Number System Tutorial I: Integers, Fractions, Prime & Composite

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How to identify if a fraction gives terminating decimal or recurring decimal

If the simplified form of a fraction consist the prime factors 2 or 5 only in its denominator, then it is terminating decimal.
OR
If the denominator of the simplified form of any fraction consist at least one factor other than 2 or 5 then the fraction gives a recurring decimal.
Which among the following fractions is/are recurring decimal?
  • `371/128`
  • `403/160`
  • `3721/1024`
  • `8431/3125`
  • `3731/4311`
When observing the prime factors of the denominators in each fraction, it is easy to identify that the denominator of option E is a multiple of 3 and its numerator is not a multiple of 3. Therefore the prime factor 3 of the denominator should be there even after the simplification ( if it is possible).
Hence answer is `3731/4311`
We can check the prime factors of the rest of the denominators.
`128 = 2^7`
`160 = 2^5 * 5`
`1024 = 2^10`
`3125 = 5^5`
Here all of these denominators have the prime factors 2 or 5 only.

 

How to convert from recurring decimal to fraction?

Here we are going to find out the corresponding fraction of given recurring decimal. Group of recurring decimals are classified in to two.

Type I group of recurring decimals

In this type of recurring decimals the repeating digit/a group of digits, starts repeating immediately after the decimal point, ie. There is not any digit in the decimal part as non-repeating. Eg: `0.bar(54), 0.bar(524), 0.bar(1234)` etc

Method of conversion

Find the corresponding fraction of `0.bar(5)`
Let `x = 0.bar(5) -> Equation(1)`
`(1) * 10 -> 10x = 5.bar(5) -> Equation(2)`
`(2)-(1) -> 9x = 5`
`x = 5/9`
Therefore `0.bar(5)=5/9`
Find the corresponding fraction of `0.bar(25)`
Let `x = 0.bar(25) -> Equation(1)`
`(1) * 100 -> 100x = 25.bar(25) -> Equation(2)`
`(2) - (1) -> 99x = 25`
`x = 25/99`
Therefore `0.bar(25) = 25/99`'

Easy approach in converting Type I recurring decimal to fraction

`0.bar(ab) = "repeating group (here ab)"/"as many 9's as the number of digits in repeating group"`

Examples in converting recurring decimal to fraction

`0.bar(13) = 13/99`
`0.bar(251) = 251/999`
`0.bar(1234) = 1234/9999`

Type II group of recurring decimals

In this type the repeating digit/group of digits starts after a digit/ some digits of non-repeating digits. Eg. `0.1bar(6) (0.1666...), 0.03bar(8) (0.03888...), 0.45bar(13) (0.45131313...)`

Method of conversion

Find the corresponding fraction of `0.1bar(6)`
Let `x = 0.1bar(6) -> Equation(1)`
Take the non-repeating digit to the left side (integral portion) of the decimal point.
`(1) * 10 -> 10x = 1.bar(6) -> Equation(2)`
Take on repeating group/digit to the left side of the decimal point.
`(2) * 10 -> 100x = 16.bar(6) -> Equation(3)`
`(3)-(2) -> 90x = 15`
`x = 15/90 = 1/6`
Therefore `0.1bar(6)=1/6`
Find the corresponding fraction of `0.12bar(345)`
Let `x = 0.12bar(345) -> Equation(1)`
Take the non-repeating digit to the left side (integral portion) of the decimal point.
`(1) * 100 -> 100x = 12.bar(345) -> Equation(2)`
Take on repeating group/digit to the left side of the decimal point.
`100000x = 12345.bar(345) -> Equation(3)`
`(3)-(2) -> 99900x = (12345-12) = 12333`
`x = 12333/99900`
Therefore `0.12bar(345)=12333/99900`

Easy approach in converting Type II recurring decimal to fraction

`0.ab bar(cde) = `("entire decimal group" - "non-repeating decimal group")/"as many 9's as the number of repeating digits in the decimal part with as many 0's as the number of non-repeating digits in the decimal part"

Examples in converting Type II recurring decimal to fraction

`0.12bar(345) = (12345 - 12)/99900 = 12333/99900`
`0.8bar(45) = (845-8)/990 = 837/990`

More examples

Find the corresponding fraction of `3.2bar(3)`
`3.2bar(3) = 3 + 0.2bar(3)`
`3 + (23 - 1)/90 = 3 + 21/90 = 3 + 7/30`
`=97/30`
Find `0.3bar(7) + 0.4bar(5) + 0.1bar(6)`
`0.3bar(7) = 34/90`
`0.4bar(5) = 41/90`
`0.1bar(6) = 15/90`
`0.3bar(7) + 0.4bar(5) + 0.1bar(6) = 34/90 + 41/90 + 15/90 = 90/90 = 1`

Fractions

Any number which can be expressed in the form p/q, where p and q are Natural Numbers is called a fraction. Example: `1/2, 3/2, 5` etc. There are three types of fractions.

Proper Fractions

If the numerator of a fraction is lesser than the denominator, then it is a proper fraction. Hence the value of a proper fraction should lie in between 0 and 1. Eg. `1/3, 2/5, 6/13` etc

Improper fractions

If the numerator of a fraction is greater than or equal to its denominator then it is an improper fraction. Hence the value of any improper fraction is greater than or equal to 1. Eg. `3/2, 10/7, 5` etc

Mixed Fractions

Basically a mixed fraction is another expression of a corresponding improper fraction. Example: `5/4` is an improper fraction. I can be expressed in the following manner too.
`5/4 = (4+1)/4 = 1 + 1/4 = 1 1/4`, Here 1 is the natural number part and `1/4` is the proper fractional part.

Comparison of fractions

Comparison of different types of fractions is a basic requirement in Data Interpretation. Some direct questions from the comparison concept also can be expected in your exam. Hence you must be familiar with different methods for the comparison of fractions.

Fraction comparison using cross multiplication

Let's look at example to see how to compare two fractions quickly using cross multiplication.

Which is greater, `3/7` or `7/16` ?

For the comparison of any two fractions, it is quite easy to apply the cross multiplication method. Here, multiply the numerator of the first fraction with the denominator of the second fraction. This product is representing the first fraction. ie. `3 xx 16 = 48`

Similarly multiply the numerator of the second fraction with the denominator of the first fraction and this product is representing the second fraction. `ie. 7 xx 7 = 49`

Here `49 > 48`, therefore `7/16 > 3/7`.
Compare `101/200` and `51/100` ?
`101 xx 100 = 10100 -> "Represents 1st Fraction"`
`51 xx 200 = 10200 -> "Represents 2nd Fraction"`
`10200 > 10100, :. 51/100 > 101/200`.

Fraction comparison using denominator equalization

In a group of fractions, if the denominators of all the factions are equal then the largest fraction consist the largest numerator.

Let's look at example to see how to compare two fractions quickly using denominator equalization.

Compare the fractions, `3/5`, `7/10` and `21/25` ?
Here we can equate the denominators of all fractions to the LCM of denominators.
LCM of denominators = LCM (5, 10, 25) = 50
`3/5 = (3xx10)/(5xx10) = 30/50`
`7/10 = (7xx5)/(10xx5) = 35/50`
`21/25 = (21xx2)/(25xx2) = 42/50`
`:. 21/25 > 7/10 > 3/5`

Fraction comparison using numerator equalization

In a group of fractions if the numerators are equal then the largest fraction consist of the least denominator.

Let's look at example to see how to compare two fractions quickly using numerator equalization.

Compare the fractions `3/5`, `7/10` and `21/60`?
Here we can equate the numerators of all fractions to the LCM of numerators.
LCM of denominators = LCM (3, 7, 21) = 21
`3/5 = (3xx7)/(5xx7) = 21/35`
`7/10 = (7xx3)/(10xx3) = 21/30`
`21/60 ("no need to change")`
`:. 7/10 > 3/5 > 21/60`

Fraction comparison using special property of proper fractions

In a set of proper fractions if the differences between the numerator and denominator in each fraction are equal, then the greatest fraction has the greatest numerator and denominator.

Let's look at an example to see how to compare two fractions quickly using special property of proper fractions.

Compare `3/7`, `11/15`, `23/27`, `25/29` and `107/111`.?
By observation, the differences between numerator and denominator in each fraction are equal to 4. Hence `107/111 > 25/29 > 23/27 > 11/15 > 3/7`.

Fraction comparison using special property of improper fractions

In a set of improper fractions if there is a common difference between the numerator and denominator in each fraction, then the largest fraction has the least numerator and denominator.

Let's look at an example to see how to compare two fractions quickly using special property of improper fractions.

Compare `19/17`, `5/3`, `103/101`, `1015/1013` and `7/5`.?
Here the differences between the numerator and denominator in each fraction are equal. Therefore, `5/3 > 7/5 > 19/17 > 103/101 > 1015/1013`.

General approaches for fraction comparison

More often it is not possible to find any particular relations between the numerators and denominators in the given set of fraction. Then we have to approach the comparison in a general way of comparison.

Approach I: Decimal Value form

In this method we are converting the given set of fractions in its corresponding decimal values.

Compare the fractions `5/11`, `1/3`, `4/9`, `3/7`, `2/5` and `5/12`.?
Convert the fractions into its decimal values.
`5/11 = 0.bar(45)`
`1/3 = 0.bar(3)`
`4/9 = 0.bar(4)`
`3/7 = 0.bar(428571)`
`2/5 = 0.4`
`5/12 = 0.41bar(6)`
`:. 5/11 > 3/7 > 5/12 > 4/9 > 2/5 > 1/3`
Which among the given fractions is greatest. `1047/523`, `501/251`, `869/436`, `2041/1021`?
In the given set of fractions it is not an easy task to find out the corresponding decimal value of each fraction. Hence we are applying another one method for finding the largest among them. From the given fractions, it is easy to approximate them in the following manner;
`1047/523 > 2`
`501/ 251 < 2`
`869/ 436 < 2`
`2041/1021 < 2`
Therefore the largest fraction is `1047/523`.
Approach II: Percentage comparison

In this method, we are finding the approximate percentage value represented by each of the fractions and then comparing them.

Arrange the following fractions in ascending order. `9/24`, `5/17`, `92/366`, `70/176`?
`9/24`: 9 is approximately 34% of 24
`5/17`: 5 is approximately 30% of 17
`92/366`: 92 is approximately 25% of 366
`70/176`: 70 is approximately 40% of 176
`:. 92/366 < 5/17 < 9/24 < 70/176`.
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