- Read and understand given problem situations.
- Make a plan to solve the problem.
- Solve the problem and check the results.
- Compare alternative approaches to solving the problem.
- Solve real-world problems using a plan.
We always think of mathematics as the subject in school where we solve lots of problems. Problem solving is necessary in all aspects of life. Buying a house, renting a car, or figuring out which is the better sale are just a few examples of situations where people use problem-solving techniques. In this book, you will learn different strategies and approaches to solving problems. In this section, we will introduce a problem-solving plan that will be useful throughout this book.
Read and Understand a Given Problem Situation
The first step to solving a word problem is to read and understand the problem. Here are a few questions that you should be asking yourself:
- What am I trying to find out?
- What information have I been given?
- Have I ever solved a similar problem?
This is also a good time to define any variables. When you identify your knowns and unknowns, it is often useful to assign them a letter to make notation and calculations easier.
Make a Plan to Solve the Problem
The next step in the problem-solving plan is to develop a strategy. How can the information you know assist you in figuring out the unknowns?
Here are some common strategies that you will learn:
- Drawing a diagram.
- Making a table.
- Looking for a pattern.
- Using guess and check.
- Working backwards.
- Using a formula.
- Reading and making graphs.
- Writing equations.
- Using linear models.
- Using dimensional analysis.
- Using the right type of function for the situation.
In most problems, you will use a combination of strategies. For example, looking for patterns is a good strategy for most problems, and making a table and drawing a graph are often used together. The “writing an equation” strategy is the one you will work with the most in your study of algebra.
Solve the Problem and Check the Results
Once you develop a plan, you can implement it and solve the problem, carrying out all operations to arrive at the answer you are seeking.
The last step in solving any problem should always be to check and interpret the answer. Ask yourself:
- Does the answer make sense?
- If you plug the answer back into the problem, do all the numbers work out?
- Can you get the same answer through another method?
Compare Alternative Approaches to Solving the Problem
Sometimes one specific method is best for solving a problem. Most problems, however, can be solved by using several different strategies. When you are familiar with all of the problem-solving strategies, it is up to you to choose the methods that you are most comfortable with and that make sense to you. In this book, we will often use more than one method to solve a problem, so we can demonstrate the strengths and weakness of different strategies for solving different types of problems.
Whichever strategy you are using, you should always implement the problem-solving plan when you are solving word problems. Here is a summary of the problem-solving plan.
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.
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 solve your problem solving plan.
Carry out the plan - Solve
This is where you solve the equation you developed in Step 2.
Look - Check and Interpret
Check to see if you used all your information. Then look to see if the answer makes sense.
Solve Real-World Problems Using a Plan
Let’s now apply this problem solving plan to a problem.
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?
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 0.5 × original price.
So, the first sale price =0.5 × original price
A savings of 40% off the new price means you pay 60% of the new price, or 0.6 × new price.
0.6×(0.5×original price)=0.3×original price is the price after the second discount.
We know that after two discounts, the final price is $21.
So 0.3×original price=$21.
Step 3: Solve
Since 0.3×original price=$21, we can find the original price by dividing $21 by 0.3.
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 $70=.5×$70=$35 savings. So the price after the first discount is original price−savings or $70−35=$35.
Then 40% of that is .4×$35=$14. So after the second discount, the price is $35−14=$21.
The answer checks out.
The problem-solving plan used here is based on the ideas of George Po´lya, who describes his useful problem-solving strategies in more detail in the book How to Solve It. Some of the techniques in the book can also be found on Wikipedia, in the entry http://en.wikipedia.org/wiki/How_to_Solve_It.
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- Mariana deposits $500 in a savings account that pays 3% simple interest per year. How much will be in her account after three years?
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- A department store has a 15% discount for employees. Suppose an employee has a coupon worth $10 off any item and she wants to buy a $65 purse. What is the final cost of the purse if the employee discount is applied before the coupon is subtracted?
- To host a dance at a hotel you must pay $250 plus $20 per guest. How much money would you have to pay for 25 guests?
- Yusef’s phone plan costs $10 a month plus $0.05 per minute. If his phone bill for last month was $25.80, how many minutes did he spend on the phone?
- It costs $12 to get into the San Diego County Fair and $1.50 per ride.
- If Rena spent $24 in total, how many rides did she go on?
- How much would she have spent in total if she had gone on five more rides?
- An ice cream shop sells a small cone for $2.95, a medium cone for $3.50, and a large cone for $4.25. Last Saturday, the shop sold 22 small cones, 26 medium cones and 15 large cones. How much money did the store earn?
- In Lise’s chemistry class, there are two midterm exams, each worth 30% of her total grade, and a final exam worth 40%. If Lise scores 90% on both midterms and 80% on the final exam, what is her overall score in the class?
- The sum of the angles in a triangle is 180 degrees. If the second angle is twice the size of the first angle and the third angle is three times the size of the first angle, what are the measures of the angles in the triangle?
- A television that normally costs $120 goes on sale for 20% off. What is the new price?
- A cake recipe calls for 134 cup of flour. Jeremy wants to make four cakes. How many cups of flour will he need?
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- Kylie is mowing lawns to earn money for a new bike. After mowing four lawns, she still needs $40 more to pay for the bike. After mowing three more lawns, she has $5 more than she needs to pay for the bike.
- How much does she earn per lawn?
- What is the cost of the bike?
- Jared goes trick-or-treating with his brother and sister. At the first house they stop at, they collect three pieces of candy each; at the next three houses, they collect two pieces of candy each. Then they split up and go down different blocks, where Jared collects 12 pieces of candy and his brother and sister collect 14 each.
- How many pieces of candy does Jared end up with?
- How many pieces of candy do all three of them together end up with?
- Marco’s daughter Elena has four boxes of toy blocks, with 50 blocks in each one. One day she dumps them all out on the floor, and some of them get lost. When Marco tries to put them away again, he ends up with 45 blocks in one box, 53 in another, 46 in a third, and 51 in the fourth. How many blocks are missing?
- A certain hour-long TV show usually includes 16 minutes of commercials. If the season finale is two and a half hours long, how many minutes of commercials should it include to keep the same ratio of commercial time to show time?
- Karen and Chase bet on a baseball game: if the home team wins, Karen owes Chase fifty cents for every run scored by both teams, and Chase owes Karen the same amount if the visiting team wins. The game runs nine innings, and the home team scores one run in every odd-numbered inning, while the visiting team scores two runs in the third inning and two in the sixth. Who owes whom how much?
- Kelly, Chris, and Morgan are playing a card game. In this game, the first player to empty their hand scores points for all the cards left in the other players’ hands as follows: aces are worth one point, face cards ten points, and all other cards are face value. When Kelly empties her hand, Morgan is holding two aces, a king, and a three; Chris is holding a five, a seven, and a queen. How many points does Kelly score?
- A local club rents out a social hall to host an event. The hall rents for $350, and they hope to make back the rental price by charging $15 admission per person. How many people need to attend for the club to break even?
- You plan to host a barbecue, and you expect 10 friends, 8 neighbors, and 7relatives to show up.
- If you expect each person (including yourself) to eat about two ounces of potato salad, how many half-pound containers of potato salad should you buy?
- If hot dogs come in ten-packs that cost $4.80 apiece and hot dog buns come in eight-packs that cost $2.80 apiece, how much will you need to spend to have hot dogs and buns for everyone?