Question

The sum of five consecutive integers is 75.

What is the third number?



13
14
15
16

Answer 1

Let the first number be x, second be x+1 and so on.. Then add and equate to 75..

Answer 2

i would guess and check

Answer 3

\(14+15+16+17+18=80\) nope, too big
lets try again

Answer 4

Guessing -> not good

2 reasons: not really faster at all and develops no critical thinking.

Moral: Try not to guess unless doing it for speed as on a test.

(x - 2) + (x - 1) + x + (x + 1) + (x + 2) = 75

Notice how the number in the left side will cancel giving:

5x = 75

Answer 5

\(13+14+15+16+17=75\) got it on the second try

Answer 6

"critical thinking" ? that means come up with a way to solve, not "find the equation and solve it"

Answer 7

Critical thinking from Michael Carroll, PhD.

"using the equation OR creating the equation"

So, eventually you'll see the -2, -1, 1, and 2 cancel and you'll see

5x = 75

in your mind.

Answer 8

@satellite73 notes on critical thinking. It helps to have a brother specializing in critical thinking AND hours of discussion with him!

Answer 9

ohhhhhhhh

Answer 10

@DD21 Are you saying that you would really rather guess than find the real way to solve? Seriously?

Answer 11

its 15

Answer 12

You can guess, but what if you keep guessing wrongly? It's up to you, but I wouldn't get in that "rut". You might end up guessing too many times. @DD21

Answer 13

So, I advise strongly against guessing and instead I advise using a good methodology.

Answer 14

So, what are your thoughts, @DD21 ?

Answer 15

i need tutoring

Answer 16

You will find good tutoring at Openstudy, but I am asking you what your thoughts are about guessing. or not.

Answer 17

you shouldnt guess..life isnt a guessing game??

Answer 18

Those are good thoughts about life and I appreciate your perspective, but just looking at this problem only for the moment, all I'm trying to point out is there are superior methods to solving this problem than just trying to guess like the other helper (wrongly) suggested. I'm just trying to show you a good technique.

Answer 19

I hope you find all the good tutoring at Openstudy that you can. It was nice working with you.

Answer 20

Thsnks but im going to need more than a single method..i need hands on learning..im not all that good in math..

Answer 21

depends on your thinking
i am rather simpleminded, so i like to think simply
if someone for some reason would ask me "add two consecutive numbers, get 37, what are the numbers?"
rather than writing an equation i would probably think "15 and 16 is too low, lets try higher ones" and arrive at 18+19 in short order

Answer 22

of course for some problems guessing and checking would not work
but positive integers are easy enough to work with

Answer 23

That can work for simple problems, but the real problem is that it develops bad habits. Extremely few people can switch back-and-forth between theory-based techniques and guessing for speed. One has to develop the theory-based techniques first and then one can have the choice on tests.

Answer 24

if some one asked me to think of two numbers whose product was 8 and whose sum was 6, i am pretty sure i would arrive at 2 and 4 before i would write \(x^2+6x+8=0\) and solve for \(x\)

Answer 25

Different problem, different technique.

Answer 26

same idea exactly
sometimes just thinking of the right answer works faster than thinking of some equation to solve to find the answer

Answer 27

sometimes not obviously

Answer 28

Think of this instead and I'll think you'll see it now. Take the problem at hand but take away the choices.

Answer 29

i was not given choices

Answer 30

Now you can't guess as easily without choices. And as you may or may not know, a lot of higher math tests are not multiple-choice.

Answer 31

the very first thing i wrote was \(14+15+16+17+18=80\) no too big

Answer 32

anywho..ppl have diffrent ways of solveing the issue their having

Answer 33

also i would like to point out that even if you feel the need to solve an equation, there are many many choices to do it, so it is not like there is one method
even for this problem i bet you could come up with a few different equations to solve

Answer 34

Other methods are fine too, as long as they are systematic and help the student think, but not guessing. Not at Openstudy. You should really re-think that. It does not help students. Not here.

As I said before, once you know how to do the problem from an abstract basis, you can choose to do it that way or not. If you never develop the ability, you'll never have the choice. And Openstudy students need to know better than guessing. Guessing is good for speed in a timed test, granted. But it doesn't promote learning, which is what we're about here.

Answer 35

i respect your opinion, but i totally disagree

we are thinking human beings, and i believe that if common sense or trial and error can solve a problem easily than that is a valid, convenient, and to my way of thinking vastly superior way of solving a problem

Answer 36

*then

Answer 37

Well, I've said my piece, and there's really nothing more I can add. If you don't see it, then you don't see it. So be it. But as a moderator, I really think you should. Anyway, have a good day. @satellite73