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

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?
371128371128
403160403160
3721102437211024
8431312584313125
3731431137314311

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 3731431137314311

We can check the prime factors of the rest of the denominators.

128=27128=27
160=25⋅5160=25⋅5
1024=2101024=210
3125=553125=55
Here all of these denominators have the prime factors 2 or 5 only.

Method of conversion

Find the corresponding fraction of 0.5¯${\displaystyle 0.\overline{5}}$

Let x=0.5¯→Equation(1)${\displaystyle x=0.\overline{5}\to Equation\left(1\right)}$
(1)⋅10→10x=5.5¯→Equation(2)${\displaystyle \left(1\right)\cdot 10\to 10x=5.\overline{5}\to Equation\left(2\right)}$
(2)−(1)→9x=5${\displaystyle \left(2\right)-\left(1\right)\to 9x=5}$
x=59${\displaystyle x=\frac{5}{9}}$
Therefore 0.5¯=59${\displaystyle 0.\overline{5}=\frac{5}{9}}$

Find the corresponding fraction of 0.25¯¯¯¯${\displaystyle 0.\overline{25}}$

Let x=0.25¯¯¯¯→Equation(1)${\displaystyle x=0.\overline{25}\to Equation\left(1\right)}$
(1)⋅100→100x=25.25¯¯¯¯→Equation(2)${\displaystyle \left(1\right)\cdot 100\to 100x=25.\overline{25}\to Equation\left(2\right)}$
(2)−(1)→99x=25${\displaystyle \left(2\right)-\left(1\right)\to 99x=25}$
x=2599${\displaystyle x=\frac{25}{99}}$
Therefore 0.25¯¯¯¯=2599${\displaystyle 0.\overline{25}=\frac{25}{99}}$'

Easy approach in converting Type I recurring decimal to fraction

0.ab¯¯¯=repeating group (here ab)as many 9's as the number of digits in repeating group${\displaystyle 0.\overline{ab}=\frac{\text{repeating group (here ab)}}{\text{as many 9's as the number of digits in repeating group}}}$

Examples in converting recurring decimal to fraction

0.13¯¯¯¯=1399${\displaystyle 0.\overline{13}=\frac{13}{99}}$
0.251¯¯¯¯¯=251999${\displaystyle 0.\overline{251}=\frac{251}{999}}$
0.1234¯¯¯¯¯¯¯=12349999${\displaystyle 0.\overline{1234}=\frac{1234}{9999}}$

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. 0.16¯(0.1666...),0.038¯(0.03888...),0.4513¯¯¯¯(0.45131313...)${\displaystyle 0.1\overline{6}(0.1666...),0.03\overline{8}(0.03888...),0.45\overline{13}(0.45131313...)}$

Method of conversion

Find the corresponding fraction of 0.16¯${\displaystyle 0.1\overline{6}}$

Let x=0.16¯→Equation(1)${\displaystyle x=0.1\overline{6}\to Equation\left(1\right)}$
Take the non-repeating digit to the left side (integral portion) of the decimal point.
(1)⋅10→10x=1.6¯→Equation(2)${\displaystyle \left(1\right)\cdot 10\to 10x=1.\overline{6}\to Equation\left(2\right)}$
Take on repeating group/digit to the left side of the decimal point.
(2)⋅10→100x=16.6¯→Equation(3)${\displaystyle \left(2\right)\cdot 10\to 100x=16.\overline{6}\to Equation\left(3\right)}$
(3)−(2)→90x=15${\displaystyle \left(3\right)-\left(2\right)\to 90x=15}$
x=1590=16${\displaystyle x=\frac{15}{90}=\frac{1}{6}}$
Therefore 0.16¯=16${\displaystyle 0.1\overline{6}=\frac{1}{6}}$

Find the corresponding fraction of 0.12345¯¯¯¯¯${\displaystyle 0.12\overline{345}}$

Let x=0.12345¯¯¯¯¯→Equation(1)${\displaystyle x=0.12\overline{345}\to Equation\left(1\right)}$
Take the non-repeating digit to the left side (integral portion) of the decimal point.
(1)⋅100→100x=12.345¯¯¯¯¯→Equation(2)${\displaystyle \left(1\right)\cdot 100\to 100x=12.\overline{345}\to Equation\left(2\right)}$
Take on repeating group/digit to the left side of the decimal point.
100000x=12345.345¯¯¯¯¯→Equation(3)${\displaystyle 100000x=12345.\overline{345}\to Equation\left(3\right)}$
(3)−(2)→99900x=(12345−12)=12333${\displaystyle \left(3\right)-\left(2\right)\to 99900x=(12345-12)=12333}$
x=1233399900${\displaystyle x=\frac{12333}{99900}}$
Therefore 0.12345¯¯¯¯¯=1233399900${\displaystyle 0.12\overline{345}=\frac{12333}{99900}}$

Easy approach in converting Type II recurring decimal to fraction

0.abcde¯¯¯¯¯=${\displaystyle 0.ab\overline{cde}=}$("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

0.12345¯¯¯¯¯=12345−1299900=1233399900${\displaystyle 0.12\overline{345}=\frac{12345-12}{99900}=\frac{12333}{99900}}$
0.845¯¯¯¯=845−8990=837990${\displaystyle 0.8\overline{45}=\frac{845-8}{990}=\frac{837}{990}}$

More examples

Find the corresponding fraction of 3.23¯${\displaystyle 3.2\overline{3}}$

3.23¯=3+0.23¯${\displaystyle 3.2\overline{3}=3+0.2\overline{3}}$
3+23−190=3+2190=3+730${\displaystyle 3+\frac{23-1}{90}=3+\frac{21}{90}=3+\frac{7}{30}}$
=9730${\displaystyle =\frac{97}{30}}$

Find 0.37¯+0.45¯+0.16¯${\displaystyle 0.3\overline{7}+0.4\overline{5}+0.1\overline{6}}$

0.37¯=3490${\displaystyle 0.3\overline{7}=\frac{34}{90}}$
0.45¯=4190${\displaystyle 0.4\overline{5}=\frac{41}{90}}$
0.16¯=1590${\displaystyle 0.1\overline{6}=\frac{15}{90}}$
0.37¯+0.45¯+0.16¯=3490+4190+1590=9090=1${\displaystyle 0.3\overline{7}+0.4\overline{5}+0.1\overline{6}=\frac{34}{90}+\frac{41}{90}+\frac{15}{90}=\frac{90}{90}=1}$

Fractions

Any number which can be expressed in the form p/q, where p and q are Natural Numbers is called a fraction. Example: 12,32,5${\displaystyle \frac{1}{2},\frac{3}{2},5}$ 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. 13,25,613${\displaystyle \frac{1}{3},\frac{2}{5},\frac{6}{13}}$ 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. 32,107,5${\displaystyle \frac{3}{2},\frac{10}{7},5}$ etc

Mixed Fractions

Basically a mixed fraction is another expression of a corresponding improper fraction. Example: 54${\displaystyle \frac{5}{4}}$ is an improper fraction. I can be expressed in the following manner too.
54=4+14=1+14=114${\displaystyle \frac{5}{4}=\frac{4+1}{4}=1+\frac{1}{4}=1\frac{1}{4}}$, Here 1 is the natural number part and 14${\displaystyle \frac{1}{4}}$ 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, 37${\displaystyle \frac{3}{7}}$ or 716${\displaystyle \frac{7}{16}}$ ?

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. 3×16=48${\displaystyle 3\times 16=48}$

Similarly multiply the numerator of the second fraction with the denominator of the first fraction and this product is representing the second fraction. ie.7×7=49${\displaystyle ie.7\times 7=49}$

Here 49>48${\displaystyle 49>48}$, therefore 716>37${\displaystyle \frac{7}{16}>\frac{3}{7}}$.

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