Understanding results of long division

[Probability]

I am just refreshing my learning of estimating and have a question or two

I have an example;

(53.4 x 70.9) / (22.2)

So on paper I write;

53 x 71 = 3763

Now I divide by 22

using long division

22 out of 3763

I say 22 out of 37 = 1

Then I say 1 x 22 = 22 and

37 - 22 = 15

then I bring down 6

22 from 156 = 7

7 x 22 = 154

156 - 154 = remainder 2

So I have an answer of 17 with a remainder of 2

If I do this on the calculator I get 171.5 as a result

my question is do I add 2 to the 17 to say that my answer is 17 + remainder 2 = 172?

The answer to the long division does not look the same as the calculated answer unless I move the remainder up to the 17 on top of the dividend

If I could use latex I would have made this alot clearer but sorry I can't use it.

any help appreciated

 

[Deleted member 4993]

I am just refreshing my learning of estimating and have a question or two

I have an example;

(53.4 x 70.9) / (22.2)

So on paper I write;

53 x 71 = 3763

Now I divide by 22

using long division

22 out of 3763

I say 22 out of 37 = 1

Then I say 1 x 22 = 22 and

37 - 22 = 15

then I bring down 6

22 from 156 = 7

7 x 22 = 154

156 - 154 = remainder 2

You are not done yet - you still have a 3 up there

bring down 3 to get 23

Now you got 171 with remainder 1

A totally different story.....

So I have an answer of 17 with a remainder of 2

If I do this on the calculator I get 171.5 as a result

my question is do I add 2 to the 17 to say that my answer is 17 + remainder 2 = 172?

The answer to the long division does not look the same as the calculated answer unless I move the remainder up to the 17 on top of the dividend

If I could use latex I would have made this alot clearer but sorry I can't use it.

any help appreciated

.

 

[Probability]

Thank you for your help. Comparison to the calculator answer using exact figures the long division is 0.5 in difference, but I would like to ask what actually happens to the remainder and what is it?

Calculator gives = 170.54 (2dp)

and long division = 171 with remainder 1

 

[stapel]

Comparison to the calculator answer using exact figures the long division is 0.5 in difference, but I would like to ask what actually happens to the remainder and what is it?

Calculator gives = 171.54 (2dp)

and long division = 171 with remainder 1

Note: I've corrected (in red) a typo in the above.

The calculator gives 171.54, being the approximation of 3763/22 to two decimal places. (A graphing calculator gives 171.0454545, which suggests a repeating decimal form, which is to be expected when one starts with whole numbers.)

The long division gives 171 R 1, but this is just shorthand for:

. . .$\displaystyle \displaystyle{\frac{3763}{22}\, =\, \frac{3762\, +\, 1}{22}\, =\, \frac{3762}{22}\, +\, \frac{1}{22}\, =\, 171\, +\, \frac{1}{22}}$

What decimal do you get when you do the long division of 1 by 22?

 

[HallsofIvy]

There's not much point in rounding off to "estimate" if rounding still leaves numbers that you cannot do immediately in your head.

I would have round these off to (50x 70)/20= 3500/20= 175.

 

[Probability]

Note: I've corrected (in red) a typo in the above.

The calculator gives 171.54, being the approximation of 3763/22 to two decimal places. (A graphing calculator gives 171.0454545, which suggests a repeating decimal form, which is to be expected when one starts with whole numbers.)

The long division gives 171 R 1, but this is just shorthand for:

. . .$\displaystyle \displaystyle{\frac{3763}{22}\, =\, \frac{3762\, +\, 1}{22}\, =\, \frac{3762}{22}\, +\, \frac{1}{22}\, =\, 171\, +\, \frac{1}{22}}$

What decimal do you get when you do the long division of 1 by 22?

I wonder if the calculators are rounding to different levels of accuracy?

53.4 x 70.9 / 22.2 = 170.54

I have tried this on three calculators one a graphics and all give same result

So getting back to my original question about remainders then, am I to take it that the remainder is something to do with the level of accuracy?

170.4 - 171.04

In your example above I won't get a remainder

 

[JeffM]

Thank you for your help. Comparison to the calculator answer using exact figures the long division is 0.5 in difference, but I would like to ask what actually happens to the remainder and what is it?

Calculator gives = 170.54 (2dp)

and long division = 171 with remainder 1 This is wrong

You took an ugly example so you rounded to make it easier, but the rounding obscures the logic. When you are trying to understand things use EASY examples.

$\displaystyle 11 \div 5 = \dfrac{11}{5} = \dfrac{10 + 1}{5} = \dfrac{10}{5} + \dfrac{1}{5} = \dfrac{2 * 5}{5} + \dfrac{1}{5} = 2 + \dfrac{1}{5}.$

The definition of a remainder is that it is the numerator of the fraction remaining that is not incorporated into your whole number or decimal quotient. If you like you can think of it as a sort of error term.

When you divide 11 by 5 you do not get a whole number. Try dividing 11 kitchen matches into five sets, each with an equal number of matches. It cannot be done. When you do the best you can, you have five sets each with two matches, but one match is left over, the remainder. That is where the word "remainder" comes from: it's what remains after division of a whole number of things into equal sets. But that word does not make much intuitive sense when you are not dividing by a whole number. However, the logic is exactly the same.

$\displaystyle 53.4 * 70.9 \div 22.2 = \dfrac{53.4 * 70.9}{22.2} = \dfrac{3786.06}{22.2}.$ That is an exact answer.

But it may not be a handy number to work with. So

$\displaystyle \dfrac{3786.06}{22.2} = \dfrac{3774 + 12.06}{22.2} = \dfrac{3774}{22.2} + \dfrac{12.06}{22.2} = \dfrac{170 * 22.2}{22.2} + \dfrac{12.06}{22.2} = 170 + \dfrac{12.06}{22.2}.$

This too is an exact answer, and one way to express it in English is to say that the answer is exactly 170 with remainder 12.06. However, this whole language of remainders makes very little sense if we are not dealing with whole numbers in the first place.

Or you can say that the answer is approximately 171 or that the answer is approximately 170.54. The only difference between the two answers is the degree of precision in the answer. The answer of 171 is less precise than the answer 170.54

 

[Probability]

You took an ugly example so you rounded to make it easier, but the rounding obscures the logic. When you are trying to understand things use EASY examples.

\(\displaystyle 11 \div 5 = \dfrac{11}{5} = \dfrac{10 + 1}{5} = \dfrac{10}{5} + \dfrac{1}{5} = \dfrac{2 * 5}{2 } + \dfrac{1}{5} = 2 + \dfrac{1}{5}.\)

Is the highlight in red a typo error or am I misunderstanding?

The definition of a remainder is that it is the numerator of the fraction remaining that is not incorporated into your whole number or decimal quotient. If you like you can think of it as a sort of error term.

When you divide 11 by 5 you do not get a whole number. Try dividing 11 kitchen matches into five sets, each with an equal number of matches. It cannot be done. When you do the best you can, you have five sets each with two matches, but one match is left over, the remainder. That is where the word "remainder" comes from, It's what remains after division of a whole number of things into equal sets. But that word does not make much intuitive sense when you are not dividing by a whole number. However, the logic is exactly the same.

$\displaystyle 53.4 * 70.9 \div 22.2 = \dfrac{53.4 * 70.9}{22.2} = \dfrac{3786.06}{22.2}.$ That is an exact answer.

But it may not be a handy number to work with. So

$\displaystyle \dfrac{3786.06}{22.2} = \dfrac{3774 + 12.06}{22.2} = \dfrac{3774}{22.2} + \dfrac{12.06}{22.2} = \dfrac{170 * 22.2}{22.2} + \dfrac{12.06}{22.2} = 170 + \dfrac{12.06}{22.2}.$

This too is an exact answer, and one way to express it in English is to say that the answer is exactly 170 with remainder 12.06. However, this whole language of remainders makes very little sense if we are not dealing with whole numbers in the first place.

Or you can say that the answer is approximately 171 or that the answer is approximately 170.54. The only difference between the two answers is the degree of precision in the answer. The answer of 171 is less precise than the answer 170.54

Thanks for the above I see that I require practice.

 

[Probability]

In keeping with long division I am interested in getting an understanding of how to work these out without a calculator.

so today I was looking at example where I was asked to convert to a decimal.

2 and 1/4 = 9 / 4 = 9 divided by 4

so I said

4 from 9 = 2, then 2 x 4 = 8 then I said 9 - 8 = 1

what I get is a solution of 2 with remainder 1 but the calculator says 2.25 as a result?

Why can't I get that using pen and paper?

 

[lookagain]

In keeping with long division I am interested in getting an understanding of how to work these out without a calculator.

so today I was looking at example where I was asked to convert to a decimal.

2 and 1/4 = 9 / 4 = 9 divided by 4 . . . Yes.

so I said

4 from 9 = 2, <----- I don't see a valid meaning with that. You could state that two fours are the most that can be subtracted from nine to leave a non-negative remainder, which in this case is one. then 2 x 4 = 8 then I said 9 - 8 = 1

> > what I get is a solution of 2 with remainder 1 but the calculator says 2.25 as a result? <<

Why can't I get that using pen and paper?

9 is the dividend, 4 is the divisor, 2 is the quotient, and 1 is the remainder. $\displaystyle \ \ \dfrac{dividend}{divisor} \ = \ quotient \ + \dfrac{remainder}{divisor}. \ \$ Specifically here, $\displaystyle \frac{9}{4} \ = \ 2 \ + \ \frac{1}{4} ** \ = \ 2 \ + \ 0.25 \ = \ 2.25 \ \ \ \ \ \$** You can treat $\displaystyle \frac{1}{4} \ \ as \ \ a \ \ long \ \ division \ \ as \ \ 4 \ \ divided \ \ into \ \ 1.00 \ \ to \ \ get \ \ 0.25.$

 

[JeffM]

Thanks for the above I see that I require practice.

Yes. I did have a typo. It has now been corrected. I apologize. An explanation with an error in it, even a fairly obvious typographical error, may be worse than no explanation at all.

 

[JeffM]

In keeping with long division I am interested in getting an understanding of how to work these out without a calculator.

so today I was looking at example where I was asked to convert to a decimal.

2 and 1/4 = 9 / 4 = 9 divided by 4

so I said

4 from 9 = 2, then 2 x 4 = 8 then I said 9 - 8 = 1

what I get is a solution of 2 with remainder 1 but the calculator says 2.25 as a result?

Why can't I get that using pen and paper?

You can and will get that if you use decimal notation instead of remainders. Your calculator gave you a decimal answer, not a whole number answer with a remainder. For practical purposes, the decimal representation is usually much more useful. The remainder notation is an abbreviation of a theoretically exact representation.

$\displaystyle 9 \div 4 = 2\ remainder\ 1\ ABBREVIATES\ the\ exact\ representation\ of\ \dfrac{9}{4} = \dfrac{8}{4} + \dfrac{1}{4} = 2 + \dfrac{1}{4} = 2.25.$

In the case above, the decimal representation exactly equals the theoretically exact representation. In the case below, the decimal representation is an approximation of the theoretically exact representation.

$\displaystyle 41 \div 9 = 4\ remainder\ 5\ ABBREVIATES\ the\ exact\ representation\ of\ \dfrac{41}{9} = \dfrac{36}{9} + \dfrac{5}{9} = 4 + \dfrac{5}{9} \approx 4.56.$