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Lesson 1: Fractions- Repeating decimal sequences upon division or con
[#permalink]
Updated on: 19 Aug 2022, 07:39
19
Kudos
4
Bookmarks
Here is my first post on topics that I am learning on the go through my practice of multiple questions and which you will generally not find in any of the GMAT courses (even the most extensive ones). I have been preparing for a year and have always taken every new concept that came my way as a new challenge and found it extremely intimidating to absorb it because even after hours of dedicated course prep, I did not know such things which are actually ESSENTIAL but still noone provides. I hope to keep posts like these coming so it becomes an extensive resource directory, however, given my personal limitations, there is only little I can be sure of circumstances but surely do keep me following along as it certainly is on my agenda for now.
Lesson 1: Fractions- Repeating decimal sequences upon division or converting repeating decimals into fraction fast
First - We use the number that is repeating and the denominator has as many "9" digits as there are different digits in the block that repeats. e.g.
0.555555 = 5/9
0.13131313 = 13/99
0.432432432432 = 432/999
You can thus remember: If a fraction (in lowest terms) can be written as a repeating decimal and the number of decimal digits repeating is n, then the denominator must be a factor of 10^n - 1.
Second - If the sequence starts to repeat after some zeros, add the same number of zeros in the denominator. e.g.
0.005555 = 5/900
0.013131313 = 13/990
0.0004324324324...=432/999000=54/124875=2/46250.0004324324324...=432/999000=54/124875=2/4625
Third - Terminating decimals + Repeating Decimals. e.g.
2.31555555=2.31+0.00555555=231/100+5/9002.31555555=2.31+0.00555555=231/100+5/900
0.745454545=0.7+0.0454545=7/10+45/9900.745454545=0.7+0.0454545=7/10+45/990
Last one - The reciprocal of a prime number "p", except 2 and 5, has a repeating sequence of p-1 digits, or a factor of p-1 digits. e.g.
1/7=0.142857142857142857...1/7=0.142857142857142857... - As you can notice, the sequence has 6 digits = p-1 = 7-1 = 6 digits.
And if you multiply this fraction by a number you will only change the beginning of the sequence. e.g.
4/7=4∗1/7=0.57142857142857...
Borrowed content from- coelholds
Additional: You must know when the decimals will be terminating or not.
1. Any integer divided by 2 or 5 or both (or their multiples) will always be a terminating decimal. Hence any number divided by 4, 8, or 10 will be a terminating decimal
2. Any integer divided by 9 will be a repeating decimal that will follow the repeating sequence mentioned in point 1 in the above section
3. Any integer divided by 3 and 7 will have a repeating sequence of 2 or 6 respectively, or factors of them (as per point 4 in above section)
4. 6 does not seem to have any defined pattern as 2/6=Repeating decimal with single-digit sequence and 3/6=0.5= Terminating decimal
Question to apply this strategy on: https://gmatclub.com/forum/if-each-of-t ... 59213.html, https://gmatclub.com/forum/a-bar-over-a ... 99427.html
Lesson 1: Fractions- Repeating decimal sequences upon division or converting repeating decimals into fraction fast
First - We use the number that is repeating and the denominator has as many "9" digits as there are different digits in the block that repeats. e.g.
0.555555 = 5/9
0.13131313 = 13/99
0.432432432432 = 432/999
You can thus remember: If a fraction (in lowest terms) can be written as a repeating decimal and the number of decimal digits repeating is n, then the denominator must be a factor of 10^n - 1.
Second - If the sequence starts to repeat after some zeros, add the same number of zeros in the denominator. e.g.
0.005555 = 5/900
0.013131313 = 13/990
0.0004324324324...=432/999000=54/124875=2/46250.0004324324324...=432/999000=54/124875=2/4625
Third - Terminating decimals + Repeating Decimals. e.g.
2.31555555=2.31+0.00555555=231/100+5/9002.31555555=2.31+0.00555555=231/100+5/900
0.745454545=0.7+0.0454545=7/10+45/9900.745454545=0.7+0.0454545=7/10+45/990
Last one - The reciprocal of a prime number "p", except 2 and 5, has a repeating sequence of p-1 digits, or a factor of p-1 digits. e.g.
1/7=0.142857142857142857...1/7=0.142857142857142857... - As you can notice, the sequence has 6 digits = p-1 = 7-1 = 6 digits.
And if you multiply this fraction by a number you will only change the beginning of the sequence. e.g.
4/7=4∗1/7=0.57142857142857...
Borrowed content from- coelholds
Additional: You must know when the decimals will be terminating or not.
1. Any integer divided by 2 or 5 or both (or their multiples) will always be a terminating decimal. Hence any number divided by 4, 8, or 10 will be a terminating decimal
2. Any integer divided by 9 will be a repeating decimal that will follow the repeating sequence mentioned in point 1 in the above section
3. Any integer divided by 3 and 7 will have a repeating sequence of 2 or 6 respectively, or factors of them (as per point 4 in above section)
4. 6 does not seem to have any defined pattern as 2/6=Repeating decimal with single-digit sequence and 3/6=0.5= Terminating decimal
Question to apply this strategy on: https://gmatclub.com/forum/if-each-of-t ... 59213.html, https://gmatclub.com/forum/a-bar-over-a ... 99427.html
Re: Lesson 1: Fractions- Repeating decimal sequences upon division or con
[#permalink]
29 Jun 2022, 01:32
1
Kudos
Elite097 wrote:
Here is my first post on topics that I am learning on the go through my practice of multiple questions and which you will generally not find in any of the GMAT courses (even the most extensive ones). I have been preparing for a year and have always taken every new concept that came my way as a new challenge and found it extremely intimidating to absorb it because even after hours of dedicated course prep, I did not know such things which are actually ESSENTIAL but still noone provides. I hope to keep posts like these coming so it becomes an extensive resource directory, however, given my personal limitations, there is only little I can be sure of circumstances but surely do keep me following along as it certainly is on my agenda for now.
Lesson 1: Fraction- Repeating decimal sequences upon division
First - We use the number that is repeating and the denominator has as many "9" digits as there are different digits in the block that repeats. e.g.
0.555555 = 5/9
0.13131313 = 13/99
0.432432432432 = 432/999
You can thus remember: If a fraction (in lowest terms) can be written as a repeating decimal and the number of decimal digits repeating is n, then the denominator must be a factor of 10^n - 1.
Second - If the sequence starts to repeat after some zeros, add the same number of zeros in the denominator. e.g.
0.005555 = 5/900
0.013131313 = 13/990
0.0004324324324...=432/999000=54/124875=2/46250.0004324324324...=432/999000=54/124875=2/4625
Third - Terminating decimals + Repeating Decimals. e.g.
2.31555555=2.31+0.00555555=231/100+5/9002.31555555=2.31+0.00555555=231/100+5/900
0.745454545=0.7+0.0454545=7/10+45/9900.745454545=0.7+0.0454545=7/10+45/990
Last one - The reciprocal of a prime number "p", except 2 and 5, has a repeating sequence of p-1 digits, or a factor of p-1 digits. e.g.
1/7=0.142857142857142857...1/7=0.142857142857142857... - As you can notice, the sequence has 6 digits = p-1 = 7-1 = 6 digits.
And if you multiply this fraction by a number you will only change the beginning of the sequence. e.g.
4/7=4∗1/7=0.57142857142857...
Borrowed content from- coelholds
Additional: You must know when the answers will be terminating or not.
1. Any integer divided by 2 or 5 or both (or their multiples) will always be a terminating decimal. Hence any number divided by 4, 8, or 10 will be a terminating decimal
2. Any integer divided by 9 will be a repeating decimal that will follow the repeating sequence mentioned in point 1 in the above section
3. Any integer divided by 3 and 7 will have a repeating sequence of 2 or 6 respectively, or factors of them (as per point 4 in above section)
4. 6 does not seem to have any defined pattern as 2/6=Repeating decimal with single-digit sequence and 3/6=0.5= Terminating decimal
Hope you liked this. Please do give me kudo so I know it helped you and gave you a useful strategy. It will also help me come up with more such posts.
Question to apply this strategy on: https://gmatclub.com/forum/if-each-of-t ... 59213.html
Lesson 1: Fraction- Repeating decimal sequences upon division
First - We use the number that is repeating and the denominator has as many "9" digits as there are different digits in the block that repeats. e.g.
0.555555 = 5/9
0.13131313 = 13/99
0.432432432432 = 432/999
You can thus remember: If a fraction (in lowest terms) can be written as a repeating decimal and the number of decimal digits repeating is n, then the denominator must be a factor of 10^n - 1.
Second - If the sequence starts to repeat after some zeros, add the same number of zeros in the denominator. e.g.
0.005555 = 5/900
0.013131313 = 13/990
0.0004324324324...=432/999000=54/124875=2/46250.0004324324324...=432/999000=54/124875=2/4625
Third - Terminating decimals + Repeating Decimals. e.g.
2.31555555=2.31+0.00555555=231/100+5/9002.31555555=2.31+0.00555555=231/100+5/900
0.745454545=0.7+0.0454545=7/10+45/9900.745454545=0.7+0.0454545=7/10+45/990
Last one - The reciprocal of a prime number "p", except 2 and 5, has a repeating sequence of p-1 digits, or a factor of p-1 digits. e.g.
1/7=0.142857142857142857...1/7=0.142857142857142857... - As you can notice, the sequence has 6 digits = p-1 = 7-1 = 6 digits.
And if you multiply this fraction by a number you will only change the beginning of the sequence. e.g.
4/7=4∗1/7=0.57142857142857...
Borrowed content from- coelholds
Additional: You must know when the answers will be terminating or not.
1. Any integer divided by 2 or 5 or both (or their multiples) will always be a terminating decimal. Hence any number divided by 4, 8, or 10 will be a terminating decimal
2. Any integer divided by 9 will be a repeating decimal that will follow the repeating sequence mentioned in point 1 in the above section
3. Any integer divided by 3 and 7 will have a repeating sequence of 2 or 6 respectively, or factors of them (as per point 4 in above section)
4. 6 does not seem to have any defined pattern as 2/6=Repeating decimal with single-digit sequence and 3/6=0.5= Terminating decimal
Hope you liked this. Please do give me kudo so I know it helped you and gave you a useful strategy. It will also help me come up with more such posts.
Question to apply this strategy on: https://gmatclub.com/forum/if-each-of-t ... 59213.html
Value of 1/9 is 0.11111.....you are manipulating this
value of 1/11 is 0.09 ......we can learn this two two division in parallel
if want to add new lesson add something really that not covered by extensive course....
1-if any no added to factorial is divisible by every no less than that factorial.
2-X/Y is less than something the anything added in both num and den is always less then that soemthing
jusT for instance i added there are lot more
Re: Lesson 1: Fractions- Repeating decimal sequences upon division or con
[#permalink]
29 Jun 2022, 01:33
3
Kudos
1
Bookmarks
Elite097 wrote:
Here is my first post on topics that I am learning on the go through my practice of multiple questions and which you will generally not find in any of the GMAT courses (even the most extensive ones). I have been preparing for a year and have always taken every new concept that came my way as a new challenge and found it extremely intimidating to absorb it because even after hours of dedicated course prep, I did not know such things which are actually ESSENTIAL but still noone provides. I hope to keep posts like these coming so it becomes an extensive resource directory, however, given my personal limitations, there is only little I can be sure of circumstances but surely do keep me following along as it certainly is on my agenda for now.
Lesson 1: Fraction- Repeating decimal sequences upon division
First - We use the number that is repeating and the denominator has as many "9" digits as there are different digits in the block that repeats. e.g.
0.555555 = 5/9
0.13131313 = 13/99
0.432432432432 = 432/999
You can thus remember: If a fraction (in lowest terms) can be written as a repeating decimal and the number of decimal digits repeating is n, then the denominator must be a factor of 10^n - 1.
Second - If the sequence starts to repeat after some zeros, add the same number of zeros in the denominator. e.g.
0.005555 = 5/900
0.013131313 = 13/990
0.0004324324324...=432/999000=54/124875=2/46250.0004324324324...=432/999000=54/124875=2/4625
Third - Terminating decimals + Repeating Decimals. e.g.
2.31555555=2.31+0.00555555=231/100+5/9002.31555555=2.31+0.00555555=231/100+5/900
0.745454545=0.7+0.0454545=7/10+45/9900.745454545=0.7+0.0454545=7/10+45/990
Last one - The reciprocal of a prime number "p", except 2 and 5, has a repeating sequence of p-1 digits, or a factor of p-1 digits. e.g.
1/7=0.142857142857142857...1/7=0.142857142857142857... - As you can notice, the sequence has 6 digits = p-1 = 7-1 = 6 digits.
And if you multiply this fraction by a number you will only change the beginning of the sequence. e.g.
4/7=4∗1/7=0.57142857142857...
Borrowed content from- coelholds
Additional: You must know when the answers will be terminating or not.
1. Any integer divided by 2 or 5 or both (or their multiples) will always be a terminating decimal. Hence any number divided by 4, 8, or 10 will be a terminating decimal
2. Any integer divided by 9 will be a repeating decimal that will follow the repeating sequence mentioned in point 1 in the above section
3. Any integer divided by 3 and 7 will have a repeating sequence of 2 or 6 respectively, or factors of them (as per point 4 in above section)
4. 6 does not seem to have any defined pattern as 2/6=Repeating decimal with single-digit sequence and 3/6=0.5= Terminating decimal
Hope you liked this. Please do give me kudo so I know it helped you and gave you a useful strategy. It will also help me come up with more such posts.
Question to apply this strategy on: https://gmatclub.com/forum/if-each-of-t ... 59213.html
Lesson 1: Fraction- Repeating decimal sequences upon division
First - We use the number that is repeating and the denominator has as many "9" digits as there are different digits in the block that repeats. e.g.
0.555555 = 5/9
0.13131313 = 13/99
0.432432432432 = 432/999
You can thus remember: If a fraction (in lowest terms) can be written as a repeating decimal and the number of decimal digits repeating is n, then the denominator must be a factor of 10^n - 1.
Second - If the sequence starts to repeat after some zeros, add the same number of zeros in the denominator. e.g.
0.005555 = 5/900
0.013131313 = 13/990
0.0004324324324...=432/999000=54/124875=2/46250.0004324324324...=432/999000=54/124875=2/4625
Third - Terminating decimals + Repeating Decimals. e.g.
2.31555555=2.31+0.00555555=231/100+5/9002.31555555=2.31+0.00555555=231/100+5/900
0.745454545=0.7+0.0454545=7/10+45/9900.745454545=0.7+0.0454545=7/10+45/990
Last one - The reciprocal of a prime number "p", except 2 and 5, has a repeating sequence of p-1 digits, or a factor of p-1 digits. e.g.
1/7=0.142857142857142857...1/7=0.142857142857142857... - As you can notice, the sequence has 6 digits = p-1 = 7-1 = 6 digits.
And if you multiply this fraction by a number you will only change the beginning of the sequence. e.g.
4/7=4∗1/7=0.57142857142857...
Borrowed content from- coelholds
Additional: You must know when the answers will be terminating or not.
1. Any integer divided by 2 or 5 or both (or their multiples) will always be a terminating decimal. Hence any number divided by 4, 8, or 10 will be a terminating decimal
2. Any integer divided by 9 will be a repeating decimal that will follow the repeating sequence mentioned in point 1 in the above section
3. Any integer divided by 3 and 7 will have a repeating sequence of 2 or 6 respectively, or factors of them (as per point 4 in above section)
4. 6 does not seem to have any defined pattern as 2/6=Repeating decimal with single-digit sequence and 3/6=0.5= Terminating decimal
Hope you liked this. Please do give me kudo so I know it helped you and gave you a useful strategy. It will also help me come up with more such posts.
Question to apply this strategy on: https://gmatclub.com/forum/if-each-of-t ... 59213.html
Kids concepts (No offence)