i have been helping my daughter with her long division homework. One of the questions was 172/6 she worked out the answer using the long division tree, and upon checking it found the answer differs when using a calculator. Can anybody answer me this.Why when 172/6 = 28.6 with a calculator, with long division it is 28.4
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3It's not $.4$. Your remainder was $4$ and you still need to divide it by $6$. So with "long division", the answer is $28+4/6\approx 28.667$. – David Mitra Jan 30 '14 at 17:12
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thanks for the reply, I am not the sharpest pencil in the case when it comes to maths, hence why I used a calculator to check, would you be able to show how that is worked out so I can try and explain to my daughter – user125004 Jan 30 '14 at 17:17
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See if can make sense of http://www.wolframalpha.com/input/?i=long+division+172%2F6 – Amzoti Jan 30 '14 at 17:20
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Here's what you're doing: $6$ goes into $172$ twenty times, and there's something left over (the remainder). So $172/6=20 +\text{remainder}/6$. The remainder is $172-20\cdot 6=52$ (this is what goes on in the first stage of the long division). So $172/6 = 20 +52/6$. Next stage: $6$ goes into $52$ eight times with a remainder of $4$. So $172/6 =20+8 +4/6$. You stop here since "$6$ doesn't go into $4$". (I hope this makes sense.) – David Mitra Jan 30 '14 at 17:22
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Is your daughter leaving the remainder or is she doing decimal division? If she is dealing only with integers than "172 divided by 6 equals 28 remainder 4", and if she is dealing with decimal division "172 divided by 6 equals 28.66666..." like your calculator has shown and the answer below – lsoranco Jan 30 '14 at 17:22
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The actual process of "long division", as used in the answer below, is just a shorthand device for doing what I did above. – David Mitra Jan 30 '14 at 17:27
4 Answers
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Long division:
28.6
-----------------
6 ) 172.000
12
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52
48
--
4 0
3 6
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4
At this point the remainder, 4, is the same as the remainder in the previous step, also 4, so everything will repeat as it did in the previous step, adding an infinite line of 6es to the quotient, which is 28.66666…, and not 28.4 nor 28.6 as you said.
MJD
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Thank you everybody your posts have been most helpful and has helped me (I think) I will now pass my new found knowledge on to my daughter (or try to anyway) – user125004 Jan 30 '14 at 17:30
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The calculater expresses the answer in decimals. Your daughter's long division expresses the answer using an actual remainder.
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I'm aware that this is a very old post, but I found it while looking for myself. I ended up finding this video, which puts the reason into simpler terms than the rest of the comments, in my opinion. https://youtu.be/27i2qdF8vX4 Just in case anyone coming across this post is confused as well. (Basically, the common mistake is assuming the remainder is already in decimal form. There is just one more step to long division for the remainder to become a decimal, since the answer to long division is expressed as a whole number and a fraction. The remainder is a fraction until converted.)
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Actually you are both wrong. It really does equal 28.4. When you divide 172 by ^6, 6 becomes base. If you were counting apples you would want to accept the product of long division , and the remainder thereof. If are dividing with a calculator the output will not factoring for base number. A simple binary example will show you why base is the most important factor.
EXAMPLE : Sally has 172 apples in a basket. David asks sally too seperate her apples into groups of 6. Sally decides that the best way to do this will be to place one apple at a time into 6 equal rows. Sally and David decided that they will place one apple at a time at the start of each row until they have no more apples left. Then the will eat the remaining apples.
HERE WE GO : I will place one at a time. Starting a new row of apples only when I have completed the previous one. In this example, each row must consist of 6 apples. `For the sake of this example we will use the arabic numeral 0 to represent each instance of an apple.
000000 row 1
000000 row 2
000000 row 3
000000 row 4
000000 row 5
000000 row 6
000000 row 7
000000 row 8
000000 row 9
000000 row 10, and we are 60 apples into the this thing
000000 row 11
000000 row 12
000000 row 13
000000 row 14
000000 just 82 apples left, you can count the 0's if you don't believe me
000000 row 15
000000 row 16
000000 row 17
000000 row 18
000000 row 19
000000 row 20
000000 row 21
000000 row 22
000000 row 23
000000 row 24
000000 row 25
000000 row 26
000000 ... row 27, Woah. David and Sally are getting tired.
000000 ... row 28. Looks familiar right. 172/6 = 28, remember
0000 hey, look it's 4 again!
... and finally, column 29, well it looks like they won't have enough apples to divide them up perfectly. So 28 x 6 = 168 remainder 4 (28.4 base 6). Hey sally, would you mind counting the apples that remain in row for 29 for us? But, what about the remainder? It is 4 even! Not sure why they aren't teaching this in school. But, this is technically why division and multiplication are not true.
Sally finds that she has exactly 4 apples left over after dividing her total supply of 172 apples into groups of 6. The remainder on your calculator is a representative fraction displayed over a hundred in base 10 as a percentage ratio . More or less, your calculator is actually wrong 100% of the time. Also, I never finished highschool but I will give you the easy way to convert the decimal number, calculator answer, to a more usable real number that is actually worth having.
Di= Dividend
Dv = Divisor (divisor = base) divisor is the same thing as you base, all you are doing when you divided is dropping number from the 10 system and adapting your dividend to fill the gaps or compress.
Quo = Quotient, base 10
[Di]÷[Dv]= Quo
172 ÷ 6 = 28.666666666666E
This .66666666666E is the part me to adjust to base 6 or else we will be looking for apples we don't have when it came to this. Unfortunately, this is difficult but the easiest way is the easiest way is to just do long division. When the remainder is a factor of the base number you can typical figure these equations out on the fly. Essential just convert the remainder to a percentage of the base. i.e. %.66666666666E of 6 if 3.99999999. So, 28.4 .
