Three Consecutive Integers

harpazo

harpazo

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The sum of the least and greatest of 3 consecutive integers is 60. What are the values of the 3 integers?

Solution:

In my opinion, this is poorly stated.

Integers x and x + 2 are equated to 60. This leads to 2 consecutive integers. I know that x + 1 is also included but not expressed in the problem.

x + x + 2 = 60

2x + 2 = 60

2x = 60 - 2

2x = 58

x = 58/2

x = 29

Now, x + 2 becomes 31 when x is 29. I must then use x + 1 and this becomes 30. I do not understand why we needed to equate x + x + 2 to 60 before including x + 1 to get the final three integers.

Answer: 29, 30, 31
 
I don't think the problem is poorly stated; but without experience, one might be confused by it (as with any math).

What you are told is, "The sum of the least and greatest of 3 consecutive integers is 60." We can restate this as, "We are given three consecutive integers, and we find that the sum of the first and the third is 60." There are three numbers in the problem; it happens that one of them is not involved in the equation we'll write.

Before writing the equation, you chose to define a variable x as the first of the three numbers; so the three numbers are:

first = x, second = x+1, and third = x+2.​

That led to the equation you solved: "First + third = 60" becomes "x + (x+2) = 60".

You could also solve the problem by defining x as the middle of the three numbers, so that the numbers are x-1, x, x+1. Then the equation becomes (x-1) + (x+1) = 60, which is easily solved: 2x = 60, x = 30. Then the numbers are 29, 30, 31. This choice of variables often helps when there are three numbers, though it is not often taught!

So you did fine on your solution, and interpreted the problem without trouble.

On the other hand, your statement, "Integers x and x + 2 are equated to 60. This leads to 2 consecutive integers," is poorly stated. It is not (each of) the two expressions that is equated to 60, but their sum. And the equation arises from the assumption of three consecutive integers. The numbers x and x+2 are not consecutive integers, if that is what you are saying. Care in language helps in clear thinking, which is necessary in these problems!
 
Dr.Peterson said:
I don't think the problem is poorly stated; but without experience, one might be confused by it (as with any math).

What you are told is, "The sum of the least and greatest of 3 consecutive integers is 60." We can restate this as, "We are given three consecutive integers, and we find that the sum of the first and the third is 60." There are three numbers in the problem; it happens that one of them is not involved in the equation we'll write.

Before writing the equation, you chose to define a variable x as the first of the three numbers; so the three numbers are:

first = x, second = x+1, and third = x+2.​

That led to the equation you solved: "First + third = 60" becomes "x + (x+2) = 60".

You could also solve the problem by defining x as the middle of the three numbers, so that the numbers are x-1, x, x+1. Then the equation becomes (x-1) + (x+1) = 60, which is easily solved: 2x = 60, x = 30. Then the numbers are 29, 30, 31. This choice of variables often helps when there are three numbers, though it is not often taught!

So you did fine on your solution, and interpreted the problem without trouble.

On the other hand, your statement, "Integers x and x + 2 are equated to 60. This leads to 2 consecutive integers," is poorly stated. It is not (each of) the two expressions that is equated to 60, but their sum. And the equation arises from the assumption of three consecutive integers. The numbers x and x+2 are not consecutive integers, if that is what you are saying. Care in language helps in clear thinking, which is necessary in these problems!
Click to expand...

The way you stated the question is easier for me to grasp.
 
I often find restating a problem to be a good first or second step in problem solving. No matter how well it might have been stated, you can bring it a little closer to the equation it will become!
 
Dr.Peterson said:
I often find restating a problem to be a good first or second step in problem solving. No matter how well it might have been stated, you can bring it a little closer to the equation it will become!
Click to expand...

True. My ultimate goal for years has been to "master" word problems at least for grades 6 to 12. Later on, I would like to learn geometry applications, trigonometry application, precalculus applications and finally calculus. SAT MATH, GRE MATH, and GMAT MATH is a totally different ball game.
 
You could have also realized that the average of the 1st and 3rd consecutive number gives the middle number. The sum of the 1st and 3rd numbers is 60, so their average is 30. Hence the middle number is 30. Therefore the three integers are 29, 30 and 31.
 
harpazo said:
The sum of the least and greatest of 3 consecutive integers is 60 …
I do not understand why we needed to equate x + x + 2 to 60 before including x + 1 …
Click to expand...
Hi. It's because we were not given information about x+1. We were given information about x and x+2 (their sum). That information allowed you to write an equation and solve for x=29, the value which you needed in order to evaluate the next two Integers x+1 and x+2.


x and x + 2 are equated to 60. This leads to 2 consecutive integers …
Click to expand...
The expressions x and x+2 do not represent two consecutive Integers. Maybe you were thinking about something else (eg: solving for x leads to values for x+1 and x+2 -- those values are two consecutive Integers).

?
 
Otis said:
Hi. It's because we were not given information about x+1. We were given information about x and x+2 (their sum). That information allowed you to write an equation and solve for x=29, the value which you needed in order to evaluate the next two Integers x+1 and x+2.



The expressions x and x+2 do not represent two consecutive Integers. Maybe you were thinking about something else (eg: solving for x leads to values for x+1 and x+2 -- those values are two consecutive Integers).

?
Click to expand...

Yes, I was thinking that solving for x and substituting into x + 1 and x + 2 leads to consecutive integers.
 
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