# Problem-Solving Models

## Use the Problem Solving plan to solve story problems

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Comparison of Problem-Solving Models

### Comparison of Problem-Solving Models

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

#### Original Versus Sale Price

A coffee maker is on sale at 50% off the regular ticket price. On the “Sunday Super Sale” the same coffee maker is on sale at an additional 40% off. If the final price is $21, what was the original price of the coffee maker? Solution Step 1: Understand We know: A coffee maker is discounted 50% and then 40%. The final price is$21.

We want: The original price of the coffee maker.

Step 2: Strategy

Let’s look at the given information and try to find the relationship between the information we know and the information we are trying to find.

50% off the original price means that the sale price is half of the original or \begin{align*}0.5 \ \times\end{align*} original price.

So, the first sale price \begin{align*}= 0.5 \ \times\end{align*} original price

A savings of 40% off the new price means you pay 60% of the new price, or \begin{align*}0.6 \ \times\end{align*} new price.

\begin{align*}0.6 \times (0.5 \times \text{original price}) = 0.3 \times \text{original price}\end{align*} is the price after the second discount.

We know that after two discounts, the final price is 21. So \begin{align*}0.3 \times \text{original price} = \21\end{align*}. Step 3: Solve Since \begin{align*}0.3 \times \text{original price} = \21\end{align*}, we can find the original price by dividing21 by 0.3.

\begin{align*}\text{Original price} = \21 \div 0.3 = \70\end{align*}.

The original price of the coffee maker was $70. Step 4: Check We found that the original price of the coffee maker is$70.

To check that this is correct, let’s apply the discounts.

50% of \begin{align*}\70 = .5 \times \70 = \35\end{align*} savings. So the price after the first discount is \begin{align*}\text{original price} - \text{savings}\end{align*} or \begin{align*}\70 - 35 = \35\end{align*}.

Then 40% of that is \begin{align*}.4 \times \35 = \14\end{align*}. So after the second discount, the price is \begin{align*}\35 - 14 = \21\end{align*}.

#### Receiving Cash

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*}

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 of10 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*}

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.

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

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.

### Example

#### Example 1

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?

We want to know the price one week before the event. We know the price six weeks before the event is $20, and that it goes up$5 each week.

Weeks before event. Price of tickets.
6 \begin{align*}\20\end{align*}
5 \begin{align*}\20+\5 = \25\end{align*}
4 \begin{align*}\25+\5=\30\end{align*}
3 \begin{align*}\30+\5 = \35\end{align*}
2 \begin{align*}\35+\5 = \40\end{align*}
1 \begin{align*}\40+\5 = \45\end{align*}

One week before the event, the tickets will cost $45. ### Review 1. Britt has$2.25 in nickels and dimes. If she has 40 coins in total, how many of each coin does she have?
2. 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?
3. 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*}
4. 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?
5. 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? 6. 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?
7. 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?
8. 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?
9. Mikhail is filling a sack with oranges.
1. If each orange weighs 5 ounces and the sack will hold 2 pounds, how many oranges will the sack hold before it bursts?
2. 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?
10. 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?
11. 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?
12. 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?
13. 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.
1. How many visitors does the park have on the seventh day?
2. How many total visitors does the park have all week?
14. 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?
15. 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?
16. Mark is three years older than Janet, and the sum of their ages is 15. How old are Mark and Janet?
17. 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?
18. 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.
1. Which pod will arrive at the islands first, and on what day?
2. How long after that will it take the other pod to reach the islands?
3. 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?
4. How far will both pods be from the islands on that day?

### Texas Instruments Resources

In the CK-12 Texas Instruments Algebra I FlexBook® resource, 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.

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