2.8: Problem-Solving Strategies: Guess and Check and Work Backwards
This lesson will expand your toolbox of problem-solving strategies to include guess and check and work backwards. Let’s begin by reviewing the four-step problem-solving plan.
Step 1: Understand the problem.
Step 2: Devise a plan – Translate.
Step 3: Carry out the plan – Solve.
Step 4: Look – Check and Interpret.
Develop and Use the Strategy: Guess and Check
The strategy for the “guess and check” method is to guess a solution and use that guess in the problem to see if you get the correct answer. If the answer is too big or too small, then make another guess that will get you closer to the goal. You continue guessing until you arrive at the correct solution. The process might sound like a long one; however, the guessing process will often lead you to patterns that you can use to make better guesses along the way.
Here is an example of how this strategy is used in practice.
Example 1: Nadia takes a ribbon that is 48 inches long and cuts it in two pieces. One piece is three times as long as the other. How long is each piece?
Solution: We need to find two numbers that add to 48. One number is three times the other number.
\begin{align*}&\text{Guess} && 5 \ \text{and} \ 15 && \text{the sum is} \ 5 + 15 = 20 && \text{which is too small}\\ &\text{Guess bigger numbers} && 6 \ \text{and} \ 18 && \text{the sum is} \ 6 + 18 = 24 && \text{which is too small}\end{align*}
However, you can see that the previous answer is exactly half of 48.
Multiply 6 and 18 by two.
\begin{align*}\text{Our next guess is} && 12 \ \text{and} \ 36 && \text{the sum is} \ 12 + 36 = 48 && \text{This is correct}.\end{align*}
Develop and Use the Strategy: Work Backwards
The “work backwards” method works well for problems in which a series of operations is applied to an unknown quantity and you are given the resulting value. The strategy in these problems is to start with the result and apply the operations in reverse order until you find the unknown. Let’s see how this method works by solving the following problem.
Example 2: Anne has a certain amount of money in her bank account on Friday morning. During the day she writes a check for $24.50, makes an ATM withdrawal of $80, and deposits a check for $235. At the end of the day, she sees that her balance is $451.25. How much money did she have in the bank at the beginning of the day?
Solution: We need to find the money in Anne’s bank account at the beginning of the day on Friday. From the unknown amount, we subtract $24.50 and $80 and we add $235. We end up with $451.25. We need to start with the result and apply the operations in reverse.
Start with $451.25. Subtract $235, add $80, and then add $24.50.
\begin{align*}451.25 - 235 + 80 + 24.50 = 320.75\end{align*}
Anne had $320.75 in her account at the beginning of the day on Friday.
Plan and Compare Alternative Approaches to Solving Problems
Most word problems can be solved in more than one way. Often one method is more straightforward than others. In this section, you will see how different problem-solving approaches compare for solving different kinds of problems.
Example 3: Nadia’s father is 36. He is 16 years older than four times Nadia’s age. How old is Nadia?
Solution: This problem can be solved with either of the strategies you learned in this section. Let’s solve the problem using both strategies.
Guess and Check Method:
We need to find Nadia’s age.
We know that her father is 16 years older than four times her age, or \begin{align*}4 \times\end{align*} (Nadia’s age) + 16.
We know her father is 36 years old.
Work Backwards Method:
Nadia’s father is 36 years old.
To get from Nadia’s age to her father’s age, we multiply Nadia’s age by four and add 16.
Working backward means we start with the father’s age, subtract 16, and divide by 4.
Solve Real-World Problems Using Selected Strategies as Part of a Plan
Example 4: Hana rents a car for a day. Her car rental company charges $50 per day and $0.40 per mile. Peter rents a car from a different company that charges $70 per day and $0.30 per mile. How many miles do they have to drive before Hana and Peter pay the same price for the rental for the same number of miles?
Solution: Hana’s total cost is $50 plus $0.40 times the number of miles.
Peter’s total cost is $70 plus $0.30 times the number of miles.
Guess the number of miles and use this guess to calculate Hana’s and Peter’s total cost.
Keep guessing until their total cost is the same.
\begin{align*}&\text{Guess} && 50 \ miles\\ &\text{Check} && \$50+\$0.40(50)=\$70 && \$70+\$0.30(50)=\$85 \\ &\text{Guess} && 60 \ miles\\ &\text{Check} && \$50+\$0.40(60)=\$74 && \$70+\$0.30(60)=\$88 \end{align*}
Notice that for an increase of 10 miles, the difference between total costs fell from $15 to $14. To get the difference to zero, we should try increasing the mileage by 140 miles.
\begin{align*}&\text{Guess} && 200 \ miles\\ &\text{Check} && \$50+\$0.40(200)=\$130 && \$70+\$0.30(200)=\$130 && \text{correct}\end{align*}
Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Problem Solving Word Problems 2 (12:20)
- Finish the problem we started in Example 3.
- Nadia is at home and Peter is at school, which is 6 miles away from home. They start traveling toward each other at the same time. Nadia is walking at 3.5 miles per hour and Peter is skateboarding at 6 miles per hour. When will they meet and how far from home is their meeting place?
- Peter bought several notebooks at Staples for $2.25 each and he bought a few more notebooks at Rite-Aid for $2 each. He spent the same amount of money in both places and he bought 17 notebooks in total. How many notebooks did Peter buy in each store?
- Andrew took a handful of change out of his pocket and noticed that he was holding only dimes and quarters in his hand. He counted that he had 22 coins that amounted to $4. How many quarters and how many dimes does Andrew have?
- Anne wants to put a fence around her rose bed that is one and a half times as long as it is wide. She uses 50 feet of fencing. What are the dimensions of the garden?
- Peter is outside looking at the pigs and chickens in the yard. Nadia is indoors and cannot see the animals. Peter gives her a puzzle. He tells her that he counts 13 heads and 36 feet and asks her how many pigs and how many chickens are in the yard. Help Nadia find the answer.
- Andrew invests $8000 in two types of accounts: a savings account that pays 5.25% interest per year and a more risky account that pays 9% interest per year. At the end of the year, he has $450 in interest from the two accounts. Find the amount of money invested in each account.
- There is a bowl of candy sitting on our kitchen table. This morning Nadia takes one-sixth of the candy. Later that morning Peter takes one-fourth of the candy that’s left. This afternoon, Andrew takes one-fifth of what’s left in the bowl and finally Anne takes one-third of what is left in the bowl. If there are 16 candies left in the bowl at the end of the day, how much candy was there at the beginning of the day?
- Nadia can completely mow the lawn by herself in 30 minutes. Peter can completely mow the lawn by himself in 45 minutes. How long does it take both of them to mow the lawn together?
Mixed Review
- Rewrite \begin{align*}\sqrt{500}\end{align*} as a simplified square root.
- To which number category does \begin{align*}\frac{-2}{13}\end{align*} belong?
- Simplify \begin{align*}\frac{1}{2}|19-65|-14\end{align*}.
- Which property is being applied? \begin{align*}16+4c+11=(16+11)+4c\end{align*}
- Is \begin{align*}\left \{(4,2),(4,-2),(9,3),(9,-3)\right \}\end{align*} a function?
- Write using function notation: \begin{align*}y=\frac{1}{12} x-5\end{align*}.
- Jordyn spent $36 on four cases of soda. How much was each case?
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