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2158/148=?
method A
148|2158
14r86======14.
148|86
0r86=======0
148|860
5r120======5
148|120
0r120======0
148|1200
8r16=======8
148|16
0r16=======0
148|160
1r12=======1
148|12
0r12=======0
148|120
0r120-------recurring loop
2158/148=14.05{08010}
method B (Microsoft Calculator)
2158/148=14.5{810}
Which result is correct and why?
Another math test:
If you take two apples from three apples, how many apples do you have?
.
Answer :
Two! You just took two apples.
I didn't ask you what three minus two is.
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The "error" is in your second line...
2158/148=?
method A
148|2158
14r86======14.
148|86
0r86=======0
148|860
5r120======5
148|120
0r120======0
148|1200
8r16=======8
148|16
0r16=======0
148|160
1r12=======1
148|12
0r12=======0
148|120
0r120-------recurring loop
2158/148=14.05{08010}method B (Microsoft Calculator)
2158/148=14.5{810}Which result is correct and why?
While it is true that 148 doesn't go into 86, that's why you had the remainder in the previous line. Putting the zero after the decimal place is incorrect. As soon as you go past the decimal place, you drop a zero at the end of the 86 and now you are dividing 860 by 148.
(It's a little easier to show you on paper.)
I hope that helps. But the calculator is correct (in this case).
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Another math test:...
How about this one...
Three guys go into a hotel and get a room. The cost of the room is $30 and the three guys think "That makes things easy -- $10 each." The bellboy takes the guys up to the room and returns to the desk. At the desk, the manager says that he forgot that they were having a special that day and the cost was actually $25. He gave the bellboy $5 to take back to the three guys. On the way back, the bellboy decided that since 3 doesn't go into 5 evenly, he would make life easier on them and he pocketed $2 as his "tip" and gave them the remaining $3. Now if each of the guys paid $10 and got one back, that means that they paid $9 for the room. 9 times 3 = 27 -- plus the two dollars the bellboy took makes $29.
What happened to the missing dollar?
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Another math test:...How about this one...
Three guys go into a hotel and get a room. The cost of the room is $30 and the three guys think "That makes things easy -- $10 each." The bellboy takes the guys up to the room and returns to the desk. At the desk, the manager says that he forgot that they were having a special that day and the cost was actually $25. He gave the bellboy $5 to take back to the three guys. On the way back, the bellboy decided that since 3 doesn't go into 5 evenly, he would make life easier on them and he pocketed $2 as his "tip" and gave them the remaining $3. Now if each of the guys paid $10 and got one back, that means that they paid $9 for the room. 9 times 3 = 27 -- plus the two dollars the bellboy took makes $29.
What happened to the missing dollar?
.............not cool.
RPG Superstar 2010 Top 16
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Another math test:...How about this one...
Three guys go into a hotel and get a room. The cost of the room is $30 and the three guys think "That makes things easy -- $10 each." The bellboy takes the guys up to the room and returns to the desk. At the desk, the manager says that he forgot that they were having a special that day and the cost was actually $25. He gave the bellboy $5 to take back to the three guys. On the way back, the bellboy decided that since 3 doesn't go into 5 evenly, he would make life easier on them and he pocketed $2 as his "tip" and gave them the remaining $3. Now if each of the guys paid $10 and got one back, that means that they paid $9 for the room. 9 times 3 = 27 -- plus the two dollars the bellboy took makes $29.
What happened to the missing dollar?
The $2 is being incorrectly added twice in the above equation to get to $29
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Another math test:...How about this one...
Three guys go into a hotel and get a room. The cost of the room is $30 and the three guys think "That makes things easy -- $10 each." The bellboy takes the guys up to the room and returns to the desk. At the desk, the manager says that he forgot that they were having a special that day and the cost was actually $25. He gave the bellboy $5 to take back to the three guys. On the way back, the bellboy decided that since 3 doesn't go into 5 evenly, he would make life easier on them and he pocketed $2 as his "tip" and gave them the remaining $3. Now if each of the guys paid $10 and got one back, that means that they paid $9 for the room. 9 times 3 = 27 -- plus the two dollars the bellboy took makes $29.
What happened to the missing dollar?
argh!
I don't have the math ability or language to explain it. And I know its going to bug me all afternoon.
Of course they really didn't pay $9 each for the room, they paid approx $8.33 each for that, and paid the bellboy approx $0.67 each as a tip, and kept (effectively) $1 each. 8.33 x 3 = 25 + (3 x 1 = 3) + (3 x 0.67 = 2) = 30 ... hmmm, can't explain it more clearly than that, am I on the right track?
EDIT: Or what Dementrius said.
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The "error" is in your second line...
2158/148=?
method A
148|2158
14r86======14.
148|86
0r86=======0
148|860
5r120======5
148|120
0r120======0
148|1200
8r16=======8
148|16
0r16=======0
148|160
1r12=======1
148|12
0r12=======0
148|120
0r120-------recurring loop
2158/148=14.05{08010}method B (Microsoft Calculator)
2158/148=14.5{810}Which result is correct and why?
While it is true that 148 doesn't go into 86, that's why you had the remainder in the previous line. Putting the zero after the decimal place is incorrect. As soon as you go past the decimal place, you drop a zero at the end of the 86 and now you are dividing 860 by 148.
(It's a little easier to show you on paper.)
I hope that helps. But the calculator is correct (in this case).
Ok But why is Zero not subject to the same rules as a non zero value if zero is a number like 1, 2...?
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Okay, different maths question, but one that bugged me a few months ago when the issue of operator precedence came up -- while the original context was in relation to computing the disagreement was about operator precedence in basic maths *not* in computing.
She expressed the opinion that in a calculation division must be performed before multiplication, and that addition must be performed before subtraction. To back this up she refered to a mnemonic she'd been taught at school -- PEDMAS -- (expand) parentheses, exponentiation, division, multiplication, addition, subtraction -- describing the order in which things should be done. A quick google suggested that indeed this mnemonic (or slight variants of it) are indeed common and popular.
I disagreed, saying that the order of resolving multiplication and division (and similarly, addition and subtraction) dosn't matter as they're essentially the same operation, and instead both multiplication and division are resolved, left-to-right, at the same time (and also suggesting that a mnemonic aimed at school children probably wasn't the most reliable source).
Thoughts? Is her position common? Correct? Am I missing something fundamental?
She expressed the opinion that in a calculation division must be performed before multiplication, and that addition must be performed before subtraction. To back this up she refered to a mnemonic she'd been taught at school -- PEDMAS -- (expand) parentheses, exponentiation, division, multiplication, addition, subtraction -- describing the order in which things should be done. A quick google suggested that indeed this mnemonic (or slight variants of it) are indeed common and popular.
I disagreed, saying that the order of resolving multiplication and division (and similarly, addition and subtraction) dosn't matter as they're essentially the same operation, and instead both multiplication and division are resolved, left-to-right, at the same time (and also suggesting that a mnemonic aimed at school children probably wasn't the most reliable source).
Thoughts? Is her position common? Correct? Am I missing something fundamental?
the left-to-right part of your post (bolding mine) threw me off a bit but other than that you are right. The order of multiplications and divisions does not matter, as long as the structure of the equation remains the same.
In order to simplify the equation as much as possible, I would generally argue for multiplcation before division rather than the other way around, as it reduces the number of operations in which you have to account for fractures.|
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Multiplicationa and division can be completed in whichever order you want, as long as they're ALL complete before any addition or subtraction. Again, addition and subtraction can be complete in any order. The reason being they're the same function in reverse order. They share equal power in the mathematical equation.
The general rule is you work from left to right across a problem completing the mathematical functions in order of precedence (which you already outlined in your post). The left to right bit is only there as a way of logical progression to minimize the chance of calculation errors throughout the process.
Mathemeticians eh, who'd have 'em? :)
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Ok But why is Zero not subject to the same rules as a non zero value if zero is a number like 1, 2...?
I'm not sure I fully understand what you mean. It is under the same rules.
In your example, you have your whole numbers with 2158 divided by 148 gives you 14 with a remainder of 86. If you want to continue the operation with decimals, you put a decimal with 2158.0. You drop the 0 down to the 86, making it 860 and you continue the operation.
Let's say that you had a remainder of 10 instead of 86. Then when you bring the zero down, it makes 100 and 148 still doesn't go into 100 -- so at that point you would put a zero at the top and make another zero and so on.
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She expressed the opinion that in a calculation division must be performed before multiplication, and that addition must be performed before subtraction. To back this up she refered to a mnemonic she'd been taught at school -- PEDMAS -- (expand) parentheses, exponentiation, division, multiplication, addition, subtraction -- describing the order in which things should be done. A quick google suggested that indeed this mnemonic (or slight variants of it) are indeed common and popular.
I disagreed, saying that the order of resolving multiplication and division (and similarly, addition and subtraction) dosn't matter as they're essentially the same operation, and instead both multiplication and division are resolved, left-to-right, at the same time (and also suggesting that a mnemonic aimed at school children probably wasn't the most reliable source).
Thoughts? Is her position common? Correct? Am I missing something fundamental?
the left-to-right part of your post (bolding mine) threw me off a bit but other than that you are right. The order of multiplications and divisions does not matter, as long as the structure of the equation remains the same.
In order to simplify the equation as much as possible, I would generally argue for multiplcation before division rather than the other way around, as it reduces the number of operations in which you have to account for fractures.
I'm sorry, but the order does matter and it needs to be done left to right. Multiplication and division have the same "weight" and so they are done together from left to right. If you do all multiplication or all division before the other one, you are subject to potential problems. The only real way to avoid that is to convert all "division" into fractions and make everything multiplication. Then you can do them in any order.
For example, if you have 9/3*5. If you try and do them in any order and try and do 3*5 first, you end up with 9/15 which equals 3/5 or 0.6. If you do it in the correct order you end up with 15. There are other examples that fairly easily show that multiplication and division have the same weight and must be done from left to right.
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RPG Superstar 2010 Top 16
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I disagreed, saying that ... both multiplication and division are resolved, left-to-right, at the same time (and also suggesting that a mnemonic aimed at school children probably wasn't the most reliable source).Thoughts? Is her position common? Correct? Am I missing something fundamental?
I've taught remedial mathematics for years, and (a) you are completely correct, (b) misconceptions are common.
Commentary under the cut:
Good mnemonics ("Please excuse my dear Aunt Sally.") are pretty useful, ut in this case don't distinguish whether operations are on the same "rank."
Part of the problem is that "order of operations" problems are a tad artifical. In any sensible situation, someone writes down an expression that makes sense to him, and then evaluates it. Nobody says, looking at books to sell:
"Well, I have three original editions and five new editions, each worth seventeen dollars. So I'll write out 3 + 5 x 17. Darn, I forgot the parentheses; I'm compelled now to multiply the 5 x 17 first and only then add the 3. Oh, fie on me!"
"Order of operations" problems only occur when someone has written an expression, left off any helpful parentheses, and then suddenly died, leaving no instructions to her next of kin as to what the expression might represent. It provides a set of default instructions for them to carry out her calculations.
You don't want to die intestate. You also don't want to write expressions without parentheses.
Here's one of my favorite math questions:
"In standard (base ten) notation, both 64 and 121 are perfect squares.
"In base eight, the number represented by "64" is (6 x 8) + (4 x 1) = 52 (base ten), which is not a perfect square. But the number represented by "121" is (1 x 64) + (2 x 8) + (1 x 1) = 81, which just so happens to be a perfect square.
"In which bases is "121" a perfect square?"
For example, if you have 9/3*5. If you try and do them in any order and try and do 3*5 first, you end up with 9/15 which equals 3/5 or 0.6. If you do it in the correct order you end up with 15. There are other examples that fairly easily show that multiplication and division have the same weight and must be done from left to right.
we mean exactly the same thing.
A*B*C == A*C*B for any value of A,B and C
and in your example,
A=9
B=1/3
C=5
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RPG Superstar 2010 Top 16
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We mean exactly the same thing.
A*B*C == A*C*B for any value of A,B and C
and in your example,
A=9
B=1/3
C=5
Well, we're talking about rational numbers, so in that context you're right, but order of operation still applies when you can't commute elements in multiplication, (2x2 matrices and quaternions, for example).
Also, order of operations tends to be taught before introducing negative numbers. So 8 - 3 - 2 needs to be treated as subtraction, rather than adding negatives.
By the way, what do you folks consider the order of operations for exponentiation? Would 2^3^3 simplify to 2^27 = 134,217,728 or 8^3 = 512 ?
Well, we're talking about rational numbers, so in that context you're right, but order of operation still applies when you can't commute elements in multiplication, (2x2 matrices and quaternions, for example).
I should have stayed away from this mess (I mean discussion) But yeah thanks chris
By the way, what do you folks consider the order of operations for exponentiation? Would 2^3^3 simplify to 2^27 = 134,217,728 or 8^3 = 512 ?
Against my better judgement I am going to try to answer this one.
I think it should evaluate to 2^27
Evaluate the arguments in the exponent before evaluating the exponent itself In other words, 2^SQR(9) = 8
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Thoughts? Is her position common? Correct? Am I missing something fundamental?
You are actually both correct. She is correct, but perhaps not necessarily for the reasons she thinks. Let’s look at another example –
3*4/2*5
Going left to right – 12/2*5 = 6*5 = 30
Doing multiplication first – 12/10 = 1.2
Doing division first – 3*2*5 = 6*5 = 30
Essentially, what she is doing is converting to fractions first and then multiplying the middle two numbers before moving on –
3*4*(1/2)*5 = 3*2*5 = 30
So mathematically, she will get the right answer her way – as will you. However, there are a number of times where there could be a rounding error if she is not converting to fractions. A simple example could be like 14*2/7. With my simple calculator, if I do division first and then multiplication, I end up with 3.99999999994. While the answer fairly clearly should be exactly 4.
The same rules pretty well hold true for subtraction and addition as well. But it is very easy to get into negative numbers if one tries to do all subtraction first. (But if you do all addition first without converting to negative numbers, you could get the wrong answer.)