problem solving reasonable answers

10 Ways to Check For Reasonableness

The next-to-last step of any problem solving should be checking your answer for reasonableness. (The last step, of course, is stating the answer clearly and completely.) Often when we’re working through a problem, especially a math problem, we can lose sight of the bigger picture of the problem we’re working on. We get down in the details of the math or other problem solving. When we come up out of these weeds and near the end of the problem, we’re so relieved we just want to get it done. At some point, we all will have a result that makes absolutely no sense in context. If we rush through in our relief, we may not notice this glaring problem. Even children know something about the world in which they live and can do this step.

Below are a few ways to approach this process of checking for reasonableness. Think of it as a choose-your-favorite ideas list. As with many of my posts, have fun with these. Use them to build your relationships through laughter and sharing jokes while still learning math. Or if you’re working by yourself, at least use them to give yourself a chuckle!

At the end of the movie Monsters Inc. , the corporation discovers that children’s laughter generates even more power for the monsters’ world than children’s screams. Connection, laughter, and enjoyment — even little tiny moments of them — are similarly more powerful for learning than fear, boredom, and doubt.

This post was originally published on mathteacherbarbie.com. If you are viewing it somewhere else, you are viewing a stolen copy.

1. Start with three estimates: too big, too small, and best guess

This suggestion comes in at the beginning of the problem solving process. Before doing any math but after reading the problem, make three guesses about the answer. Make one guess that’s bigger than you think the answer could possibly be (I recommend making it hilariously big and having a good laugh trying to imagine it). Make one guess that’s smaller than you think the answer could possibly big (I also recommend trying to imagine this one for a chuckle). Then make one final guess that you suspect might be about right, or at least reasonable. At the end, you can compare your solution to these three guesses. Hopefully it’s closer to your middle “reasonable” guess than to the extremes.

2. Read the solution aloud

Put your answer into an answer sentence. Then read the sentence aloud to yourself. Does it sound right? Does it sound reasonable? If someone said that sentence to you, would you look at them funny because the number sounded so weird in that context?

3. Call your grandma (or relative, friend, etc)

They’ll love hearing from you! And you can tell them you need to read them an answer and ask whether it makes sense. Then, just like for number 2 above, read your solution in an answer sentence. If they laugh or act surprised at the number, then that’s good feedback that the answer might not be reasonable.

4. Look for clues in the problem

Is there anything in the problem statement that gives you clues about what types of answers might be reasonable? (If Jim and Jane each have 7 apples, it doesn’t make sense that they would have 2,783 apples together. The “correct” answer is probably a lot fewer digits than that.)

5. Do research

Maybe it’s a context you don’t know much about. Do a tiny bit of research. Whether this be actually performing jumping jacks yourself to find out what’s a reasonable number to do in a minute (if that’s the context of the problem, of course), looking something up on the internet, a simple brainstorm session, or asking someone, there are more ways than ever before of finding out information about what’s reasonable in a given scenario.

6. Take a break and come back later with fresh eyes

Sometimes we need a fresh perspective. That sense of “I’m almost done” relief can be a powerful thing. It can even overwhelm our sense of reality. Doing something else, whether just another problem, having a snack, or sleeping, can refresh our sense of reality and reasonableness. Reread your solution after this break. If it still makes sense, great! If not, you might need to check your work while you still can.

7. Make sure you answer the question asked

Sometimes we get sidetracked or misread the question that was actually asked in the problem. Go back and check that question. Does your answer make sense in response to that request? Do your processes and steps make sense in response to that request? We all do it. We all end up answering different questions other than the ones asked sometimes. It probably tells you something about the context of the problem, even if it doesn’t answer the question directly.

8. Estimate

Use rounding and other estimation strategies to figure out an approximate answer quickly with simpler arithmetic. It’s often easier to keep our perspective when the arithmetic and other steps in the middle are easier to do. Is your final answer close to the estimate? If so, that’s a great sign! (Note: in a quick web search, this comes up as by far the most frequent suggestion for checking reasonableness. Those results lead me to believe this is likely the one your child is being directly taught in class. I wanted to give a few out-of-the-box ideas here as well for those times estimating fails you or is simply not quite enough.)

9. “Plug in” your solution to check for a true statement

Unlike most of the other strategies on this list, this one applies to very specific problems, not all of them. However, also unlike the other suggestions, it can give you a definitive “yes” or “no” answer for whether your solution is correct. If you were asked to “solve” an equation (or inequality) for an unknown number, then you can check by plugging your solution back into the original equation (or inequality) to make sure that it creates a true statement.

10. Try a similar problem

If you can, make up a similar problem — either similar context or different context but similar math structure. Hopefully it’s a problem that you find easier to solve. Solving that problem can give insights about the original problem and what might be right and what might be wrong in the original work and answer.

Whatever strategies you choose, establishing these habits of checking reasonableness serve us all well throughout life, not just math class. And if you’re a regular reader, you know that’s one of the themes I love!

You’ve Got This!

Barbie has taught math, supported students, and volunteered in classrooms for over 20 years. Her daughter is currently learning math in a Common Core state.

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How to Check for Physically Reasonable Answers When Solving Physics Problems

Physics i: 501 practice problems for dummies (+ free online practice).

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Because physics describes reality, your solutions to any physics problems you tackle should be able to describe reality, too. You can avoid many mistakes by checking that your answers have the following properties:

They have the right units. If a problem asks you to find a speed and you get 5 kilograms, you know you made a mistake somewhere. (Note that this check only works if you keep track of your units throughout the whole problem.)

They’re the right size. If you calculate that the mass of a planet is 53 grams, that the speed of a soccer ball is 3 trillion meters per second, or that the temperature of ice is 350 degrees Celsius, start searching for the mistake.

They point in the right direction. When you’re looking for a vector, sometimes you know roughly what direction it should point.

They have the right sign. If you find that the density of a liquid is –1,200 kilograms per cubic meter, you made a sign error along the way.

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Reasonableness in Math

What is reasonableness in math, how do you calculate reasonableness, solved examples, practice problems, frequently asked questions, reasonableness: introduction.

Have you ever come across a situation where you are attempting a math problem and need to check whether you did it right or not? When you find the solution, a good way to check if the problem is solved correctly is to check for reasonableness. 

While solving a problem, you should always ask yourself whether your answer is logical and appropriate or simply whether it makes sense or not.

Let’s see how we can use the concept of reasonableness.

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What does reasonable mean in math? Well, all it means is being moderate or fair while finding a solution and not excessive than the actual number or what is reasonable within the context of the given factors or values. This is the simple meaning of reasonableness.

When solving a math problem, we can check if the answer we have derived is reasonable or not, based on an estimate. A reasonable estimate does not exceed the original numbers in a problem. Let us look at a reasonable estimate example:

Example 1: Suppose you want to divide $\$418$ among 4 people. What would be the reasonable estimate? 

We can round down 418 to the nearest hundreds, 420.

$420 \div 4=105$

Actual answer $= 104.5$

Thus, our solution is reasonable.

Example: Find $65 − 29$.

Actual subtraction:

So, $65 − 29 = 36$

How would you check if it is reasonable or correct?

Add 36 and 29. 

$36 + 29 = 65$

So, the answer is correct!

Reasonableness: Definition

In math, reasonableness can be defined as checking or verifying whether the result of the solution or the calculation of the problem is correct or not. We can do it by either estimating or plugging in your result to check it. 

We use convenient numbers to find an estimate and then compare this estimate to the actual answer to check for reasonableness. Sometimes, this method may not tell you if you have the correct answer, but it will definitely tell you if you are close.

There are various situations where we use reasonableness. We use it to cross-verify our addition, multiplication, division, and even other complex mathematical problems. 

How to Verify Multiplication

Students often make mistakes while carrying out multiplication of two large numbers. This method helps gain confidence and makes sure that the answer is not outrageous.

For example, let’s multiply 51 and 41.

In the first step, round 51 to 50 and 41 to 40. 

Multiply 50 and 40.

$50 \times 40 = 2000$.

The actual product is given by

$50 \times 40 = 2091$.

Now we subtract 2000 from 2091, the answer will be 91.

This shows that the difference between them is very less; thus, our answer is correct. If we had got a more considerable difference, then we would have had to consider it wrong.

How to Verify Division Problems

We can solve and verify the division problems using estimation. Round the divisor and the dividend to the closer and convenient value and check whether the actual answer is reasonable or not.

For example, let’s find the answer to $\frac{2100}{1518}$. 

• Round up the dividend to 2000 and the divisor to 1500. 
• Here you can estimate that the solution to $\frac{2000}{1500}$ will be close to 1 because $\frac{20}{15} \approx 1.3$
• After doing the actual calculation for $\frac{2100}{1518}$, the answer is about 1.35.

Sometimes try to look for patterns that are familiar to you and can help you in solving different problems. 

For example, let’s solve $\frac{63}{7}$. 

• The square of 8 is 64; this will allow you to estimate the approximate value quickly to 8. 
• Here, the actual answer is 9, which is very close to 8.

We can examine the divisor, simply it and check for multiples that are close to the value of the dividend. 

For example, in $\frac{8544}{45}$, there are two options: 

• If you are rounding the dividend to 9000, then your estimate for the quotient will be 200 based on $45 \times 200 = 9000$
• If you are rounding the divisor to 50, then the estimate will be 171 on the basis of $171 \times 50 = 8550$.

 The actual quotient is 189.8. So, both the estimates are valid; we can use either.

• Rounding numbers, making numbers compatible, and properties of operations are some strategies that are used to check the reasonableness of answers.

In this article, we learned about the concept of reasonableness. We came across some conditions like multiplication and division and how we use reasonableness in such cases. We can now look at some examples and solve some practice problems to better understand reasonableness. 

1. Solve $45 \times 5$ using reasonableness.

Solution: 

On calculating $45 \times 5$, we get 225. Now, for the sake of reasonableness, if we divide 225 by 5, 

$\frac{225}{5} = 45$. 

Hence, we verified using reasonableness.

2. Anne bought 3.8 pounds of grains. The grains cost her $\$1.99$ per pound. What was the total cost of Anne’s grains? Also, find a reasonable estimate.

The grains cost $\$1.99$ per pound.

To find the cost of 3.8 pounds, we have to multiply 1.99 and 3.8 as represented below.

If we round the answer 7.562 to the nearest hundredth place, we get 7.56.

Hence, the cost of 3.8 pounds of grains is $\$7.56$.

Now, let us find a reasonable estimate.

Round 1.99 to the nearest whole number, that is, 2.

Round 3.8 to the nearest whole number, that is, 4. 

Multiplying the whole numbers 2 and 4.

That is, $2 \times 4 = 8$

The answer is reasonable, because 8 is close to 7.56.

3. A certain plant grows 3.75 inches per month. If the plant continues to grow at this rate, how much will the plant grow in 6.25 months ?

A certain plant grows in one month  $=  3.75$ inches.

To find the amount of plant grown in 6.25 months, we have to multiply 3.75 and 6.25 as given below. 

Hence, the plant will grow 23.4375 inches in 6.25 months.

Now, let us estimate to check if our answer is reasonable.

Round off 3.75 to the nearest whole number, that is, 4.

Round off 6.25 to the nearest whole number, that is, 6. 

Multiply the whole numbers 4 and 6, that is, $4 \times 6 = 24$

Our answer is reasonable, because 24 is closer to 23.437.

4. Tanya bought 3.4 pounds of yogurt that cost $\$6.95$ per pound. How much did she spend on yogurt?

The cost of one pound of yogurt $=  \$6.95$

To find the cost of 3.4 pounds of yogurt, we have to multiply 3.4 and 6.95.

$3.4 \times 6.95  =  23.63$

Hence, the cost of 3.4 pounds of yogurt is $\$23.63$

Now, let us estimate to check whether our answer is reasonable.

Rounding off 3.4 to the nearest whole number-

Rounding off 6.95 to the nearest whole number-

That is, 7 

Multiplying the whole numbers 3 and 7.

That is, $3 \times 7  =  21$

Our answer is reasonable. Because 21 is much closer to 23.63.

5. Cameron earns $\$9.40$ per hour working at a dairy farm. How much money will he earn for 18.5 hours of work ?

Cameron’s earning in one hour  $=  \$9.40$

To find his earning of 18.5 hours, we have to multiply 18.5 and 9.40

$18.5 \times 9.4  =  173.90$

Hence, his earning for 18.5 hours is $\$173.90$.

Rounding off 18.5 to the nearest whole number, that is, 19.

Rounding off 9.40 to the nearest whole number, that is, 9. 

Multiplying the whole numbers 19 and 9.

That is, $19 \times 9 = 171$

Our answer is reasonable. Because 171 is much closer to 173.90

Attend this quiz & Test your knowledge.

Choose a reasonable estimate for $95 \times 16$.

Jen calculated the sum $x + y$ to be z. in which case will the answer be correct, how to verify $56$ $-$ $14$, rounding 12.99 to the nearest whole number gives us, $24 \div 5$ is approximately equal to.

How do you check reasonableness in math?

You can check for reasonableness either by rounding or by estimation or simply by substituting the values to check if it works.

Where do we use reasonableness ?

We use reasonableness in situations where we need to verify the answers or to check for correctness of solutions.

How to check if the sum of two decimals is reasonable?

Round both the decimals to the nearest whole numbers and compare the actual answer to the estimated sum.

What are outrageous solutions?

The solutions to a math problem which are not reasonable or which are nowhere in the fair range of the actual answer are outrageous solutions.

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9.1 Use a Problem Solving Strategy

Learning objectives.

• Approach word problems with a positive attitude
• Use a problem solving strategy for word problems
• Solve number problems

Be Prepared 9.1

Before you get started, take this readiness quiz.

• Translate “6 “6 less than twice x ” x ” into an algebraic expression. If you missed this problem, review Example 2.25 .
• Solve: 2 3 x = 24 . 2 3 x = 24 . If you missed this problem, review Example 8.16 .
• Solve: 3 x + 8 = 14 . 3 x + 8 = 14 . If you missed this problem, review Example 8.20 .

Approach Word Problems with a Positive Attitude

The world is full of word problems. How much money do I need to fill the car with gas? How much should I tip the server at a restaurant? How many socks should I pack for vacation? How big a turkey do I need to buy for Thanksgiving dinner, and what time do I need to put it in the oven? If my sister and I buy our mother a present, how much will each of us pay?

Now that we can solve equations, we are ready to apply our new skills to word problems. Do you know anyone who has had negative experiences in the past with word problems? Have you ever had thoughts like the student in Figure 9.2 ?

When we feel we have no control, and continue repeating negative thoughts, we set up barriers to success. We need to calm our fears and change our negative feelings.

Start with a fresh slate and begin to think positive thoughts like the student in Figure 9.3 . Read the positive thoughts and say them out loud.

If we take control and believe we can be successful, we will be able to master word problems.

Think of something that you can do now but couldn't do three years ago. Whether it's driving a car, snowboarding, cooking a gourmet meal, or speaking a new language, you have been able to learn and master a new skill. Word problems are no different. Even if you have struggled with word problems in the past, you have acquired many new math skills that will help you succeed now!

Use a Problem-solving Strategy for Word Problems

In earlier chapters, you translated word phrases into algebraic expressions, using some basic mathematical vocabulary and symbols. Since then you've increased your math vocabulary as you learned about more algebraic procedures, and you've had more practice translating from words into algebra.

You have also translated word sentences into algebraic equations and solved some word problems. The word problems applied math to everyday situations. You had to restate the situation in one sentence, assign a variable, and then write an equation to solve. This method works as long as the situation is familiar to you and the math is not too complicated.

Now we'll develop a strategy you can use to solve any word problem. This strategy will help you become successful with word problems. We'll demonstrate the strategy as we solve the following problem.

Example 9.1

Pete bought a shirt on sale for $18 , $18 , which is one-half the original price. What was the original price of the shirt?

Step 1. Read the problem. Make sure you understand all the words and ideas. You may need to read the problem two or more times. If there are words you don't understand, look them up in a dictionary or on the Internet.

• In this problem, do you understand what is being discussed? Do you understand every word?

Step 2. Identify what you are looking for. It's hard to find something if you are not sure what it is! Read the problem again and look for words that tell you what you are looking for!

• In this problem, the words “what was the original price of the shirt” tell you that what you are looking for: the original price of the shirt.

Step 3. Name what you are looking for. Choose a variable to represent that quantity. You can use any letter for the variable, but it may help to choose one that helps you remember what it represents.

• Let p = p = the original price of the shirt

Step 4. Translate into an equation. It may help to first restate the problem in one sentence, with all the important information. Then translate the sentence into an equation.

Step 5. Solve the equation using good algebra techniques. Even if you know the answer right away, using algebra will better prepare you to solve problems that do not have obvious answers.

Step 6. Check the answer in the problem and make sure it makes sense.

• We found that p = 36 , p = 36 , which means the original price was $36 . $36 . Does $36 $36 make sense in the problem? Yes, because 18 18 is one-half of 36 , 36 , and the shirt was on sale at half the original price.
• Step 7. Answer the question with a complete sentence.
• The problem asked “What was the original price of the shirt?” The answer to the question is: “The original price of the shirt was $36 .” $36 .”

If this were a homework exercise, our work might look like this:

Joaquin bought a bookcase on sale for $120 , $120 , which was two-thirds the original price. What was the original price of the bookcase?

Two-fifths of the people in the senior center dining room are men. If there are 16 16 men, what is the total number of people in the dining room?

We list the steps we took to solve the previous example.

Problem-Solving Strategy

• Step 1. Read the word problem. Make sure you understand all the words and ideas. You may need to read the problem two or more times. If there are words you don't understand, look them up in a dictionary or on the internet.
• Step 2. Identify what you are looking for.
• Step 3. Name what you are looking for. Choose a variable to represent that quantity.
• Step 4. Translate into an equation. It may be helpful to first restate the problem in one sentence before translating.
• Step 5. Solve the equation using good algebra techniques.
• Step 6. Check the answer in the problem. Make sure it makes sense.

Let's use this approach with another example.

Example 9.2

Yash brought apples and bananas to a picnic. The number of apples was three more than twice the number of bananas. Yash brought 11 11 apples to the picnic. How many bananas did he bring?

Guillermo bought textbooks and notebooks at the bookstore. The number of textbooks was 3 3 more than the number of notebooks. He bought 5 5 textbooks. How many notebooks did he buy?

Gerry worked Sudoku puzzles and crossword puzzles this week. The number of Sudoku puzzles he completed is seven more than the number of crossword puzzles. He completed 14 14 Sudoku puzzles. How many crossword puzzles did he complete?

In Solve Sales Tax, Commission, and Discount Applications , we learned how to translate and solve basic percent equations and used them to solve sales tax and commission applications. In the next example, we will apply our Problem Solving Strategy to more applications of percent.

Example 9.3

Nga's car insurance premium increased by $60 , $60 , which was 8% 8% of the original cost. What was the original cost of the premium?

Pilar's rent increased by 4% . 4% . The increase was $38 . $38 . What was the original amount of Pilar's rent?

Steve saves 12% 12% of his paycheck each month. If he saved $504 $504 last month, how much was his paycheck?

Solve Number Problems

Now we will translate and solve number problems . In number problems, you are given some clues about one or more numbers, and you use these clues to build an equation. Number problems don't usually arise on an everyday basis, but they provide a good introduction to practicing the Problem Solving Strategy . Remember to look for clue words such as difference , of , and and .

Example 9.4

The difference of a number and six is 13 . 13 . Find the number.

The difference of a number and eight is 17 . 17 . Find the number.

The difference of a number and eleven is −7 . −7 . Find the number.

Example 9.5

The sum of twice a number and seven is 15 . 15 . Find the number.

The sum of four times a number and two is 14 . 14 . Find the number.

Try It 9.10

The sum of three times a number and seven is 25 . 25 . Find the number.

Some number word problems ask you to find two or more numbers. It may be tempting to name them all with different variables, but so far we have only solved equations with one variable. We will define the numbers in terms of the same variable. Be sure to read the problem carefully to discover how all the numbers relate to each other.

Example 9.6

One number is five more than another. The sum of the numbers is twenty-one. Find the numbers.

Try It 9.11

One number is six more than another. The sum of the numbers is twenty-four. Find the numbers.

Try It 9.12

The sum of two numbers is fifty-eight. One number is four more than the other. Find the numbers.

Example 9.7

The sum of two numbers is negative fourteen. One number is four less than the other. Find the numbers.

Try It 9.13

The sum of two numbers is negative twenty-three. One number is 7 7 less than the other. Find the numbers.

Try It 9.14

The sum of two numbers is negative eighteen. One number is 40 40 more than the other. Find the numbers.

Example 9.8

One number is ten more than twice another. Their sum is one. Find the numbers.

Try It 9.15

One number is eight more than twice another. Their sum is negative four. Find the numbers.

Try It 9.16

One number is three more than three times another. Their sum is negative five. Find the numbers.

Consecutive integers are integers that immediately follow each other. Some examples of consecutive integers are:

Notice that each number is one more than the number preceding it. So if we define the first integer as n , n , the next consecutive integer is n + 1 . n + 1 . The one after that is one more than n + 1 , n + 1 , so it is n + 1 + 1 , n + 1 + 1 , or n + 2 . n + 2 .

Example 9.9

The sum of two consecutive integers is 47 . 47 . Find the numbers.

Try It 9.17

The sum of two consecutive integers is 95 . 95 . Find the numbers.

Try It 9.18

The sum of two consecutive integers is −31 . −31 . Find the numbers.

Example 9.10

Find three consecutive integers whose sum is 42 . 42 .

Try It 9.19

Find three consecutive integers whose sum is 96 . 96 .

Try It 9.20

Find three consecutive integers whose sum is −36 . −36 .

Links To Literacy

Section 9.1 exercises, practice makes perfect.

In the following exercises, use the problem-solving strategy for word problems to solve. Answer in complete sentences.

Two-thirds of the children in the fourth-grade class are girls. If there are 20 20 girls, what is the total number of children in the class?

Three-fifths of the members of the school choir are women. If there are 24 24 women, what is the total number of choir members?

Zachary has 25 25 country music CDs, which is one-fifth of his CD collection. How many CDs does Zachary have?

One-fourth of the candies in a bag of are red. If there are 23 23 red candies, how many candies are in the bag?

There are 16 16 girls in a school club. The number of girls is 4 4 more than twice the number of boys. Find the number of boys in the club.

There are 18 18 Cub Scouts in Troop 645 . 645 . The number of scouts is 3 3 more than five times the number of adult leaders. Find the number of adult leaders.

Lee is emptying dishes and glasses from the dishwasher. The number of dishes is 8 8 less than the number of glasses. If there are 9 9 dishes, what is the number of glasses?

The number of puppies in the pet store window is twelve less than the number of dogs in the store. If there are 6 6 puppies in the window, what is the number of dogs in the store?

After 3 3 months on a diet, Lisa had lost 12% 12% of her original weight. She lost 21 21 pounds. What was Lisa's original weight?

Tricia got a 6% 6% raise on her weekly salary. The raise was $30 $30 per week. What was her original weekly salary?

Tim left a $9 $9 tip for a $50 $50 restaurant bill. What percent tip did he leave?

Rashid left a $15 $15 tip for a $75 $75 restaurant bill. What percent tip did he leave?

Yuki bought a dress on sale for $72 . $72 . The sale price was 60% 60% of the original price. What was the original price of the dress?

Kim bought a pair of shoes on sale for $40.50 . $40.50 . The sale price was 45% 45% of the original price. What was the original price of the shoes?

In the following exercises, solve each number word problem.

The sum of a number and eight is 12 . 12 . Find the number.

The sum of a number and nine is 17 . 17 . Find the number.

The difference of a number and twelve is 3 . 3 . Find the number.

The difference of a number and eight is 4 . 4 . Find the number.

The sum of three times a number and eight is 23 . 23 . Find the number.

The sum of twice a number and six is 14 . 14 . Find the number.

The difference of twice a number and seven is 17 . 17 . Find the number.

The difference of four times a number and seven is 21 . 21 . Find the number.

Three times the sum of a number and nine is 12 . 12 . Find the number.

Six times the sum of a number and eight is 30 . 30 . Find the number.

One number is six more than the other. Their sum is forty-two. Find the numbers.

One number is five more than the other. Their sum is thirty-three. Find the numbers.

The sum of two numbers is twenty. One number is four less than the other. Find the numbers.

The sum of two numbers is twenty-seven. One number is seven less than the other. Find the numbers.

A number is one more than twice another number. Their sum is negative five. Find the numbers.

One number is six more than five times another. Their sum is six. Find the numbers.

The sum of two numbers is fourteen. One number is two less than three times the other. Find the numbers.

The sum of two numbers is zero. One number is nine less than twice the other. Find the numbers.

One number is fourteen less than another. If their sum is increased by seven, the result is 85 . 85 . Find the numbers.

One number is eleven less than another. If their sum is increased by eight, the result is 71 . 71 . Find the numbers.

The sum of two consecutive integers is 77 . 77 . Find the integers.

The sum of two consecutive integers is 89 . 89 . Find the integers.

The sum of two consecutive integers is −23 . −23 . Find the integers.

The sum of two consecutive integers is −37 . −37 . Find the integers.

The sum of three consecutive integers is 78 . 78 . Find the integers.

The sum of three consecutive integers is 60 . 60 . Find the integers.

Find three consecutive integers whose sum is −3 . −3 .

Everyday Math

Shopping Patty paid $35 $35 for a purse on sale for $10 $10 off the original price. What was the original price of the purse?

Shopping Travis bought a pair of boots on sale for $25 $25 off the original price. He paid $60 $60 for the boots. What was the original price of the boots?

Shopping Minh spent $6.25 $6.25 on 5 5 sticker books to give his nephews. Find the cost of each sticker book.

Shopping Alicia bought a package of 8 8 peaches for $3.20 . $3.20 . Find the cost of each peach.

Shopping Tom paid $1,166.40 $1,166.40 for a new refrigerator, including $86.40 $86.40 tax. What was the price of the refrigerator before tax?

Shopping Kenji paid $2,279 $2,279 for a new living room set, including $129 $129 tax. What was the price of the living room set before tax?

Writing Exercises

Write a few sentences about your thoughts and opinions of word problems. Are these thoughts positive, negative, or neutral? If they are negative, how might you change your way of thinking in order to do better?

When you start to solve a word problem, how do you decide what to let the variable represent?

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

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• Resource Library
• 7th Grade Mathematics

Education Standards

Maryland college and career ready math standards.

Learning Domain: Expressions and Equations

Standard: Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations as strategies to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.

Standard: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.

Standard: Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, The perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?

Common Core State Standards Math

Cluster: Solve real-life and mathematical problems using numerical and algebraic expressions and equations

Reasonable Estimations & Exact Solutions

Students solve real-world problems by writing and solving equations. Students estimate the solution and determine if the estimate is reasonable before finding the exact solution. They write the solution as a complete sentence.

Students complete a Self Check.

Key Concepts

Students solve real-world problems by first estimating the solution and assessing the reasonableness of the solution. Next, they write an equation to solve the problem and then use the properties of equality to solve the equation. Students write the solution to the problem as a complete sentence.

Goals and Learning Objectives

• Write equations to solve multi-step real-life problems involving rational numbers.
• Solve equations using addition, subtraction, multiplication, and division of rational numbers.
• Use estimations strategies to estimate the solution and determine if the estimate is reasonable.
• Write the solution as a complete sentence.

Lesson Guide

Have students watch the video with a partner. Point out to students that their task is to write an equation to solve the problem posed in the video. Students can watch the video as many times as needed in order to find all information needed and write the equation.

As students are watching and listening to the video, they should take notes about any amounts given and label those amounts. They should make note of what they need to find out and use that information to define the variable in their equation.

Possible equation:

Let x equal the price in dollars of one pound of carrot raisin salad.

( 2 1 2 ⋅ 6.10 ) + ( 1 3 4 ⋅ 5 ) + 1 2 x = 28.25

SWD: Students with disabilities may have difficulty determining the relevant information for this task. Before they watch the video, provide students with a note-taking organizer that includes space for notes on how to write an equation.

Watch the video.

• Write an equation to find the cost per pound for the carrot raisin salad.

VIDEO: Best Deli

Math Mission

Discuss the Math Mission. Students will write equations to represent problem situations, make estimates, and solve the equations.

ELL: After defining the term estimation , explain that in solving math problems, we sometimes look for answers that are precise while other times estimation is sufficient. The context of the problem determines whether the answer should be precise or estimated.

Write equations, make estimates, and solve the equations.

Carrot Raisin Salad

Have students work in pairs. Ask a few questions to make sure that students understand the task.

SWD: Make sure all students understand the first task. Have students restate the task back to you in their own words so you can assess their understanding.

Preparing for Ways of Thinking

As students are working, have them record the ways in which they make their estimates so that they can share these during the Ways of Thinking discussion. Try to identify students who seem to have good estimation skills so that they can provide a good model for the class.

Remind students to always begin by identifying the variable in the equation.

Mathematical Practices

Mathematical Practice 1: Make sense of problems and persevere in solving them.

When students have finished solving the problem, they should go back to the problem and use the context of the problem to check their final answer. If the answer does not check, students need to rethink all of their steps. It may be that the equation they have written does not accurately represent the problem, or it may be they made an error in solving the equation.

Interventions

Student does not know what the variable should represent.

• What does the question ask you to find?

Student does not know where to begin to make an estimate.

• What are you being asked to find?
• Can you round 6.10 to a whole number?
• Think about 2 • 6.
• Think about  1 2 of 6.

Student does not understand how to check their answer.

• Read the problem again.
• Use the answer you got and calculate the total cost using the information in the problem.
• Does the total cost you calculated match the total cost in the problem?

Possible Answers

• Estimate: Total cost is $28.25. The fruit salad costs about $15; the potato salad costs about $8.00, so the carrot raisin salad should cost around $5.
• Equation solution: Let x equal the cost per pound in dollars of the carrot raisin salad.

( 2 1 2 ⋅ 6.10 ) + ( 1 3 4 ⋅ 5 ) + 1 2 x = $ 28.25 ( 15.25 ) + ( 8.75 ) + 1 2 x = $ 28.25 24.00 + 1 2 x = $ 28.25 1 2 x = $ 4.25 x = $ 8.50

The carrot raisin salad costs $8.50 per pound.

Look at the equation you wrote for the carrot raisin salad.

• Estimate the solution and decide whether your estimate makes sense.
• Solve the equation and write the answer as a complete sentence.

Use rounding and mental strategies to help you make an estimate.

The Deli and Other Situations

Have students work in pairs.

• What amounts could you round to help you estimate?
• 0.8 x + (2.5)(6.70) = 22.75
• Estimate: The total cost is about $23. The ham costs about $17, so 0.8 lb of turkey should cost about $6. Turkey should cost about $8 per lb.
• Equation solution: Let x equal the cost per pound of turkey.

0.8 x + ( 2.5 ) ( 6.70 ) = 22.75 0.8 x + 16.75 = 22.75 0.8 x = 6 x = 7.50

The turkey costs $7.50 per pound.

Mrs. Ortiz buys 0.8 lb of turkey and 2.5 lb of ham. The total cost is $22.75. The ham costs $6.70 per lb. What is the price per pound of the turkey?

• Write an equation to represent the problem.

Use the question in the problem to help you decide what quantity the variable in your equation should represent.

Three Salads

• 3(5.95 + 2 x ) + 3(5.95 + x ) = $48.75
• Estimate: The total cost of the salads and extra ingredients is $48.75. The salads alone cost about $36 ($6 × 6). The extra ingredients cost about $13. There were 9 extra ingredients, so each ingredient costs about $1.
• Let x equal the cost of each extra ingredient

3 ( 5.95 + 2 x ) + 3 ( 5.95 + x ) = $ 48.75 17.85 + 6 x + 17.85 + 3 x = $ 48.75 35.70 + 9 x = $ 48.75 9 x = $ 13.05 x = $ 1.45

Each extra ingredient costs $1.45.

Challenge Problem

• 3 x + 5 = 4 x + 1.25
• Estimate: The difference in the tips was about $4. Since one boy babysat for $3 per hour and the other for $4 per hour (a difference of $1 per hour), the difference in the tip should be the same as the number of hours each boy babysat. So, Jack and Marcus each babysat for about 4 hours.
• Let x equal the number of hours each boy babysat.

3 x + 5 = 4 x + 1.25 3 x − 3 x + 5 = 4 x − 3 x + 1.25 5 = x + 1.25 3.75 = x

Each boy babysat 3.75 hours.

A salad costs $5.95, plus an extra charge for each additional ingredient. You order 3 salads that each have 2 additional ingredients, and 3 salads that each have 1 additional ingredient. The total cost is $48.75. What is the cost of 1 additional ingredient?

Jack and Marcus both babysat for the same number of hours on Thursday and they both earned the same amount of money. However, Jack was paid $3 per hour and received a $5 tip, and Marcus was paid $4 per hour and received a $1.25 tip. How many hours did the boys babysit?

You will use the variable more than once in your equation.

Make Connections

Mathematics.

Facilitate the discussion to help students understand the mathematics of the lesson informally. As you discuss each of the problems, ask questions such as the following:

• How did you arrive at your estimate?
• Does your estimate make sense?
• How did you determine what quantity should be represented by the variable?
• What steps did you use to solve the equation?
• What properties of equality did you use in solving the equation?
• Does your solution make sense?
• How does your estimate compare to the solution?
• Is there another way to solve the problem?

ELL: During class discussions, make sure you provide wait time (5–10 seconds) and acknowledge student responses, both verbally and with gestures.

Performance Task

Ways of thinking: make connections.

Take notes about your classmates’ equations, estimates, and solutions.

As your classmates present, ask questions such as:

• How did you come up with your estimate?
• How close is your estimate to the final solution?
• How did you decide which quantity in the problem should be represented by the variable?
• Can you explain the steps you used to solve the equation?
• Does the solution to the equation make sense in terms of the problem situation?
• Why do you sometimes need to use both the addition property of equality and the multiplication property of equality to solve equations?

Write and Solve Equations

Have each student write a summary of the math in this lesson; then write a class summary. When done, if you think the summary is helpful, share it with the class.

A Possible Summary

In this lesson, we not only solved problems using equations, we first estimated the answers. To do this, we sometimes rounded decimal or fraction numbers, we sometimes had to rewrite numbers in a different form, and we used mental math.

We had several ways to check our answers. We could first compare our answer to our estimate, we could check that the solution to the equation made the equation true, and we could go back to the word problem and see if our answer matches the situation.

To solve equations like 4 x = 6, you need to use the multiplication property of equality; to solve equations like x + 6 = 10, you need to use the addition property of equality; but to solve equations like 3 x + 6 = 12, you need to use both properties.

Formative Assessment

Summary of the math: write and solve equations.

Write a summary about writing and solving equations.

Check your summary:

• Do you describe how to make an estimate?
• Do you discuss ways to check answers to problems?
• Do you explain how to determine which quantity in a problem should be represented by the variable?

What Is the Number?

This task allows you to assess students’ work and determine what difficulties they are having. The results of the Self Check will help you determine which students should work on the Gallery and which students would benefit from review before the assessment. Have students work on the Self Check individually.

Have students submit their work to you. Make notes on what their work reveals about their current levels of understanding and their different problem-solving approaches.

Do not score students’ work. Share with each student the most appropriate Interventions to guide their thought process. Also note students with a particular issue so that you can work with them in the Putting It Together lesson that follows.

Student applies operations in the wrong order—for example, chooses 4 x + 7 = 80 as an appropriate equation.

• In this expression, what is the first thing that happens to the number I am thinking of? Then what happens?
• What does x represent? What are you adding 7 to?

Student does not recognize all relevant expressions, for example, the student chooses 4( x + 7) = 80 as the only appropriate equation.

• How else could you write the expression 4( x + 7)?

Student calculates an incorrect value for x .

• If you substitute your value of x into the equation, do you get a true equation?
• How will you check whether your value for x is correct?
• 4( x + 7) = 80 and 4 x + 28 = 80 both represent the problem.
• The value of x represents the original number (in the statement, "I am thinking of a number...")

4 ( x + 7 ) = 80 4 x + 28 = 80 4 x + 28 − 28 = 80 − 28 4 x = 52 4 x 4 = 52 4 x = 13

Complete this Self Check by yourself.

I am thinking of a number. When I add 7 and then multiply by 4, the result is 80. What is my number?

• Which of the following equations represent this problem? Select all that apply, and justify your choices.

x + 28 = 80

4( x + 7) = 80

4 x + 7 = 80

4 x + 28 = 80

• For each equation that you identified, find the value of x and explain what it represents.

Three Consecutive Numbers

Student assumes that the three numbers are equal. For example, the student selects Total = x + 2 x + 3 x as an appropriate equation.

• What does consecutive mean?
• What does x represent?
• Can you try some numbers to check that this works?

Student does not multiply all terms in the parentheses. For example, the student selects Total = x + (2 x + 1) = (3 x + 2) as an appropriate equation.

• How do you write “one more than x ” using algebra? Now read the question again: What happens next? What happens if you add two of these numbers together?

Student does not correctly interpret the solution of the equation to solve the problem.

• You have found that x = 27. Read the question again. What are the three consecutive numbers?
• Both x + 2 x + 2 + 3 x + 6 and x + 2( x + 1) + 3( x + 2) correctly represent the situation.
• The value of x represents the first number, so the three numbers are 27, 28, and 29.

x + 2 ( x + 1 ) + 3 ( x + 2 ) = 170 x + 2 x + 2 + 3 x + 6 = 170 6 x + 8 − 8 = 170 − 8 6 x = 162 6 x 6 = 162 6 x = 27

The numbers 5, 6, and 7 are examples of consecutive numbers —that is, each number follows the previous one.

Suppose three consecutive numbers are used in the following way to get a total. The first number plus two times the second number plus three times the third number equals the total.

• Which of the following expressions represent this situation? Select all that apply, and justify your choices.

Total = x + 2 x + 3 x

Total = x + 2 x + 2 + 3 x + 6

Total = x + 2( x + 1) + 3( x + 2)

Total = x + (2 x + 1) + (3 x + 2)

• The total is 170. What are the three consecutive numbers? Explain your answer.

Reflect On Your Work

Have each student do a quick reflection before the end of the class. Review the reflections to determine where students might need some additional support in writing equations to solve problems.

Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.

Something I still don’t understand about writing equations to represent situations is …

The Art of Effective Problem Solving: A Step-by-Step Guide

Author: Daniel Croft

Daniel Croft is an experienced continuous improvement manager with a Lean Six Sigma Black Belt and a Bachelor's degree in Business Management. With more than ten years of experience applying his skills across various industries, Daniel specializes in optimizing processes and improving efficiency. His approach combines practical experience with a deep understanding of business fundamentals to drive meaningful change.

Whether we realise it or not, problem solving skills are an important part of our daily lives. From resolving a minor annoyance at home to tackling complex business challenges at work, our ability to solve problems has a significant impact on our success and happiness. However, not everyone is naturally gifted at problem-solving, and even those who are can always improve their skills. In this blog post, we will go over the art of effective problem-solving step by step.

You will learn how to define a problem, gather information, assess alternatives, and implement a solution, all while honing your critical thinking and creative problem-solving skills. Whether you’re a seasoned problem solver or just getting started, this guide will arm you with the knowledge and tools you need to face any challenge with confidence. So let’s get started!

Problem Solving Methodologies

Individuals and organisations can use a variety of problem-solving methodologies to address complex challenges. 8D and A3 problem solving techniques are two popular methodologies in the Lean Six Sigma framework.

Methodology of 8D (Eight Discipline) Problem Solving:

The 8D problem solving methodology is a systematic, team-based approach to problem solving. It is a method that guides a team through eight distinct steps to solve a problem in a systematic and comprehensive manner.

The 8D process consists of the following steps:

• Form a team: Assemble a group of people who have the necessary expertise to work on the problem.
• Define the issue: Clearly identify and define the problem, including the root cause and the customer impact.
• Create a temporary containment plan: Put in place a plan to lessen the impact of the problem until a permanent solution can be found.
• Identify the root cause: To identify the underlying causes of the problem, use root cause analysis techniques such as Fishbone diagrams and Pareto charts.
• Create and test long-term corrective actions: Create and test a long-term solution to eliminate the root cause of the problem.
• Implement and validate the permanent solution: Implement and validate the permanent solution’s effectiveness.
• Prevent recurrence: Put in place measures to keep the problem from recurring.
• Recognize and reward the team: Recognize and reward the team for its efforts.

Download the 8D Problem Solving Template

A3 Problem Solving Method:

The A3 problem solving technique is a visual, team-based problem-solving approach that is frequently used in Lean Six Sigma projects. The A3 report is a one-page document that clearly and concisely outlines the problem, root cause analysis, and proposed solution.

The A3 problem-solving procedure consists of the following steps:

• Determine the issue: Define the issue clearly, including its impact on the customer.
• Perform root cause analysis: Identify the underlying causes of the problem using root cause analysis techniques.
• Create and implement a solution: Create and implement a solution that addresses the problem’s root cause.
• Monitor and improve the solution: Keep an eye on the solution’s effectiveness and make any necessary changes.

Subsequently, in the Lean Six Sigma framework, the 8D and A3 problem solving methodologies are two popular approaches to problem solving. Both methodologies provide a structured, team-based problem-solving approach that guides individuals through a comprehensive and systematic process of identifying, analysing, and resolving problems in an effective and efficient manner.

Step 1 – Define the Problem

The definition of the problem is the first step in effective problem solving. This may appear to be a simple task, but it is actually quite difficult. This is because problems are frequently complex and multi-layered, making it easy to confuse symptoms with the underlying cause. To avoid this pitfall, it is critical to thoroughly understand the problem.

To begin, ask yourself some clarifying questions:

• What exactly is the issue?
• What are the problem’s symptoms or consequences?
• Who or what is impacted by the issue?
• When and where does the issue arise?

Answering these questions will assist you in determining the scope of the problem. However, simply describing the problem is not always sufficient; you must also identify the root cause. The root cause is the underlying cause of the problem and is usually the key to resolving it permanently.

Try asking “why” questions to find the root cause:

• What causes the problem?
• Why does it continue?
• Why does it have the effects that it does?

By repeatedly asking “ why ,” you’ll eventually get to the bottom of the problem. This is an important step in the problem-solving process because it ensures that you’re dealing with the root cause rather than just the symptoms.

Once you have a firm grasp on the issue, it is time to divide it into smaller, more manageable chunks. This makes tackling the problem easier and reduces the risk of becoming overwhelmed. For example, if you’re attempting to solve a complex business problem, you might divide it into smaller components like market research, product development, and sales strategies.

To summarise step 1, defining the problem is an important first step in effective problem-solving. You will be able to identify the root cause and break it down into manageable parts if you take the time to thoroughly understand the problem. This will prepare you for the next step in the problem-solving process, which is gathering information and brainstorming ideas.

Step 2 – Gather Information and Brainstorm Ideas

Gathering information and brainstorming ideas is the next step in effective problem solving. This entails researching the problem and relevant information, collaborating with others, and coming up with a variety of potential solutions. This increases your chances of finding the best solution to the problem.

Begin by researching the problem and relevant information. This could include reading articles, conducting surveys, or consulting with experts. The goal is to collect as much information as possible in order to better understand the problem and possible solutions.

Next, work with others to gather a variety of perspectives. Brainstorming with others can be an excellent way to come up with new and creative ideas. Encourage everyone to share their thoughts and ideas when working in a group, and make an effort to actively listen to what others have to say. Be open to new and unconventional ideas and resist the urge to dismiss them too quickly.

Finally, use brainstorming to generate a wide range of potential solutions. This is the place where you can let your imagination run wild. At this stage, don’t worry about the feasibility or practicality of the solutions; instead, focus on generating as many ideas as possible. Write down everything that comes to mind, no matter how ridiculous or unusual it may appear. This can be done individually or in groups.

Once you’ve compiled a list of potential solutions, it’s time to assess them and select the best one. This is the next step in the problem-solving process, which we’ll go over in greater detail in the following section.

Step 3 – Evaluate Options and Choose the Best Solution

Once you’ve compiled a list of potential solutions, it’s time to assess them and select the best one. This is the third step in effective problem solving, and it entails weighing the advantages and disadvantages of each solution, considering their feasibility and practicability, and selecting the solution that is most likely to solve the problem effectively.

To begin, weigh the advantages and disadvantages of each solution. This will assist you in determining the potential outcomes of each solution and deciding which is the best option. For example, a quick and easy solution may not be the most effective in the long run, whereas a more complex and time-consuming solution may be more effective in solving the problem in the long run.

Consider each solution’s feasibility and practicability. Consider the following:

• Can the solution be implemented within the available resources, time, and budget?
• What are the possible barriers to implementing the solution?
• Is the solution feasible in today’s political, economic, and social environment?

You’ll be able to tell which solutions are likely to succeed and which aren’t by assessing their feasibility and practicability.

Finally, choose the solution that is most likely to effectively solve the problem. This solution should be based on the criteria you’ve established, such as the advantages and disadvantages of each solution, their feasibility and practicability, and your overall goals.

It is critical to remember that there is no one-size-fits-all solution to problems. What is effective for one person or situation may not be effective for another. This is why it is critical to consider a wide range of solutions and evaluate each one based on its ability to effectively solve the problem.

Step 4 – Implement and Monitor the Solution

When you’ve decided on the best solution, it’s time to put it into action. The fourth and final step in effective problem solving is to put the solution into action, monitor its progress, and make any necessary adjustments.

To begin, implement the solution. This may entail delegating tasks, developing a strategy, and allocating resources. Ascertain that everyone involved understands their role and responsibilities in the solution’s implementation.

Next, keep an eye on the solution’s progress. This may entail scheduling regular check-ins, tracking metrics, and soliciting feedback from others. You will be able to identify any potential roadblocks and make any necessary adjustments in a timely manner if you monitor the progress of the solution.

Finally, make any necessary modifications to the solution. This could entail changing the solution, altering the plan of action, or delegating different tasks. Be willing to make changes if they will improve the solution or help it solve the problem more effectively.

It’s important to remember that problem solving is an iterative process, and there may be times when you need to start from scratch. This is especially true if the initial solution does not effectively solve the problem. In these situations, it’s critical to be adaptable and flexible and to keep trying new solutions until you find the one that works best.

To summarise, effective problem solving is a critical skill that can assist individuals and organisations in overcoming challenges and achieving their objectives. Effective problem solving consists of four key steps: defining the problem, generating potential solutions, evaluating alternatives and selecting the best solution, and implementing the solution.

You can increase your chances of success in problem solving by following these steps and considering factors such as the pros and cons of each solution, their feasibility and practicability, and making any necessary adjustments. Furthermore, keep in mind that problem solving is an iterative process, and there may be times when you need to go back to the beginning and restart. Maintain your adaptability and try new solutions until you find the one that works best for you.

• Novick, L.R. and Bassok, M., 2005.  Problem Solving . Cambridge University Press.

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Daniel Croft

Hi im Daniel continuous improvement manager with a Black Belt in Lean Six Sigma and over 10 years of real-world experience across a range sectors, I have a passion for optimizing processes and creating a culture of efficiency. I wanted to create Learn Lean Siigma to be a platform dedicated to Lean Six Sigma and process improvement insights and provide all the guides, tools, techniques and templates I looked for in one place as someone new to the world of Lean Six Sigma and Continuous improvement.

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Determining Reasonableness of Solutions (System of Equations)

Let's get started.

Let's investigate how a system of equations applies to real world situations, and what the solution represents.

TEKS Standards and Student Expectations

A(5)  Linear functions, equations, and inequalities. The student applies the mathematical process standards to solve, with and without technology, linear equations and evaluate the reasonableness of their solutions. The student is expected to:

A(5)(C) solve systems of two linear equations with two variables for mathematical and real-world problems

Resource Objective(s)

Students will use a variety of methods to find the solution to a system of equations. They will determine the reasonableness of a solution in a real world problem.

Essential Questions

What types of real world situations can be represented by a system of equations?

How can substitution be used to solve a system of equations?

How can a graph be used to determine the solution to a system of equations?

How can you determine if a solution is reasonable?

• Elimination
• Intersecting Lines
• Linear Function
• Rate of Change
• Reasonableness
• System of Linear Equations

Systems of Equations and Solutions

A system of linear equations is two or more linear equations that use two or more variables.

A solution to a system of linear equations is a pair of numbers for x and y that make both equations true.

To find a solution to a system of linear equations, one of three methods can be used: substitution, elimination, or graphing. 

To determine if a solution is reasonable, verify whether the pair of values satisfies the conditions given in the verbal situation.

Iris enjoys making friendship bracelets. She is able to braid a purple bracelet at a rate of 5 cm per day, and a green bracelet at a rate of 4 cm per day. The purple bracelet is already 5 cm long, and the green bracelet is already 11 cm long. Iris estimates that the two bracelets would be the same length after four days. Is her estimate correct?

Follow the steps to determine if the estimate is reasonable.

To determine the number of days needed for the bracelets to reach the same length you can solve by substitution.

Solve by substitution: 5 x   +   5   =   4 x   +   11                     - 4 x                     - 4 x   x   +   5 =   11                           - 5       - 5                           x   =   6

From this we that the bracelets would be the same length after 6 days.

Purple bracelet:  y = 5(6) + 5 = 35 cm. Green bracelet:  y  = 4(6) + 11 = 35 cm. The bracelets would each be 35 cm after 6 days.

Read each of the following questions, and determine if the proposed solution is reasonable.

Systems of Linear Equations: Real World Applications

A system of equations can also be solved by elimination. In this method, you add or subtract the equations in order to eliminate one of the variables, and solve for the other variable. 

A local department store sells a bag of 30 mini candy bars and 20 snack-size candy bars for $7. They will also sell a bag of 10 mini candy bars and 50 snack-size candy bars for $11. Marla estimated that in both bags, the mini candy bars cost $0.10 each, and the snack-size candy bars cost $0.20 each. Was Marla's estimate reasonable?

Follow the steps below to solve the system of equations by elimination, and determine if the estimate is reasonable. 

Systems of equations can also be represented in a graph. In this case, the intersection shows the solution.

The following graph shows the cost of two different cell phone plans from two different companies. 

Linear Equations and Linear Systems in the Real World

A set of tables can also be used to show a system of equations. In this case, it is helpful to extend the table to find missing pieces of data.

The tables below show the amount of money that three friends have saved over a period of 9 months.

Break Even Points

A system of equations can also be used to find break even points. In these problems there is specific amount of money that has been invested in a product, and a certain amount that the product is being sold for in order to try and make a profit. When finding the break even point you are looking for the point where the money invested and the profit are equivalent. 

Joe's Cookie Stand pays $480 for rent. They incur an expense of $0.08 per cookie. They sell the cookies for $0.32 each. How many cookies do they have to sell to break even?

Read each of the following questions to determine if the proposed solution is reasonable.

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Problem-Solving Basics for One-Dimensional Kinematics

Learning objectives.

By the end of this section, you will be able to:

• Apply problem-solving steps and strategies to solve problems of one-dimensional kinematics.
• Apply strategies to determine whether or not the result of a problem is reasonable, and if not, determine the cause.

Figure 1. Problem-solving skills are essential to your success in Physics. (credit: scui3asteveo, Flickr)

Problem-solving skills are obviously essential to success in a quantitative course in physics. More importantly, the ability to apply broad physical principles, usually represented by equations, to specific situations is a very powerful form of knowledge. It is much more powerful than memorizing a list of facts. Analytical skills and problem-solving abilities can be applied to new situations, whereas a list of facts cannot be made long enough to contain every possible circumstance. Such analytical skills are useful both for solving problems in this text and for applying physics in everyday and professional life.

Problem-Solving Steps

While there is no simple step-by-step method that works for every problem, the following general procedures facilitate problem solving and make it more meaningful. A certain amount of creativity and insight is required as well.

Examine the situation to determine which physical principles are involved . It often helps to draw a simple sketch at the outset. You will also need to decide which direction is positive and note that on your sketch. Once you have identified the physical principles, it is much easier to find and apply the equations representing those principles. Although finding the correct equation is essential, keep in mind that equations represent physical principles, laws of nature, and relationships among physical quantities. Without a conceptual understanding of a problem, a numerical solution is meaningless.

Make a list of what is given or can be inferred from the problem as stated (identify the knowns) . Many problems are stated very succinctly and require some inspection to determine what is known. A sketch can also be very useful at this point. Formally identifying the knowns is of particular importance in applying physics to real-world situations. Remember, “stopped” means velocity is zero, and we often can take initial time and position as zero.

Identify exactly what needs to be determined in the problem (identify the unknowns) . In complex problems, especially, it is not always obvious what needs to be found or in what sequence. Making a list can help.

Find an equation or set of equations that can help you solve the problem . Your list of knowns and unknowns can help here. It is easiest if you can find equations that contain only one unknown—that is, all of the other variables are known, so you can easily solve for the unknown. If the equation contains more than one unknown, then an additional equation is needed to solve the problem. In some problems, several unknowns must be determined to get at the one needed most. In such problems it is especially important to keep physical principles in mind to avoid going astray in a sea of equations. You may have to use two (or more) different equations to get the final answer.

Substitute the knowns along with their units into the appropriate equation, and obtain numerical solutions complete with units . This step produces the numerical answer; it also provides a check on units that can help you find errors. If the units of the answer are incorrect, then an error has been made. However, be warned that correct units do not guarantee that the numerical part of the answer is also correct.

Check the answer to see if it is reasonable: Does it make sense? This final step is extremely important—the goal of physics is to accurately describe nature. To see if the answer is reasonable, check both its magnitude and its sign, in addition to its units. Your judgment will improve as you solve more and more physics problems, and it will become possible for you to make finer and finer judgments regarding whether nature is adequately described by the answer to a problem. This step brings the problem back to its conceptual meaning. If you can judge whether the answer is reasonable, you have a deeper understanding of physics than just being able to mechanically solve a problem.

When solving problems, we often perform these steps in different order, and we also tend to do several steps simultaneously. There is no rigid procedure that will work every time. Creativity and insight grow with experience, and the basics of problem solving become almost automatic. One way to get practice is to work out the text’s examples for yourself as you read. Another is to work as many end-of-section problems as possible, starting with the easiest to build confidence and progressing to the more difficult. Once you become involved in physics, you will see it all around you, and you can begin to apply it to situations you encounter outside the classroom, just as is done in many of the applications in this text.

Unreasonable Results

Physics must describe nature accurately. Some problems have results that are unreasonable because one premise is unreasonable or because certain premises are inconsistent with one another. The physical principle applied correctly then produces an unreasonable result. For example, if a person starting a foot race accelerates at 0.40 m/s 2 for 100 s, his final speed will be 40 m/s (about 150 km/h)—clearly unreasonable because the time of 100 s is an unreasonable premise. The physics is correct in a sense, but there is more to describing nature than just manipulating equations correctly. Checking the result of a problem to see if it is reasonable does more than help uncover errors in problem solving—it also builds intuition in judging whether nature is being accurately described.

Use the following strategies to determine whether an answer is reasonable and, if it is not, to determine what is the cause.

Solve the problem using strategies as outlined and in the format followed in the worked examples in the text . In the example given in the preceding paragraph, you would identify the givens as the acceleration and time and use the equation below to find the unknown final velocity. That is,

Check to see if the answer is reasonable . Is it too large or too small, or does it have the wrong sign, improper units, …? In this case, you may need to convert meters per second into a more familiar unit, such as miles per hour.

This velocity is about four times greater than a person can run—so it is too large.

If the answer is unreasonable, look for what specifically could cause the identified difficulty . In the example of the runner, there are only two assumptions that are suspect. The acceleration could be too great or the time too long. First look at the acceleration and think about what the number means. If someone accelerates at 0.40 m/s 2 , their velocity is increasing by 0.4 m/s each second. Does this seem reasonable? If so, the time must be too long. It is not possible for someone to accelerate at a constant rate of 0.40 m/s 2 for 100 s (almost two minutes).

Section Summary

The six basic problem solving steps for physics are:

• Step 1 . Examine the situation to determine which physical principles are involved.
• Step 2 . Make a list of what is given or can be inferred from the problem as stated (identify the knowns).
• Step 3 . Identify exactly what needs to be determined in the problem (identify the unknowns).
• Step 4 . Find an equation or set of equations that can help you solve the problem.
• Step 5 . Substitute the knowns along with their units into the appropriate equation, and obtain numerical solutions complete with units.
• Step 6 . Check the answer to see if it is reasonable: Does it make sense?

Conceptual Questions

1. What information do you need in order to choose which equation or equations to use to solve a problem? Explain. 2. What is the last thing you should do when solving a problem? Explain.

• College Physics. Authored by : OpenStax College. Located at : http://cnx.org/contents/[email protected]:aNsXe6tc@2/Problem-Solving-Basics-for-One . License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected].

5.7. Problem Solving: Reasonable Answers

Nguyen Nu Khanh Ngoc

Problem Solving, reasonable answers

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Chapter 8, Lesson 4: Problem-Solving Investigation: Reasonable Answers

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Problem Solving: Reasonable Answers

In this reasonable answers learning exercise, students use their logical thinking skills to answer 3 math questions that involve problem solving.

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McGraw Hill My Math Grade 5 Chapter 9 Lesson 8 Answer Key Problem-Solving Investigation: Determine Reasonable Answers

All the solutions provided in McGraw Hill Math Grade 5 Answer Key PDF Chapter 9 Lesson 8 Problem-Solving Investigation: Determine Reasonable Answers  will give you a clear idea of the concepts.

McGraw-Hill My Math Grade 5 Answer Key Chapter 9 Lesson 8 Problem-Solving Investigation: Determine Reasonable Answers

Apply the Strategy

Determine a reasonable answer to solve each problem.

Question 2. Mathematical PRACTICE 6 Explain to a Friend A grocer sells 12 pounds of apples. Of those, 5\(\frac{3}{4}\) pounds are green and 3\(\frac{1}{4}\) pounds are golden. The rest are red. Which is a more reasonable estimate for how many pounds of red apples the grocer sold: 3 pounds or 5 pounds? Explain. Answer: The above-given: The number of a grocer sells = 12 The number of pounds is green apples = 5 3/4 The number of pounds is golden apples = 3 1/4 The number of pounds is red apples = x Plan: Use estimation to find a reasonable answer. Solve: we can write 5 3/4 as 6.  { 3/4 is pretty close to 1. Here we have 5 3/4 = 5 + 1 = 6} We can write 3 1/4 as 3.  { 1/4 is pretty close to 0. Here we have 3 1/4 = 3 + 0 = 3} Combining both the grocer sold 6 + 3 = 9 That means he sold 6 pounds of green apples and 3 pounds of gold apples. The remaining = total apples – (number of pounds of green apples and gold apples) The remaining = 12 – 9 The remaining = 3 Therefore, the rest of the red apples are 3 pounds Question 3. A puzzle book costs $4.25. A novel costs $9.70 more than a puzzle book. Which is the most reasonable estimate for the total cost of both items: $14, $16, or $18? Answer: The above-given: The cost of the puzzle book = $4.25 The cost of a novel = $9.70 + novel book Plan: Use estimation to find a reasonable answer. Solve: we can write $4.25 as 4 We can write $9.70 as 10 Here, the novel book is $10 more than the puzzle book. so, the cost of the novel book is 10 + 4 = $14 Finally, the total cost of both books = 14 + 4 = 18. Therefore, the answer is $18. Question 4. Thirty students from the Netherlands set up a record of 1,500,000 dominoes. Of these, 1,138,101 were toppled by one push. Which is a more reasonable estimate for how many dominoes remained standing after that push: 350,000 or 400,000? Answer: The above-given: The total number of students from the Netherlands is 30 The number of dominoes set up a record = 1,500,000 The number of dominoes was toppled = 1,138,101 Plane: Estimation Solve: 1138101 can be written as 1,150,000 If we subtract these two values: 1500000 – 1150000 = 350000 The remaining dominoes are 350000 standing after that push.

Review the strategies

Use any strategy to solve each problem.

• Determie reasonable answers.
• Look for a pattern.
• Solve a simpler problem.
• Find an estimate or exact answer.

Question 7. A high jumper starts the bar at 48 inches and raises the bar \(\frac{1}{2}\) inch after each jump. How high will the bar be after the seventh jump? Answer: The above-given: The number of inches a high jumper starts the bar = 48 The number of inches raises after each jump = 1/2 The number of inches raises after the seventh jump = x Solve: we can write up to seven jumps: 48, 48.5, 49, 49.5, 50, 50.5, 51, 51.5 The seventh jump is 51.5 Therefore, the value of the x is 51.5

Question 8. Ms. Kennedy wants to buy a new interactive whiteboard for her classroom that costs $989. So far, Ms. Kennedy has collected $485 in donations, and $106 in fundraising. About how much more money does she need to buy the interactive whiteboard? Answer: The above-given: The cost of the whiteboard = $989 The amount she collected in donations = $485 The fund she collected = $106 The amount she needed to buy the interactive whiteboard = x Solve: add: 485 + 106 = 591 The total amount she has = $591 The remaining amount = cost of the whiteboard – the amount she has The remaining amount = $989 – $591 The remaining amount = $398. Therefore, she needs $398 more to buy an interactive whiteboard.

McGraw Hill My Math Grade 5 Chapter 9 Lesson 8 My Homework Answer Key

Problem Solving

Mathematical PRACTICE 3 Check for Reasonableness Determine a reasonable answer to solve each problem.

Question 1. Alyssa needs 7\(\frac{5}{8}\) inches of ribbon for one project and 4\(\frac{7}{8}\) inches of ribbon for another project. If she has 11 inches of ribbon, will she have enough to complete both projects? Explain. Answer: The above-given: The number of inches of ribbon Alyssa needs for one project = 7 5/8 The number of inches of ribbon Alyssa needs for another project = 4 7/8 We need to find out if she has enough to complete both projects = x Plan: Estimate Solve: We can write 7 5/8 as 8 We can write 4 7/8 as 5 The above-given 11 inches. Now add: 8 + 5 = 13 For both projects, she needs 13 inches of ribbon but she only has 11 inches. Therefore, she does not have enough ribbon with her.

Question 2. Josiah has a piece of wood that measures 10\(\frac{1}{8}\)– feet. He wants to make 5 shelves. If each shelf is 1\(\frac{3}{4}\) feet long, does he have enough to make 5 shelves? Explain. Answer: The above-given: The number of feet Josiah has a piece of wood = 10 1/8 The number of shelves he wants to make is 5 The number of feet long a feet shelf = 1 3/4 Plan: Estimate Solve: we can write 10 1/8 as 10 we can write 1 3/4 as 1. 10/5 = 2 Yes, (2 > 1) he has enough wood to make 5 shelves.

Question 3. After school, Philipe spent 1\(\frac{3}{4}\)– hours at baseball practice, 2\(\frac{1}{4}\) hours on homework, and \(\frac{1}{4}\) hour getting ready for bed. Which is the most reasonable estimate for how long he spent on his activities: 3 hours, 4 hours, or 5 hours? Explain. Answer: The above-given: The number of hours Philipe spent at basketball practice = 1 3/4 The number of hours he spent on homework = 2 1/4 The number of hours he spent on getting ready for bed = 1/4 The total hours he spent on his activities = x Plan: Estimate Solve: we can write 1 3/4 as 2 We can write 2 1/4 as 2 we can write 1/4 as 0. x = 2 + 2 + 0 x = 4 Therefore, he spent 4 hours on his activities.

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