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# 1.17: Comparison of Problem-Solving Models

Difficulty Level: At Grade Created by: CK-12
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What if you were given a real-world problem with two unknowns like "You have only dimes and nickels in your pocket that total $1.25. You have a total of 14 coins in your pocket. How many nickels and dimes do you have?" How could you devise a problem-solving plan to solve it? After completing this Concept, you'll be able to make a table or look for patterns to help you solve problems like this one. ### Watch This ### Guidance 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. #### Example A 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 is21.

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 dividing 21 by 0.3. \begin{align*}\text{Original price} = \21 \div 0.3 = \70\end{align*}. The original price of the coffee maker was70.

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*}. The answer checks out. #### Example B Andrew cashes a180 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 twelve10 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 a180 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*}

### Practice

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 of120 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 a150 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, 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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