Fractions and Decimals basic concepts part 1

Fractions and decimal numbers


Fractions are not a separate type of numbers. Rather a fraction is a special representation of a number or other mathematical entities in general. Examples of number fractions are,

$\displaystyle\frac{5}{7}$, $\displaystyle\frac{20}{8}$,$\displaystyle\frac{2}{1}$, $\displaystyle\frac{4}{5}$. 

In contrast, decimal numbers represent a very important form of numbers one part of which is less than 1 but greater than zero. Examples of decimal numbers are,

$0.25$, $1.5674$, $0.33333...$.

Decimal numbers

A decimal number, in general has two parts:

The integer part on the left of the decimal point, and the fractional part on the right of the decimal point.

For example, in $72.3241$, the integer part is $72$ and the fractional part is, $0.3241$.

Very formally the fractional part is called the mantissa, while the integer part is called the characteristic. But being practical people, we will always call the integer part as the integer part and the part of the decimal number excluding the integer part as, the fractional part.

The special characteristic of the fractional part is, its value is always greater than zero and less than 1. In other words,

$0 \lt \text{Fractional part} \lt 1$.

It cannot be equal to 0 or 1. In case that happens, the fractional part doesn't remain any more a fractional part.

Place values of the fractional part of a decimal number

We know the place values of an integer follow the pattern,

$10^0$, $10^1$, $10^2$, $10^3$,...., the power of $10$ increasing by 1 from right to left all the way up indefinitely.

For the fractional part a similar and complementary system makes the place value system whole.

The power of 10 in place values for the fractional part decreases by 1 starting from $-1$ all the way down indefinitely as a mirror image of the positive powers of 10 in the place values for integers, but in this case of fractional part, from left to right.

Thus the first digit on the left of the decimal point, the unit's digit with power of 10 as $0$, is the center point.

All place values on its left have monotonously increasing (each increment step $+1$) positive powers of 10. Conversely, all place values on its right have monotonously decreasing (each decrement step $-1$) negative powers of 10.

The fractional part place values are,

$10^{-1}$, $10^{-2}$, $10^{-3}$, $10^{-4}$, $10^{-5}$, $10^{-6}$.... decreasing all the way down the negative part of the number line.

Thus the integer part has place values with powers of 10 as positive Whole numbers, [0, 1, 2, 3, 4, 5, ....] and the fractional part has place values with powers as negative of the natural numbers, [-1, -2, -3, -4, ......]. These two parts then perfectly join together to form place value powers of 10 as the complete number line with non-fraction values.

An example,

$43.82 = 4\times{10^1} + 3\times{10^0} + 8\times{10^{-1}} + 2\times{10^{-2}}$.

With this well-defined place value system, all the rules and methods of mathematical operations work in the same way for the fractional part as the integer part.

A point to note: For addition or subtraction of decimal numbers, you need to be careful in aligning the two numbers (or operands) on the decimal point. This is obvious, but still needs to be remembered to avoid chances of error.

In these sessions of fractions and decimals, we will naturally be focusing on the fractional part of the decimals and rarely on the integer part.

And in this first session we will deal with more of fractions because there are more of relevant practical concepts to explain on fractions compared to decimals.


Structure of a fraction

A fraction has two components: a numerator at the top and a denominator at the bottom. The two are separated by a dividing line. In case a fraction needs to be evaluated, the division operation needs to be completed.

For example, in the fraction $\displaystyle\frac{2}{5}$, the numerator is $2$ and the denominator is $5$.

The following section on types of fractions is basically of academic interest and may be skipped.

Types of fractions

This concept is of academic interest, but still it is better to know.

In a fraction when the numerator is less than the denominator with no common factor between them, the fraction is called a proper fraction in minimized form.


$\displaystyle\frac{4}{5}$, $\displaystyle\frac{3}{13}$, $\displaystyle\frac{45}{64}$, are all proper fractions

We will be dealing mostly with proper fractions in minimized form, that is, without any common factor between numerator and denominator and numerator less than the denominator.

In contrast, in a fraction, if the numerator is equal to or larger than the denominator, the fraction is called an improper fraction.


$\displaystyle\frac{14}{5}$, $\displaystyle\frac{53}{13}$, are both improper fractions.

A fraction with an integer and fraction part is called a mixed fraction.

$2\displaystyle\frac{4}{5}$, $4\displaystyle\frac{1}{13}$, are both mixed fractions.

In fact these two fractions are same as the two examples of improper fractions. Bothway conversions between the two forms may easily be carried out.

When converting an improper fraction to mixed fraction form, just divide the numerator by the denominator with the quotient forming the integer part and the remainder forming the new numerator.

When both the numerator and the denominator are integers we have simple fractions. All the fraction examples we have seen till now are simple fractions.

In compound fractions, the numerator and denominator will be expressions in fractions themselves. An example,

$\displaystyle\frac{\displaystyle\frac{1}{2} + \displaystyle\frac{1}{3}}{\displaystyle\frac{4}{5} +\displaystyle\frac{3}{7} }$

A fraction is not always evaluated

It is important to note, not always we need to evaluate the value of the fraction by carrying out the division operation. In such cases, the fraction form itself holds some special additional information. For example, when we say, "the ratio of the ages of the son and the father is 1 : 4 or, $\displaystyle\frac{1}{4}$, the fraction represents the multiplicative comparison between the son's age and the father's age in the form of a ratio which is a separate topic in itself. 

In this ratio, the fraction is not to be evaluated to its value of $0.25$. It simply means here, the father's age is four times the son's age, not more nor any less.

Otherwise, when fractions are meant to be evaluated, in those cases also the evaluation is delayed as much much as possible and simplification is carried out by cancellation of factors common between numerator and denominator. Only when the fraction reaches its simplest form, we evaluate it, thus saving much calculation labor. This in fact is our Delayed evaluaion technique, which is a very useful rich algebraic technique that can be applied in any type of simplification.

In many cases, we just leave the result in fraction form without evaluating it. The reason for this general practice is,

We are more comfortable dealing with the fraction formed by two integers rather than its evaluated decimal form.

For example, it is much easier to deal with the fraction, $\displaystyle\frac{1}{3}$ rather than its evaluated non-terminating decimal equivalent of $0.3333333...$.

So in general we observe,

Fractions are manipulated and transformed following mathematical rules but usually are not evaluated to their equivalent decimal values.

Meaning of a fraction

Let us take the fraction $\displaystyle\frac{2}{5}$. What does it mean?

A single integer represents a measure of some quantity. It may be age of a person, length of a straight line or for that matter number of neurons in your brain. But what about a fraction? what does it represent?

The basic core meaning of a number fraction $\displaystyle\frac{2}{5}$ is just,

Two parts out of 5.

In other words, the numerator represents the portions or number of parts which the fraction itself represents, while the denominator represents the whole. In a rather clever way, taking the help of two integers, through a single representation, relation between the "part and the whole" is expressed through a fraction.

For example when we encounter a statement like, "Ravi gave away two fifths of his month's salary to the School for the mentally challenged", we know immediately, Ravi gave away two portions of his whole amount of month's salary. Or, more mathematically, "Ravi gave away, $\displaystyle\frac{2}{5}$ th of his month's salary". 

A fraction thus represents the number of parts of the whole (number of parts), it does not usually represent the actual measure of quantity.

Evaluation of fractions to decimals

If a number fraction such as $ \displaystyle\frac{5 + 1}{2 + 3}$ is evaluated, as a general rule, first the numerator is evaluated and converted to a single number and the same is done for the denominator. At the second step the numerator is divided by the denominator.

$ \displaystyle\frac{5 + 1}{2 + 3}=\displaystyle\frac{6}{5} = 1.2$.

On evaluation of fractions in which both the numerator and the denominatore are integers, two types of decimals are generated.

Terminating decimal:

When the decimal part is terminating, it is called a terminating decimal. Examples,

$ \displaystyle\frac{10}{8}=\frac{5}{4}=1.25$,

Or a second fraction, $ \displaystyle\frac{5}{8}= 0.625$

In general, in a fraction, only if the denominator contains 2 or 5 or powers of 2 or 5, and no other prime number, the resulting decimal always terminates.

The above two fraction evaluations are the examples.

Non-terminating repeating decimal:

On evaluation of the fraction, if the decimal does not terminate but continues indefinitely to the right, we get a non-terminating decimal.

When evaluating a fraction, if a non-terminating decimal is generated, it must always have a group of digits that repeats continuously.

These form the non-terminating repeating decimals. Example,

$\displaystyle\frac{1}{3}=0.33333333..... $,


In the first, the digit 3 repeats indefinitely to the right and is written in short form as,

$ \displaystyle\frac{1}{3}= 0.\bar{3}$.

For the second fraction evaluaion the group of digits, $428571$ repeats indefinitely to the right and the decimal is written similarly as,


Condition for non-terminating repeating decimal:

If in a fraction the denominator contains any prime number as a factor other than 2 and 5, the evaluated decimal will be a non-terminating repeating decimal. Examples,

$ \displaystyle\frac{2}{7}= 0.\overline{285714}$, $ \displaystyle\frac{1}{11}=0.\overline{09}$, $\displaystyle\frac{1}{13}=0.\overline{076923}$

For each of these decimal numbers the overlined sequence of digits repeat again and again in the result indefinitely.

These two types of decimal numbers and all integers form, what we call, the set of rational numbers.

Rational numbers

Rational numbers are those numbers that can always be expressed in the form of a fraction,

$\text{A rational number} = \displaystyle\frac{p}{q} $ where $p$ and $q$ are two non-zero integers.


In a rational number, if there is a decimal part, the decimal part either would terminate or would be repeating non-terminating type.


$ 3$, $456$, $2.5$, $-7$, $1.\overline{3}$, $0.\overline{09} $

Irrational numbers

There are decimal numbers where the decimal part is non-terminating but also non-repeating. These are a special class of numbers that are not amenable to easy evaluation or mathematical operations. The special class of numbers are called Irrational numbers.

An important characteristic of these numbers is, an irrational number cannot be expressed as a proper fraction,

$\text{An irrational number} \neq \displaystyle\frac{p}{q}$, where $p$ and $q$ are two non-zero integers.

Examples of irrational numbers are,

$ 0.01001000100001....$, $\sqrt{2}$, $\displaystyle\frac{1}{1 +\sqrt{13}}$


$\sqrt{2}$,$\sqrt{3}$ or square root of any prime number are commonly known as Surds and form an important class of known irrational numbers.

Basic mathematical operations on irrational numbers:


When an irrational number is added to a rational number, result is always an irrational number.


$\sqrt{2} + 1$, an irrational number.

$(\sqrt{2} + \sqrt{3}) + 435$, an irrational number.

When an irrational number is added to another irrational number the result may either be an irrational number or a rational number.

$ (3-\sqrt{2}) + (3 + \sqrt{2})= 2\times{3} = 6$, a rational number.

$ (3-\sqrt{2}) + (3 + 2\sqrt{2})=6 + \sqrt{2}$, an irrational number.


when an irrational number is subtracted from another rational number, the result is always an irrational number.


$ 5 - \sqrt{3}$, an irrational number.

But when an irrational number is subtracted from another irrational number, the result may be either a rational number or an irrational number.


$(3+\sqrt{2}) - (3-\sqrt{2})=2\times{\sqrt{2}}$, an irrational number.

$(5+\sqrt{2}) - (3 + \sqrt{2})= 5 - 3=2$, a rational number.


When an irrational number is multiplied with a rational number, result is always an irrational number.


$\sqrt{3}\times{3}=3\sqrt{3}$, an irrational number.

But when an irrational number is multiplied with another irrational number, the result may be either an irrational number or in special cases, a rational number.


$ \sqrt{3}\times{\sqrt{2}}=\sqrt{3\times{2}}=\sqrt{6}$, an irrational number.

$\sqrt{3}\times{\sqrt{3}}=\sqrt{3\times{3}}=\sqrt{9}=3$, a rational number.


$=5^{_2} -(\sqrt{3})^{_2}$


$=22$, a rational number.


$=5^{_2} -5\sqrt{2} + 5\sqrt{3} -\sqrt{3\times{2}}$

$=25+ 5(\sqrt{3}-\sqrt{2})-\sqrt{6}$, an irrational number.

Surd rationalization technique

To convert the irrational numerator or denominator of a fraction of the form $a+\sqrt{b}$, where a and b are rational numbers, it is to be multiplied with $a-\sqrt{b}$ resulting $a^2 - b$ which is a rational number. This is called Surd rationalization technique.

Example: Rationalize $\displaystyle\frac{1}{2 - \sqrt{3}}$.

$\displaystyle\frac{1}{2 - \sqrt{3}}=\displaystyle\frac{2 + \sqrt{3}}{(2 - \sqrt{3})(2 + \sqrt{3})}$

$=\displaystyle\frac{2 + \sqrt{3}}{2^2 - 3}$

$=2 + \sqrt{3}$, the denominator is eliminated.

In contrast, all the basic mathematical operations on rational numbers produce only rational number results.

This is where rational numbers differ significantly from irrational numbers.

Real numbers

The set of rational numbers and irrational numbers together form the set of real numbers.

Basic mathematical operations on fractions

As both fractions and most of the decimals are rational numbers, the four basic mathematical operations as well as the concepts of factorization, HCF and LCM hold good for these in ways similar to integers. But as structures of fractions and decimals are a little different from integers, the actual methods differ slightly, with underlying method and dependence on place value mechanism remaining same.

Addition of fractions:


$ \displaystyle\frac{1}{4} + \displaystyle\frac{3}{10} = \displaystyle\frac{1\times{5}}{4\times{5}} + \displaystyle\frac{3\times{2}}{10\times{2}}$

The first step of adding two fractions is to make the denominator of the two factors same.

This equalization we have named as base equalization technique and covered in details in our post on Base equalization technique is indispensable for solving fraction problems.

To equalize the denominators, two Rules are used.

Rule 1: In any fraction, you can always multiply or divide each of the numerator and denominator by the same number without changing the value of the fraction. Example,

$\displaystyle\frac{1}{4} = \frac{1\times{5}}{4\times{5}} = \frac{5}{20}=\frac{1}{4}$

Rule 2: to add or subtract two fractions, the denominators of the fractions must be made equal.

The reason is, after making the denominator same, $(1\div{\text{denominator}})$ can be taken as a common factor out of the two fractions leaving the integer numerators for straightforward addition or subtraction.

To make the denominators of two fractions reach a common value, the target value is taken logically as the LCM of the two denominators.

In the above example, LCM of 4 and 10 is 20. So 20 is the target denominator value for the first fraction. As the first denominator is 4, it is then to be multiplied with 5 (and also the numerator is to be multiplied with 5) to make the denominator 20 without changing the value of the fraction. For the second fraction, the multiplying factor is 2 as the target LCM is 20 and the denominator is 10.

Mathematically the transformed numerator is,


while the transformed denominator is the LCM of the two original denominators.

Thus at the next step, the addition becomes simple,

$\displaystyle\frac{1}{4} + \displaystyle\frac{3}{10} $

$\hspace{5mm}= \displaystyle\frac{1\times{5}}{4\times{5}} + \displaystyle\frac{3\times{2}}{10\times{2}}$

$\hspace{5mm}=\displaystyle\frac{5}{20} + \displaystyle\frac{6}{20}$



Subtraction of fractions:

This is similar to addition and follows the same mechanisms. Example,

$ \displaystyle\frac{3}{10} - \displaystyle\frac{1}{4}$

$\hspace{5mm}= \displaystyle\frac{3\times{2}}{10\times{2}} - \displaystyle\frac{1\times{5}}{4\times{5}}$

$\hspace{5mm}=\displaystyle\frac{6}{20} - \displaystyle\frac{5}{20}$



Multiplication of fractions:

This is simply multiplying the two numerators and the the two denominators to form the result fraction. Though technically this is simple, in practice multiplication of two or more fractions involve a second step of canceling the common factors between the numerator and denominator of the result. Sometimes this becomes a little complex and error-prone process. That's why we would take it up under simplification later.



The first step is ok, but before actually carrying out the multplication we invariably go for canceling the common factor between the numerator and denominator first. This effectively is,

Canceling out the HCF of numerator and denominator.

$ \displaystyle\frac{5}{18}\times{\frac{27}{35}}$



At this step we have caneled the common 9 between 27 in numerator and 18 in denominator.


It is a good practice to cancel only one large common factor in one step. Chances of error reduce this way. In the first step we have canceled out common factor 9, and in the second step we will cancel out the second common factor 5 between the numerator and the denominator.

$ \displaystyle\frac{5}{18}\times{\frac{27}{35}}$



$\hspace{5mm}=\displaystyle\frac{1\times{3}}{2\times{7}} $

So finally we get,


Division of fractions:

The division is exactly same as the multiplication except that in the first step you need to invert the second fraction which is the divisor.


$\displaystyle \frac{5}{18}\div{\frac{35}{27}}$



Comparison of fractions:

A rather frequently demanded ability is to compare two fractions to decide which is the larger and which smaller. In two ways you can do this,

Method 1: In the first method you need to make the two denominators equal and you know how to do this by fixing the target number as the LCM of the two numbers and then multiplying by suitable factors. This method can very well be applied on more than two fractions also. Example,

Problem: Rank the fractions in descending order.

$ \displaystyle\frac{2}{5}$, $\displaystyle\frac{3}{7}$, $\displaystyle\frac{7}{20}$

By transforming the denominators of three fractions to their LCM 140, we get,

$ \displaystyle\frac{2}{5}$, $\displaystyle\frac{3}{7}$, $\displaystyle\frac{7}{20}$

$\hspace{5mm}\Rightarrow\displaystyle\frac{56}{140}$, $\displaystyle\frac{60}{140}$, $\displaystyle\frac{49}{140}$

As the denominators are same these can be ignored altogether and the fractions can be ranked by comparison of the numerators only. Thus, the second fraction turns out to be largest, the first one next highest and the last one smallest,

$ \displaystyle\frac{2}{5}, \displaystyle\frac{3}{7}, \displaystyle\frac{7}{20}$

$\hspace{5mm}\Rightarrow\displaystyle\frac{56}{140}, \displaystyle\frac{60}{140}, \displaystyle\frac{49}{140}$

$\hspace{5mm}\Rightarrow\displaystyle\frac{60}{140} \gt \displaystyle\frac{56}{140} \gt \displaystyle\frac{49}{140}$

$\hspace{5mm}\Rightarrow\displaystyle\frac{3}{7} \gt \displaystyle \frac{2}{5} \gt \displaystyle\frac{7}{20}$

This is the natural and simpler method of comparing fractions as the basic method is similar to addition and subtraction.

Method 2: In this alternate method which you may need to use sometimes, instead of the denominator values the numerator values of the given fractions are equalized to the target value of LCM of the numerator values and correspondingly the denominators are modified.

$ \displaystyle\frac{2}{5}, \displaystyle\frac{3}{7}, \displaystyle\frac{7}{20}$

$\hspace{5mm}\Rightarrow\displaystyle\frac{42}{105}, \displaystyle\frac{42}{98}, \displaystyle\frac{42}{120} $

We have transformed now the numerators to the LCM of 2, 3 and 7, that is, 42. Accordingly the denominators have also been modified. But in this case, with equal numerators, the larger denominator will indicate a smaller value of fraction. It is just the reverse of the previous case. So we have the fractions in descending order of values as,

$ \displaystyle\frac{2}{5}, \displaystyle\frac{3}{7}, \displaystyle\frac{7}{20}$

$\hspace{5mm}\Rightarrow\displaystyle\frac{42}{105}, \displaystyle\frac{42}{98}, \displaystyle\frac{42}{120}$

$\hspace{5mm}\Rightarrow\displaystyle\frac{42}{98} \gt \displaystyle\frac{42}{105} \gt \displaystyle\frac{42}{120}$

$\hspace{5mm}\Rightarrow\displaystyle\frac{3}{7} \gt\displaystyle\frac{2}{5} \gt \displaystyle\frac{7}{20}$

We get the same result.


Try to use the first method of equalizing the denominators as this is natural and less error-prone method.

Decimals encompass all real numbers

As integers can be expressed as decimals such as $43 = 43.00$, the set of decimal numbers may be considered as equivalent to all real numbers. This is more so because irrational numbers, though are decimals, do not fall under any other real number category.

A question for you

What does the banner picture at the top represent? Is the representation completely correct?

We may discuss this point in the next session.

At the end we leave you with a bunch of problems to solve. Answers are at the end.

Problem exercise

Recommended time to complete the exercise is 18 minutes

Problem 1.

How much more is $\sqrt{12} + \sqrt{18}$ compared to $\sqrt{3} + \sqrt{2}$?

  1. $\sqrt{3} + 2\sqrt{2}$
  2. $2(\sqrt{3} - \sqrt{2})$
  3. $\sqrt{2} - 4\sqrt{3}$
  4. $2(\sqrt{3} + \sqrt{2})$

Problem 2.

The value of $\displaystyle\frac{1}{30} + \displaystyle\frac{1}{42} + \displaystyle\frac{1}{56} + \displaystyle\frac{1}{72} + \displaystyle\frac{1}{90} + \displaystyle\frac{1}{110}$ is,

  1. $\displaystyle\frac{1}{9}$
  2. $\sqrt{2}\displaystyle\frac{2}{27}$
  3. $\displaystyle\frac{6}{55}$
  4. $\displaystyle\frac{5}{27}$

Problem 3.

$3.\overline{36} - 2.\overline{05} + 1.\overline{33}$ equals,

  1. $2.64$
  2. $2.\overline{64}$
  3. $2.60$
  4. $2.\overline{61}$

Problem 4.

$\displaystyle\frac{1\displaystyle\frac{1}{4} \div{1\displaystyle\frac{1}{2}}}{\displaystyle\frac{1}{15} + 1 -\displaystyle\frac{9}{10}}$ is equal to,

  1. $5$
  2. $6$
  3. $3$
  4. $\displaystyle\frac{2}{5}$

Problem 5:

$\displaystyle\frac{1}{3 - \sqrt{8}} - \displaystyle\frac{1}{\sqrt{8} - \sqrt{7}} + \displaystyle\frac{1}{\sqrt{7} - \sqrt{6}}$

$\hspace{10mm}- \displaystyle\frac{1}{\sqrt{6} - \sqrt{5}} + \displaystyle\frac{1}{\sqrt{5} - 2} =$

  1. 2
  2. 3
  3. 4
  4. 5

Problem 6.

$(159\times{21} + 53\times{87} + 25\times{106})$ equals

  1. 16000
  2. 1060
  3. 60100
  4. 10600

Problem 7.

$\displaystyle\frac{0.8\overline{3} \div{7.5}}{2.3\overline{21} - 0.0\overline{98}}$ equals

  1. 0.1
  2. 0.6
  3. 0.05
  4. 0.06

Problem 8.

The value of $(\sqrt{72} - \sqrt{18}) \div{\sqrt{12}}$ is,

  1. $\displaystyle\frac{\sqrt{3}}{2}$
  2. $\displaystyle\frac{\sqrt{6}}{2}$
  3. $\displaystyle\frac{\sqrt{2}}{3}$
  4. $\sqrt{6}$

Problem 9.

The value of $\sqrt{900} + \sqrt{0.09}  - \sqrt{0.000009}$ is,

  1. 30.27
  2. 30.097
  3. 30.197
  4. 30.297

Problem 10.

The value of $(3 + \sqrt{8}) + \displaystyle\frac{1}{3 - \sqrt{8}} - (6 + 4\sqrt{2})$ is,

  1. $0$
  2. $1$
  3. $8$
  4. $\sqrt{2}$

Answers to Problem exercise

Problem 1: a: $\sqrt{3} + 2\sqrt{2}$

Problem 2: c: $\displaystyle\frac{6}{55}$

Problem 3: b: $2.\overline{64}$

Problem 4: a: 5

Problem 5: d: 5

Problem 6: d: 10600

Problem 7: c: 0.05

Problem 8: b: $\displaystyle\frac{\sqrt{6}}{2}$

Problem 9: d: 30.297

Problem 10: a: 0.

Note: Look out for the detailed solutions for these problems which we will publish shortly. Meanwhile you should try to solve the problems that basically are of SSC CGL competitive test level and are not easy. If you take more time to solve, don't bother, in any way solve yourself. Remember that's the best way to learn.