Number Properties- Decimals & Remainders
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Mon Oct 10, 2011 10:29 am
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Whitney Garner
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Mon Oct 10, 2011 12:14 pm
Siete_Cien wrote:
If we think about remainders and decimals, we start to understand that the remainder of a division problem can be expressed as the value AFTER the decimal. For example:
When a positive integer x is divided by a positive integer y, the remainder is 9. If x/y=96.12 what is the value of y.
How does one go about solving this question?
How does one go about solving this question?
9/2 = 4 with a remainder of 1, but it also equals 4.5. How can we see that a remainder of 1 here is the same as .5? Well, the decimal is just the Remainder/Divisor. So here, the decimal is 1/2 or .5.
Try with a couple of others:
- 12/5 = 2.4 OR 2 with a remainder of 2. But this makes sense because .4 = 2/5.
- 86/40 = 2.15 or 2 with a remainder of 6. And again, 6/40 = 3/20 = .15
Okay, but that is going from a situation where we KNOW the numbers and finding the decimal and remainder forms of the solutions. How the heck do we move backwards??
So what about if we were just given the number 2.15 and asked to find the values we were dividing? I know that the remainder is .15 and therefore equivalent to 3/20, but as we saw in the last example, the actual remainder might not be 3 with a divisor of 20. It could have been a divisor of 40 with a remainder of 6 (because 6/40 = 3/20). In fact, we could find the value 2.15 from an INFINITE number of fractions. To prove it to yourself, grab a calculator and check these: 43/20, 86/40, 129/60...
You'll quickly see that all of these are true because they are all equivalent fractions (they all reduce to the same thing). Okay, so SUPER LONG explanation but back to the original problem - what good does all of this discussion do me???
So what if we give you more info for the 2.15 question. Now, they tell us that we know the actual value of the remainder. In this case let's say it is 6 again. Well, I can take the decimal 0.15 and convert it to a fraction:
15/100 (and then reduce completely) = 3/20.
From here, I can build ANY fraction I want just by using equivalent fractions. I want a remainder of 6 (remember, when we convert from remainder to decimal, we put the remainder over the original divisor), so I need to make the numerator 6 and then the new denominator would be my divisor. So we ask the question:
3/20 = 6/??
This one is pretty easy to see = 3/20 = 6/40, so the original divisor was 40.
***************************************************
Apply all of this to the original problem:
***************************************************
Siete_Cien wrote:
0.12 = 12/100 = 6/50 = 3/25 (so the remainder/divisor would have to be multiples of 3/25).
When a positive integer x is divided by a positive integer y, the remainder is 9. If x/y=96.12 what is the value of y.
The original remainder is 9, so:
3/25 = 9/??
3/25 = 9/75, so the original divisor (Y) is 75 for the remainder to be 9.
Whit
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Fri Dec 02, 2011 7:40 am
Whitney Garner wrote:
9/2 = 4 with a remainder of 1, but it also equals 4.5. How can we see that a remainder of 1 here is the same as .5? Well, the decimal is just the Remainder/Divisor. So here, the decimal is 1/2 or .5.
Try with a couple of others:
- 12/5 = 2.4 OR 2 with a remainder of 2. But this makes sense because .4 = 2/5.
- 86/40 = 2.15 or 2 with a remainder of 6. And again, 6/40 = 3/20 = .15
Okay, but that is going from a situation where we KNOW the numbers and finding the decimal and remainder forms of the solutions. How the heck do we move backwards??
So what about if we were just given the number 2.15 and asked to find the values we were dividing? I know that the remainder is .15 and therefore equivalent to 3/20, but as we saw in the last example, the actual remainder might not be 3 with a divisor of 20. It could have been a divisor of 40 with a remainder of 6 (because 6/40 = 3/20). In fact, we could find the value 2.15 from an INFINITE number of fractions. To prove it to yourself, grab a calculator and check these: 43/20, 86/40, 129/60...
You'll quickly see that all of these are true because they are all equivalent fractions (they all reduce to the same thing). Okay, so SUPER LONG explanation but back to the original problem - what good does all of this discussion do me???
So what if we give you more info for the 2.15 question. Now, they tell us that we know the actual value of the remainder. In this case let's say it is 6 again. Well, I can take the decimal 0.15 and convert it to a fraction:
15/100 (and then reduce completely) = 3/20.
From here, I can build ANY fraction I want just by using equivalent fractions. I want a remainder of 6 (remember, when we convert from remainder to decimal, we put the remainder over the original divisor), so I need to make the numerator 6 and then the new denominator would be my divisor. So we ask the question:
3/20 = 6/??
This one is pretty easy to see = 3/20 = 6/40, so the original divisor was 40.
***************************************************
Apply all of this to the original problem:
***************************************************
The original remainder is 9, so:
3/25 = 9/??
3/25 = 9/75, so the original divisor (Y) is 75 for the remainder to be 9.
Whit
One of the most useful shortcuts I have seen for number properties. Thanks Whit!Siete_Cien wrote:
If we think about remainders and decimals, we start to understand that the remainder of a division problem can be expressed as the value AFTER the decimal. For example:
When a positive integer x is divided by a positive integer y, the remainder is 9. If x/y=96.12 what is the value of y.
How does one go about solving this question?
How does one go about solving this question?
9/2 = 4 with a remainder of 1, but it also equals 4.5. How can we see that a remainder of 1 here is the same as .5? Well, the decimal is just the Remainder/Divisor. So here, the decimal is 1/2 or .5.
Try with a couple of others:
- 12/5 = 2.4 OR 2 with a remainder of 2. But this makes sense because .4 = 2/5.
- 86/40 = 2.15 or 2 with a remainder of 6. And again, 6/40 = 3/20 = .15
Okay, but that is going from a situation where we KNOW the numbers and finding the decimal and remainder forms of the solutions. How the heck do we move backwards??
So what about if we were just given the number 2.15 and asked to find the values we were dividing? I know that the remainder is .15 and therefore equivalent to 3/20, but as we saw in the last example, the actual remainder might not be 3 with a divisor of 20. It could have been a divisor of 40 with a remainder of 6 (because 6/40 = 3/20). In fact, we could find the value 2.15 from an INFINITE number of fractions. To prove it to yourself, grab a calculator and check these: 43/20, 86/40, 129/60...
You'll quickly see that all of these are true because they are all equivalent fractions (they all reduce to the same thing). Okay, so SUPER LONG explanation but back to the original problem - what good does all of this discussion do me???
So what if we give you more info for the 2.15 question. Now, they tell us that we know the actual value of the remainder. In this case let's say it is 6 again. Well, I can take the decimal 0.15 and convert it to a fraction:
15/100 (and then reduce completely) = 3/20.
From here, I can build ANY fraction I want just by using equivalent fractions. I want a remainder of 6 (remember, when we convert from remainder to decimal, we put the remainder over the original divisor), so I need to make the numerator 6 and then the new denominator would be my divisor. So we ask the question:
3/20 = 6/??
This one is pretty easy to see = 3/20 = 6/40, so the original divisor was 40.
***************************************************
Apply all of this to the original problem:
***************************************************
Siete_Cien wrote:
0.12 = 12/100 = 6/50 = 3/25 (so the remainder/divisor would have to be multiples of 3/25).
When a positive integer x is divided by a positive integer y, the remainder is 9. If x/y=96.12 what is the value of y.
The original remainder is 9, so:
3/25 = 9/??
3/25 = 9/75, so the original divisor (Y) is 75 for the remainder to be 9.
Whit
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"Perfection is achieved, not when there is nothing more to add, but when there is nothing left to take away." ~ Antoine de Saint-Exupery
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