### Guess and Check and Working Backwards

In this section, you will learn about the methods of **Guess and Check** and **Working Backwards**. These are very powerful strategies in problem solving and probably the most commonly used in everyday life. Let’s review our problem-solving plan.

**Step 1**

Understand the problem.

Read the problem carefully. Then list all the components and data involved, and assign your variables.

**Step 2**

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.

**Step 3**

Carry out the plan – Solve

This is where you solve the equation you came up with in Step 2.

**Step 4**

Look – Check and Interpret

Check that the answer makes sense.

Let’s now look at some strategies we can use as part of this plan.

**Develop and Use the Strategy “Guess and Check”**

The strategy for the method “Guess and Check” is to guess a solution and then plug the guess back into the problem to see if you get the correct answer. If the answer is too big or too small, make another guess that will get you closer to the goal, and continue guessing until you arrive at the correct solution. The process might sound long, but often you will find patterns that you can use to make better guesses along the way.

#### Using the "Guess and Check" Strategy

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?

**Step 1: Understand**

We need to find two numbers that add up to 48. One number is three times the other number.

**Step 2: Strategy**

We guess two random numbers, one three times bigger than the other, and find the sum.

If the sum is too small we guess larger numbers, and if the sum is too large we guess smaller numbers.

Then, we see if any patterns develop from our guesses.

**Step 3: Apply Strategy/Solve**

\begin{align*}& \text{Guess} \qquad 5\ \text{and}\ 15 \qquad 5 + 15 = 20 \qquad \text{sum is too small}\\
& \text{Guess} \qquad 6\ \text{and}\ 18 \qquad 6 + 18 = 24 \qquad \text{sum is too small}\end{align*}

Our second guess gives us a sum that is exactly half of 48. What if we double that guess?

\begin{align*}12 + 36 = 48\end{align*}

**There’s our answer.** The pieces are 12 and 36 inches long.

**Step 4: Check**

\begin{align*} 12 + 36 & = 48 \qquad \quad \ \text{The pieces add up to 48 inches}.\\
36 & = 3(12) \qquad \text{One piece is three times as long as the other}.\end{align*}

The answer checks out.

**Develop and Use the Strategy “Work Backward”**

The “Work Backward” method works well for problems where a series of operations is done on an unknown number and you’re only given the result. To use this method, start with the result and apply the operations in reverse order until you find the starting number.

#### Using the "Work Backward" Strategy

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?

**Step 1: Understand**

We need to find the money in Anne’s bank account at the beginning of the day on Friday.

She took out $24.50 and $80 and put in $235.

She ended up with $451.25 at the end of the day.

**Step 2: Strategy**

We start with an unknown amount, do some operations, and end up with a known amount.

We need to start with the result and apply the operations in reverse.

**Step 3: Apply Strategy/Solve**

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.

**Step 4: Check**

\begin{align*}& \text{Anne starts with} \qquad \qquad \quad \quad \quad \ \ \$320.75 \\ & \text{She writes a check for}\ \$24.50. \qquad \$320.75 - 24.50 = \$296.25 \\ & \text{She withdraws}\ \$80. \qquad \qquad \quad \quad \ \ \$296.25 - 80 = \$216.25 \\ & \text{She deposits}\ \$235. \qquad \qquad \qquad \quad \$216.25 + 235 = \$451.25\end{align*}

The answer checks out.

**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, but which method is best can depend on what kind of problem you are facing.

#### Comparing Strategies

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 it using both strategies.

**Guess and Check Method**

**Step 1: Understand**

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 (\text{Nadia's age}) + 16\end{align*}.

We know her father is 36 years old.

**Step 2: Strategy**

We guess a random number for Nadia’s age.

We multiply the number by 4 and add 16 and check to see if the result equals 36.

If the answer is too small, we guess a larger number, and if the answer is too big, we guess a smaller number.

We keep guessing until we get the answer to be 36.

**Step 3: Apply strategy/Solve**

\begin{align*}& \text{Guess Nadia's age} && 10 && 4(10) + 16 = 56 && \text{too big for her father's age}\\ & \text{Guess a smaller number} && 9 \ && 4(9) + 16 = 52 && \text{still too big}\end{align*}

Guessing 9 for Nadia’s age gave us a number that is 16 years too great to be her father’s age. But notice that when we decreased Nadia’s age by one, her father’s age decreased by four. That suggests that we can decrease our final answer by 16 years if we decrease our guess by 4 years.

4 years less than 9 is 5. \begin{align*}4(5) + 16 = 36\end{align*}, which is the right age.

**Answer:** Nadia is 5 years old.

**Step 4: Check**

Nadia is 5 years old. Her father’s age is \begin{align*}4(5) + 16 = 36\end{align*}. This is correct. **The answer checks out.**

You will be asked to try the Work Backward method in the Guided Practice section.

### Example

#### Example 1

Nadia’s *father is 36. He is 16 years older than four times Nadia’s age. Determine Nadia's age by using the Work Backward method.*

**Work Backward Method**

**Step 1: Understand**

We need to find Nadia’s age.

We know her father is 16 years older than four times her age, or \begin{align*}4 \times \text{Nadia's age} + 16\end{align*}.

We know her father is 36 years old.

**Step 2: Strategy**

To get from Nadia’s age to her father’s age, we multiply Nadia’s age by four and add 16.

Working backwards means we start with the father’s age, subtract 16 and divide by 4.

**Step 3: Apply Strategy/Solve**

\begin{align*}& \text{Start with the father's age} \qquad 36\\ & \text{Subtract}\ 16 \qquad \qquad \qquad \qquad \ 36 - 16 = 20 \\ & \text{Divide by}\ 4 \qquad \qquad \qquad \qquad \ 20 \div 4 = 5\end{align*}

**Answer** Nadia is 5 years old.

**Step 4: Check**

Nadia is 5 years old. Her father’s age is \begin{align*}4(5) + 16 = 36\end{align*}. This is correct. **The answer checks out.**

You see that in this problem, the “Work Backward” strategy is more straightforward than the Guess and Check method. The Work Backward method always works best when we know the result of a series of operations, but not the starting number. In the next chapter, you will learn algebra methods based on the Work Backward method.

### Review

- Nadia is at home and Peter is at school which is 6 miles away from home. They start traveling towards 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; then 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 all. How many notebooks did Peter buy in each store?
- Andrew took a handful of change out of his pocket and noticed that he was only holding dimes and quarters in his hand. He counted and found 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 can see 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.
- 450 tickets are sold for a concert: balcony seats for $35 each and orchestra seats for $25 each. If the total box office take is $13,000, how many of each kind of ticket were sold?
- There is a bowl of candy sitting on our kitchen table. One morning Nadia takes one-sixth of the candy. Later that morning Peter takes one-fourth of the candy that’s left. That 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?
- Three monkeys spend a day gathering coconuts together. When they have finished, they are very tired and fall asleep. The following morning, the first monkey wakes up. Not wishing to disturb his friends, he decides to divide the coconuts into three equal piles. There is one left over, so he throws this odd one away, helps himself to his share, and goes home. A few minutes later, the second monkey awakes. Not realizing that the first has already gone, he too divides the coconuts into three equal heaps. He finds one left over, throws the odd one away, helps himself to his fair share, and goes home. In the morning, the third monkey wakes to find that he is alone. He spots the two discarded coconuts, and puts them with the pile, giving him a total of twelve coconuts.
- How many coconuts did the first two monkeys take?
- How many coconuts did the monkeys gather in all?

- Two prime numbers have a product of 51. What are the numbers?
- Two prime numbers have a product of 65. What are the numbers?
- The square of a certain positive number is eight more than twice the number. What is the number?
- Is 91 prime? (Hint: if it’s
*not*prime, what are its prime factors?) - Is 73 prime?
- Alison’s school day starts at 8:30, but today Alison wants to arrive ten minutes early to discuss an assignment with her English teacher. If she is also giving her friend Sherice a ride to school, and it takes her 12 minutes to get to Sherice’s house and another 15 minutes to get to school from there, at what time does Alison need to leave her house?
- At her retail job, Kelly gets a raise of 10% every six months. After her third raise, she now makes $13.31 per hour. How much did she make when she first started out?
- Three years ago, Kevin’s little sister Becky had her fifth birthday. If Kevin was eight when Becky was born, how old is he now?
- A warehouse is full of shipping crates; half of them are headed for Boston and the other half for Philadelphia. A truck arrives to pick up 20 of the Boston-bound crates, and then another truck carries away one third of the Philadelphia-bound crates. An hour later, half of the remaining crates are moved onto the loading dock outside. If there are 40 crates left in the warehouse, how many were there originally?
- Gerald is a bus driver who takes over from another bus driver one day in the middle of his route. He doesn’t pay attention to how many passengers are on the bus when he starts driving, but he does notice that three passengers get off at the next stop, a total of eight more get on at the next three stops, two get on and four get off at the next stop, and at the stop after that, a third of the passengers get off.
- If there are now 14 passengers on the bus, how many were there when Gerald first took over the route?
- If half the passengers who got on while Gerald was driving paid the full adult fare of $1.50, and the other half were students or seniors who paid a discounted fare of $1.00, how much cash was in the bus’s fare box at the beginning of Gerald’s shift if there is now $73.50 in it?
- When Gerald took over the route, all the passengers currently on the bus had paid full fare. However, some of the passengers who had previously gotten on and off the bus were students or seniors who had paid the discounted fare. Based on the amount of money that was in the cash box, if 28 passengers had gotten on the bus and gotten off before Gerald arrived (in addition to the passengers who had gotten on and were still there when he arrived), how many of those passengers paid the discounted fare?
- How much money would currently be in the cash box if all the passengers throughout the day had paid the full fare?

### Review (Answers)

To view the Review answers, open this PDF file and look for section 2.11.

### 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/9612.*