Number Problem 2

[mathdad]

The sum of two numbers is 41. The larger number is 1 less than twice the smaller number. Find the numbers.

Set up:

Let x be the larger number and y the smaller number.

x + y = 41...Equation A
x = 2y - 1...Equation B

Plug B into A and solve for y.

2y - 1 + y = 41

3y - 1 = 41

3y = 41 + 1

3y = 42

y = 42/3

y = 14

Plug y = 14 into either A or B to find x.

I will use A.

x + 14 = 41

x = 41 - 14

x = 27

The numbers are 14 and 27....

 

[Dr.Peterson]

The sum of two numbers is 41. The larger number is 1 less than twice the smaller number. Find the numbers.

The numbers are 14 and 27....

Correct.

And to check, as I like to do with the words of the original problem rather than my equations (in case the latter are wrong):

• The sum of 14 and 27 is 41. 14 + 27 = 41: Yes.
• The larger number, 27, is 1 less than twice the smaller number, 14. 27 = 2(14) - 1: Yes.

In most cases, such a check is enough to be sure you are right. (It isn't, when a problem really has more than one solution, and you only found one.) If you're very uncertain, it can be appropriate to ask someone to check whether your method, rather than just your answer, is appropriate.

 

[mathdad]

Correct.

And to check, as I like to do with the words of the original problem rather than my equations (in case the latter are wrong):

• The sum of 14 and 27 is 41. 14 + 27 = 41: Yes.
• The larger number, 27, is 1 less than twice the smaller number, 14. 27 = 2(14) - 1: Yes.

In most cases, such a check is enough to be sure you are right. (It isn't, when a problem really has more than one solution, and you only found one.) If you're very uncertain, it can be appropriate to ask someone to check whether your method, rather than just your answer, is appropriate.

Dr. Peterson,

Creating an equation(s) has been my struggle in terms of math ever since I could remember. Solving the equation is not an issue. The art of setting up the correct equation is the real magic, you know, what separates math people from the rest of society. Questions will increase in complexity as the weeks go by.

 

[Dr.Peterson]

In fact, my demonstration of the check can be part of the answer to your ultimate question, how to set up the equation.

Often, people have trouble working immediately with variables, but can see how to translate to symbols more easily when they work with specific numbers. So one way to develop this skill is to first practice what the check will look like, using randomly chosen numbers. So here you might pick two numbers, say 10 and 20, and turn each sentence into an equation using those numbers rather than variables, as if you had found 10 and 20 to be your answer:

• The sum of 10 and 20 is 41. 10 + 20 = 41? Not true, but we've seen what the sentence means.
• The larger number, 20, is 1 less than twice the smaller number, 10. 20 = 2(10) - 1? Not true, but ...

Now you can just replace 10 and 20 with x and y in each equation, and you have the equations you'll need to solve:

• x + y = 41
• y = 2x - 1
• (Here I took x as the smaller number and y as the larger number, because I'd written the numbers in that order.)

This doesn't help everyone, of course, but for some it can be a big help, especially when you get to statements that are harder to translate from English to "Algebraish". I wouldn't do this with every problem, but perhaps with the first couple you try of an unfamiliar type, to help get a feel for how they work.

 

[mathdad]

In fact, my demonstration of the check can be part of the answer to your ultimate question, how to set up the equation.

Often, people have trouble working immediately with variables, but can see how to translate to symbols more easily when they work with specific numbers. So one way to develop this skill is to first practice what the check will look like, using randomly chosen numbers. So here you might pick two numbers, say 10 and 20, and turn each sentence into an equation using those numbers rather than variables, as if you had found 10 and 20 to be your answer:

• The sum of 10 and 20 is 41. 10 + 20 = 41? Not true, but we've seen what the sentence means.
• The larger number, 20, is 1 less than twice the smaller number, 10. 20 = 2(10) - 1? Not true, but ...

Now you can just replace 10 and 20 with x and y in each equation, and you have the equations you'll need to solve:

• x + y = 41
• y = 2x - 1
• (Here I took x as the smaller number and y as the larger number, because I'd written the numbers in that order.)

This doesn't help everyone, of course, but for some it can be a big help, especially when you get to statements that are harder to translate from English to "Algebraish". I wouldn't do this with every problem, but perhaps with the first couple you try of an unfamiliar type, to help get a feel for how they work.

What about creating a table leading to the right equation often seen with distance applications? Good idea? Bad idea?

 

[Dr.Peterson]

Tables can be useful, particularly in organizing information so you see what you have to work with. When I teach the technique, though, I still find that students struggle with writing the equation in problems that differ just a little from examples they have seen, because they are stuck in a routine mindset, not thinking deeply about the meaning of the problem. (When I first started teaching the topic from a book that emphasized tables, I found myself trying to follow a routine I could teach to students, depending too much on the table, and making silly mistakes because I was not actually thinking.) So I recommend using the table only as a tool. The steps I suggest for these distance problems are:

1. Understand the problem, e.g. make a diagram of who moves where when.
2. Write a template for the equation, expressing the condition that must be satisfied, e.g. "Amy's distance - Betty's distance = 10 miles".
3. Make the table as a means to finding expressions to use in that equation, e.g. Amy's distance = 35t, Betty's distance = 45(t-2).
4. Fill in the template to make an equation.
5. Solve the equation.
6. Answer the question.

 

[mathdad]

Tables can be useful, particularly in organizing information so you see what you have to work with. When I teach the technique, though, I still find that students struggle with writing the equation in problems that differ just a little from examples they have seen, because they are stuck in a routine mindset, not thinking deeply about the meaning of the problem.

1. Math beyond fifth grade is not for everyone. For example, I do not drive. It is easy for someone with 10 years of driving experience to say: COME ON, DRIVING IS VERY EASY. IF I PASSED THE ROAD TEST, SO CAN YOU.

2. Maybe you should not switch applications a little bit. Why not simply switch the numbers within the word problems?

(When I first started teaching the topic from a book that emphasized tables, I found myself trying to follow a routine I could teach to students, depending too much on the table, and making silly mistakes because I was not actually thinking.)

It is not that students do not want to think their way through the problem. The problem is THEY HAVE NO IDEA HOW TO REASON THEIR WAY TO THE ANSWER. Schools today focus on TEACHING THE TEST. Students are taught how to pass standardized exams not how to learn. I was a per diem teacher in NYC for 8 years. Why per diem? Long story to share another day. Most students enter college not knowing how to reason.

So I recommend using the table only as a tool. The steps I suggest for these distance problems are:

1. Understand the problem, e.g. make a diagram of who moves where when.
2. Write a template for the equation, expressing the condition that must be satisfied, e.g. "Amy's distance - Betty's distance = 10 miles".
3. Make the table as a means to finding expressions to use in that equation, e.g. Amy's distance = 35t, Betty's distance = 45(t-2).
4. Fill in the template to make an equation.
5. Solve the equation.
6. Answer the question.

I totally get what you are saying but honestly, read again what I said above. Dr. Peterson, I consider myself a victim of public school education. Here is the sad reality of my life.

I have three college diplomas in areas other than math. Yet, I lack the confidence a person with THREE degrees should have to pass standardized exams. In 1995, I passed the ASVAB to join the Navy. My test score was 55 percent. Not bad but I took the test after graduating from three different colleges. I also recall studying for 6 months prior to taking the military entrance exam. See my point? In other words, I should have scored a lot higher, at least 80 percent.

 

[JeffM]

Dr. Peterson,

Creating an equation(s) has been my struggle in terms of math ever since I could remember. Solving the equation is not an issue. The art of setting up the correct equation is the real magic, you know, what separates math people from the rest of society. Questions will increase in complexity as the weeks go by.

In fact, solving equations (of the types that have known algorithms for solution) is purely mechanical. The intellectual part of applied math is translating problems into the language of math. All the rest machines can do.

 

[JeffM]

What about creating a table leading to the right equation often seen with distance applications? Good idea? Bad idea?

Tables are useful, but if they are a substitute for understanding and logical thinking, they do nothing to let you apply math to different kinds of problem.

Tables organize information. That can be extremely useful. But if you don't know what information is necessary, how can you put together the proper table?