that are in the neighborhood of 3.3 inches.
Meanwhile, our first answer checks out if we want it rounded to zero decimal places, but a more accurate answer is 3.3 inches.
Review Questions
- Go back and find the solution to the problem in Example 1.
- Britt has $2.25 in nickels and dimes. If she has 40 coins in total, how many of each coin does she have?
- Jeremy divides a 160-square-foot garden into plots that are either 10 or 12 square feet each. If there are 14 plots in all, how many plots are there of each size?
- A pattern of squares is put together as shown. How many squares are in the \begin{align*}12^{th}\end{align*} diagram? \begin{align*}\;\end{align*}
- In Harrisville, local housing laws specify how many people can live in a house or apartment: the maximum number of people allowed is twice the number of bedrooms, plus one. If Jan, Pat, and their four children want to rent a house, how many bedrooms must it have?
- A restaurant hosts children’s birthday parties for a cost of $120 for the first six children (including the birthday child) and $30 for each additional child. If Jaden’s parents have a budget of $200 to spend on his birthday party, how many guests can Jaden invite?
- A movie theater with 200 seats charges $8 general admission and $5 for students. If the 5:00 showing is sold out and the theater took in $1468 for that showing, how many of the seats are occupied by students?
- Oswald is trying to cut down on drinking coffee. His goal is to cut down to 6 cups per week. If he starts with 24 cups the first week, then 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?
- 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?
- Mikhail is filling a sack with oranges.
- If each orange weighs 5 ounces and the sack will hold 2 pounds, how many oranges will the sack hold before it bursts?
- Mikhail plans to use these oranges to make breakfast smoothies. If each smoothie requires \begin{align*}\frac{3}{4}\end{align*} cup of orange juice, and each orange will yield half a cup, how many smoothies can he make?
- Jessamyn takes out a $150 loan from an agency that charges 12% of the original loan amount in interest each week. If she takes five weeks to pay off the loan, what is the total amount (loan plus interest) she will need to pay back?
- 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?
- 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 will it take him to catch up with Grace?
- A new theme park opens in Milford. On opening day, the park has 120 visitors; on each of the next three days, the park has 10 more visitors than the day before; and on each of the three days after that, the park has 20 more visitors than the day before.
- How many visitors does the park have on the seventh day?
- How many total visitors does the park have all week?
- 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?
- Quizzes in Keiko’s history class are worth 20 points each. Keiko scored 15 and 18 points on her last two quizzes. What score does she need on her third quiz to get an average score of 17 on all three?
- Tickets to an event go on sale for $20 six weeks before the event, and go up in price by $5 each week. What is the price of tickets one week before the event?
- Mark is three years older than Janet, and the sum of their ages is 15. How old are Mark and Janet?
- In a one-on-one basketball game, Jane scored \begin{align*}1 \frac{1}{2}\end{align*} times as many points as Russell. If the two of them together scored 10 points, how many points did Jane score?
- Scientists are tracking two pods of whales during their migratory season. On the first day of June, one pod is 120 miles north of a certain group of islands, and every day thereafter it gets 15 miles closer to the islands. The second pod starts out 160 miles east of the islands on June 3, and heads toward the islands at a rate of 20 miles a day.
- Which pod will arrive at the islands first, and on what day?
- How long after that will it take the other pod to reach the islands?
- Suppose the pod that reaches the islands first immediately heads south from the islands at a rate of 15 miles a day, and the pod that gets there second also heads south from there at a rate of 25 miles a day. On what day will the second pod catch up with the first?
- How far will both pods be from the islands on that day?
Texas Instruments Resources
In the CK-12 Texas Instruments Algebra I FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9611.