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Guess and Check, Work Backward

Solve story problems using estimation and evaluation

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Guess and Check, Work Backward

Let's Think About It

Car wash

Credit: melodi2
Source: http://www.freeimages.com/photo/car-wash-1433397
License: CC BY-NC 3.0

Jenn is washing cars with her volleyball team to raise money for the state competition. At first, she kept careful track of the number of cars the team washed per hour, but then things got really busy, and she lost count. She knows that the team earned $2.00 per car, and that they earned $144 total by the end of the day on Sat. If Jenn knows from her notes that they washed 5 cars the first hour, 5 the second hour, and 10 the third hour, how many did they average per hour for the last 4 hours?

In this concept, you will learn how to solve problems using the strategy 'Guess, Check and Revise.'

Guidance

Sometimes, the most efficient way to solve a problem is to simply make an 'educated guess' and see how close it comes to being correct. Then you can apply what you learned from the first guess to improve future guesses until you have the right answer.

The trick to making the Guess and Check method work well is to pay attention to how each new guess affects the answer. This way, even if it requires multiple guesses, they each get closer to being correct instead of you just hoping that a wild guess will happen to work (which may never happen)!

Guided Practice

Betsy picked a bunch of apples. She wanted to give some to her neighbor. She kept a bowl of apples for herself and gave her neighbor five more apples than she kept for herself.

If the total number of apples picked was 25, how many did Betsy keep for herself?

To solve this problem, you can write the following expression.

\begin{align*} x + (x + 5) = 25\end{align*}

In this case, the \begin{align*}x\end{align*} represents the number of apples Betsy kept for herself. 

Next, simply guess and check until you get the correct answer.

Guess that Betsy kept 7 apples for herself:

\begin{align*}7+\left(7+5\right)=&25 \quad \text{Is this true?}\\ 7+12=&25\\ 19=&25 \quad \text{No, this is not true.}\\ \end{align*}

The answer you want is 25, not 19. So Betsy kept more than 7 apples for herself.

Guess that she kept 12 apples:

\begin{align*}12 + (12 + 5) = & 25 \quad \text{Is this true?}\\ 12+17= & 25\\ 29=&25 \quad \text{No, this is not true.}\\\end{align*}

29 is too many, but it is closer, since 29 is only 4 more than 25.

Try a number between 7 and 12, that is a little closer to 12 than 7.

Try guessing that Betsy kept 10 apples for herself.

\begin{align*}10 + (10 + 5) =& 25 \quad \text{Is this true?}\\ 10+15=&25\\ 25 = & 25 \quad \text{Yes, this is true!}\\\end{align*}

This works!

The answer is Betsy kept 10 apples and gave her neighbor 15 apples.

Examples

Example 1

Kyle caught 15 fish in one day. In his first bucket he put 1 more fish than his second. How many fish were in his first bucket?

First, consider the information given in the problem. There are 15 fish in two buckets, and the second bucket has one less fish than the first bucket.

Next, write an equation to represent the information. If \begin{align*}f\end{align*} represents the number of fish in the first bucket, then \begin{align*}f-1\end{align*} would be the number in the second bucket.

\begin{align*}f+(f-1)=15\end{align*} 

Then, make an educated guess about the number of fish in the first bucket. You know the number of fish in each bucket is nearly the same, so try a number close to half of 15. Try 8:

\begin{align*}8+(8-1)=& 15 \quad \text{Is this true?}\\ 8+7=&15\\ 15 = &15 \quad \text{Yes, this is true.}\\\end{align*}

The answer is 8 fish.

Example 2

How many fish were in his second bucket?

The answer is 7 fish.

Example 3

Kyle's friend Jane went fishing also. If the number of fish they caught were all put together, there would be 33. How many fish did Jane catch?

First, write an expression to show the information given in the problem. Use \begin{align*}j\end{align*} to represent Jane's fish.

\begin{align*}15 + j = 33\end{align*}

Next, make an educated guess about the number of fish Jane caught. Since 15 times 2 would only be 30, Jane must have caught at least a few more fish than Kyle. Try 19:

 \begin{align*}15+19=&33 \quad \text{Is this true?}\\ 34 = &33 \quad \text{No, this is not true. The number is a little too high.}\\\end{align*}

Jane could not have caught 19 fish, since that would have made the total just a bit too high. Try 18:

\begin{align*}15 + 18 = & 33 \quad \text{Is this true?}\\ 33 = & 33 \quad \text{Yes, this is true.}\\\end{align*}

Jane caught 18 fish.

Follow Up

Car Wash

Credit: melodi2
Source: http://www.freeimages.com/photo/car-wash-1433397
License: CC BY-NC 3.0

Remember Jenn and her car wash cash conundrum? She knows that the team earned $2.00 per car, that they earned $144 total, that they washed 5 cars the first hour, 5 the second hour, and 10 the third hour. She wants to know how many cars they washed (on average) per hour for the last 4 hours.

First, write an expression to represent the number of cars the team washed, based on the money they earned. $2 times the number of cars Jenn wrote down plus 4 times the average for each of the four hours she didn't, equals $144.

\begin{align*}$2(5+5+10+4x)=$144\end{align*}

Then, make an educated guess at the number of cars the team washed per hour during the last 4 hours. Since the team washed 10 cars during the last hour Jenn recorded, start by guessing 10:

\begin{align*}$2(5+5+10+4 \times 10)=& $144 \quad \text{Is this true?}\\ $2(5+5+10+40) =& $144\\ $2(60)= & $144\\ $120= &$144 \quad \text{No, this is not true. It is too low}\\\end{align*}

If the team had washed 10 cars per hour for the last 4 hours, they would have only earned $120. Since they earned more than that (although not a lot more), try a slightly larger number, 13 cars per hour.

\begin{align*}$2(5+5+10+4\times13)= & $144 \quad \text{Is this true?}\\ $2(5+5+10+52)=& $144\\ $2(72) = & $144\\ $144= & $144 \quad \text{Yes, this is true.}\\\end{align*}

Jenn's team washed an average of 13 cars per hour for the last four hours.

Video Review

Explore More

Solve each of the following problems using the four part problem solving plan.

A small lion weighs in at 330 pounds. A large lion weighs in at 500 pounds.

1. If there are four large lions in the habitat, how much do the lions weigh in all?

2. If there are five small lions in the habitat, what is the total weight of the small lions?

3. If a lion can sleep 20 hours in one day, how many hours is a lion asleep over a period of five days?

4. If a lion sleeps this much, how many hours is the lion awake over a period of three days?

5. A Burchell’s zebra is smaller than a Grevy’s zebra, the Burchell’s zebra weighs about 550 pounds. What is the difference between the small zebra and the large Grevy's zebra weighing 990 pounds?

6. What is the weight difference between a small Grevy’s zebra and a zebra weighing 880 pounds?

7. What is the weight difference between a small Grevy’s zebra and a large zebra weighing 900 pounds?

8. An adult African male elephant weighs 15,400 pounds. What is the difference between its weight and the weight of a large Grevy’s zebra?

9. What is the difference between the African elephant’s weight and the weight of the small Burchell's zebra?

10. Dana caught twenty-eight fish. She wants to divide the fish into four baskets. If she does this, how many fish will be in each basket? Can she put the same number of fish in each basket?

11. Carl also went fishing. He caught five fish on the first day and four fish on the next day. If he continues this pattern on what day will he not catch any fish?

12. Jessie loves to cook fish after she catches them. She is having ten people over for dinner. If each person eats a half of a fish, how many fish will she need to cook to feed all ten people?

13. Cass takes people out on a fishing boat to go deep sea fishing. With his strategies, people often catch double the amount of fish that they do regularly. If someone normally catches three fish in a day, how many fish will they catch using Cass’ strategy?

14. If you were someone who usually caught thirty-five fish in one week, how many fish would you catch on average per day?

15. If you caught thirty - five fish in one week, how much would that be in one month?

Vocabulary

Difference

Difference

The result of a subtraction operation is called a difference.
Product

Product

The product is the result after two amounts have been multiplied.
Quotient

Quotient

The quotient is the result after two amounts have been divided.
Sum

Sum

The sum is the result after two or more amounts have been added together.
Word Problem

Word Problem

A word problem is a problem that uses verbal language to explain a mathematical situation.

Image Attributions

  1. [1]^ Credit: melodi2; Source: http://www.freeimages.com/photo/car-wash-1433397; License: CC BY-NC 3.0
  2. [2]^ Credit: melodi2; Source: http://www.freeimages.com/photo/car-wash-1433397; License: CC BY-NC 3.0

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