Decimals
A decimal is a number with a decimal point and it includes a whole number part and a decimal part. 1.2, 1.8, 3.425 are examples of decimals.
Consider the following blocks, each of which is divided into 10 equal parts. Each one of these parts represents $\frac{1}{10}$ of the whole unit, the whole block.
10 parts of Block 1 and 2 parts of Block 2 are shaded in red. In other words, 10 out of 10 parts are shaded in Block 1 and 2 out of 10 parts are shaded in Block 2. This is represented as 1.2 in decimal form, which means one whole and twotenths of a whole. This can be written as 1.2 = 1 + $\frac{2}{10}$
A decimal number is a representation of a fraction and vice versa. Consider the number line. The unit length between 1 and 2 is divided into 10 equal parts. We take 8 parts as shown in the following figure.
This point represents eight tenth more than one. We represent this as a decimal number, 1.8.
Here, 1.8 can be written as 1 + eight tenth
That is, 1.8 = 1 + $\frac{8}{10}$
We can see that a decimal number is a representation of a fraction and vice versa.
Examples on Decimals
Given below are some examples on decimals
Example 1:
3.245 is a decimal number where 3 is the whole number and 0.245 is the decimal part.
We can write 3.245 = 3 + two tenths + 4 hundredths + 5 thousandths
Example 2:
51. 9807 is a decimal number, where 51 represents the whole part and 0.9807 the decimal part.
51. 9807 = 51 + 9 tenths + 8 hundredths + 0 thousandths + 7 tenthousandths
= 51 + 9 tenths + 8 hundredths + 7 ten thousandths
If there is no whole number part, we write a 0 to the left of the decimal point. In the decimal number, 0. 234, there is no whole number part.
Example 3:
Write the fraction and decimal number representing the following figure.
Here, a box is made of 10 equal parts and 7 of them are shaded in blue. We can represent this as $\frac{7}{10}$ as a fraction and 0.7 as the decimal point.
The concept of comparing a number like 1000 and 1 gives us an basic idea that although 1000 is the same shape as 1 but quite bigger. This gives rise to decimals.
If 1000 is the same as 1 but a thousand times bigger, then 2.210 would be same as 2210 but of course a thousand times smaller.
A decimal number has two parts, a whole number part and a decimal part. Each digit of a decimal number has a place value in terms of powers of 10.
Place Value Decimals
The value of a digit depends upon its location or place in a number. The specific position of a digit in a number is called its place value. In the number 23, the digit 3 represents 3 ones, or 3; while the same digit might have a different value in some other number. Place value refers to the specific value of a digit in a decimal.
How is the place value decimal depicted?
Place value can be best depicted using a chart.
In the chart, each of the column heading represents a place value. If we move one more column to the left, the place value is increased ten times. If we move one column to the right, the place value is decreased ten times or is one tenth of the previous value. Our system of arithmatic is based on the number ten.
We create all numbers by collecting smaller numbers into groups of ten or multiples of ten. For example, 34 could be considered as 3 groups of tens and and a group of 4 ones.
34 = 3 tens + 4 ones
Example of Place Value
Let us try another example:
765.743
7 has been placed in the hundreds place
6 has been placed in the tens place
5 has been placed in the ones place
7 has been placed in the tenths place
similarly
4 has been placed in hundredths place
Like wise
3 has been placed in the thousandths place
We can also represent these as
700 + 60 + 5 + $\frac{7}{10}$ + $\frac{4}{100}$ + $\frac{3}{1000}$
Place value of digits in whole numbers:
We assign the place value to the digits from the right end starting the values as "units". The second last digit has a value 10. The third last digit has a value 100 and so on.
That is, for a whole number the right end digit has a place value 1, and the place value increases by powers of 10 towards left.
For example, look at the place values of digits in the following number:
1287652 = (1 x 1000000) + (2 x 100000) + (8 x 10000) + (7 x 1000) + (6 x 100) + (5 x 10) + (2 x 1)
Place value of digits in a decimal number:
The decimal part also has place values as powers of 10, where these powers will be negative. In a decimal number, the first digit to the right of the decimal point is called as a tenths place digit. That is, this digit has a place value onetenth ($\frac{1}{10}$). The second digit to the right of the decimal point is called a hundredths place digit.
That is, it has a place value onehundredth ( $\frac{1}{100}$ ). In this way, the place value decreases by powers of ten towards the right.
Consider a decimal number 12.345.
The whole number part is 12 and the decimal part is 0.345.
The whole number 12 has two units and one ten.
The digit "3" in the decimal part has the place value onetenth, 4 has a place value onehundredth and 5 has a place value onethousandth.
That is, three $\frac{1}{10}$ , four $\frac{1}{100}$ and five $\frac{1}{1000}$ are there in the given decimal number.
In other words, we write, 12.34 = 12 + 0.345
$= 12 + 3 \times \frac{1}{10} + 4 \times \frac{1}{100} + 5 \times \frac{1}{1000}$
Decimal Place Value Chart:
The following table illustrates the place value of decimal numbers. The digits in the whole number part have place values starting from units and increasing by powers of ten towards the left. The digits in the decimal part have place values starting from tenths and decreasing by powers of ten towards right.
Hundreds 
Tens 
Units 
Decimal point 
Tenths 
Hundredths 
Thousandths 
Tenthousandths 
Hundredthousandths 
Millionths 
1 
2 
5 
. 
5 
9 




5 
0 
6 
. 
8 





8 
9 
6 
. 
0 
0 
0 
5 


1 
2 
4 
. 
1 
0 
0 
0 
5 

0 
0 
0 
. 
5 
4 
8 
7 
2 
1 
Examples for Decimal Place Value:
Given below are some of examples for decimal place value.
Example 1:In the first row of the above table, we have the decimal number 125. 59, where the place values are represented as,
In other words, we can write,
$125.59 = (1 \times 100) + (2 \times 10) + (5 \times 1) + (5 \times \frac{1}{10}) + (9 \times \frac{1}{100})$
Similarly, 124.10005 can be written as,
$124.10005 = (1 \times 100) + (2 \times 10) + (4 \times 1) + (1 \times \frac{1}{10}) + (0 \times \frac{1}{100} ) + (0 \times \frac{1}{1000} ) + (0 \times \frac{1}{10000} ) + (5 \times \frac{1}{100000} )$
→ Read MoreAddition of decimal numbers is similar to addition of whole numbers. The decimal points must be in a line, so that all the digits in the decimal numbers should be in a line according to their place value.
Steps to Add Two Decimals
Step 1: Write the two decimal numbers one below the other keeping the decimal points lined up.
Step 2: Start adding from the right end, such that we add the digits at the same place together. This is similar to the addition of whole numbers.
Step 3: The decimal point will be in the same column.
Examples on Adding Decimals
Given below are some of examples that explain how to add decimal numbers.Example 1:
Add 11. 23 and 21. 42
Solution:
We write the numbers one below the other keeping the decimal points lined up
11.23 +
21.43
Now, we have to start adding from the right end.
11.23 +
21.43

32.66
Thus, 11. 23 + 21. 43 = 32. 66
Example 2:
Find the sum of 81.39 + 2.81 + 23.45
Solution:
We first write the numbers one below the other.
81.39 +
2.81
23.45
Now, we start adding the numbers from the right end similar to the addition of whole numbers
81.39 +
2.81
23.45

107.65
Fact about addition of decimal numbers
When we add two decimal numbers, the answer will have the same number of decimal digits as the given decimal numbers.Addition of a decimal number with a whole number
We can add a whole number and the whole number part of the decimal number keeping the decimal part the same.
Examples on addition of a decimal number with a whole number:
Given below are some examples that explain how to add a decimal number with a whole number.
Example 1:
Add 15 and 12.23
Solution:
To add a whole number and a decimal number, we have to add the whole number with the whole number part of the decimal number.
That is, 15 + 12 = 27
We leave the decimal part the same in the given decimal number
Thus, 15 + 12.23 = 27.23
Example 2:
Find the sum 21.346 + 28
Solution:
21+ 28 = 49
Thus, 21.346 + 28 = 49.346
Subtraction of decimal numbers is similar to the subtraction of whole numbers. The decimal points must be in a line, so that all the digits in the decimal numbers are in a line according to their place value.
Steps to Subtract Decimals
Step 1: If the decimal numbers are uneven numbers, put the required number of 0's at the right end of the decimal number such that both the decimal numbers have the same number of digits to the right of the decimal point.Step 2: Write the two decimal numbers one below the other, keeping the larger decimal number at the top and the smaller one below with the decimal points lined up.
Step 3: Subtract the digits starting from the right. This is similar to subtraction in whole numbers.
Examples on Subtracting Decimals
Given below are some examples that explain how to subtract decimal numbers.
Example 1:
Find 44.87  12.34
Solution:
Step 1: If the decimal numbers are uneven numbers, put the required number of 0's at the right end of the decimal number such that both the decimal numbers have the same number of digits to the right of the decimal point.
Here, both the numbers have same number of digits to the right of the decimal point
Step 2: Write the two decimal numbers one below the other, keeping the larger decimal number at the top and the smaller one below with the decimal points lined up
44.87 
12.34

Step 3: Subtract the digits starting from the right. This is similar to subtraction in whole numbers
44.87 
12.34

32.53
Multiplying decimal numbers is same as multiplying whole numbers. The product of two decimal numbers will be a decimal number.
Multiplying decimals by a whole number
To multiply a decimal number by a whole number, we ignore the decimal point and multiply the two whole numbers. Then put a decimal point in the product such that this product must have same number of digits to the right of decimal point as in the original decimal number.
Example 1:
Find the product 12 X 1.1
Solution:
We ignore the decimal point in the given decimal number and multiply with the whole number.
That is, 12 X 11=132
We have one decimal digit in the given decimal number. In other words, in 1.1, there is only one digit after the decimal point. Hence, the product must have a digit to the right of the decimal point. That is, we put a decimal point in 132 such that, there will be a decimal digit, 13.2
Hence, 12 X 1.1 = 13.2
Steps to Multiply Decimal Numbers
Step 1: Write the decimal numbers as whole numbers by removing the decimal point.Step 2: Multiply the whole numbers obtained from step 1.
Step 3: Count the total number of digits after the decimal point in both the decimal numbers.
Step 4: Take the product from step 2. Count from the right end and put a decimal point such that this product should have as many decimal digits (digits after the decimal point) as both the decimal numbers given.
Examples on Multiplying Decimals
Here are some examples that explains how to multiply decimal numbers.
Example 1:
Multiply 12.1 X 2.3
Solution:
Step 1: Write the decimal numbers as whole numbers by removing the decimal point 121 X 23
Step 2: Multiply the whole numbers obtained from step 1, 121 X 23= 2783
Step 3: Count the total number of digits after the decimal point in both the decimal numbers
12.1 One digit after the decimal point
2.3 One digit after the decimal point
So, there must be (1+1) digits to the right of the decimal point in the product.
Step 4: Take the product from step 2. Count from the right end and put a decimal point such that this product should have as many decimal digits (digits after the decimal point) as both the decimal numbers given.
12.1 X 2.3 = 27.83
Example 2:
Multiply the decimal numbers 34.52 and 7.1
Solution:
Following the steps of multiplication of decimal numbers we get,
3452 X 71 = 245092
Hence, 34.52 X 7.1 = 245.092
Multiplication of numbers by powers of 10
Observe the product 2.68 X 10.
We get 268 X 10 = 2680, and 2.68 X 10 = 26.80
That is, when we multiply 2.68 by 10, we get the same digits as in the decimal number. But, the decimal point is moved towards right by one place.
When we multiply a decimal number by 10, 100, 1000,.. (Powers of 10), the digits in the product will be same as that in the given decimal number but the decimal point will be shifted to the right side as many places as the power of 10(or as many places as there are zeros over one).Examples on multiplication of numbers by powers of 10
Given below are some examples on multiplying a number by powers of 10.
Example 1:
5.689 x 100
Solution:
Here, there are two zeros in 100. So, the decimal point will be shifted 2 places towards the right.
So, 5.689 X 100 = 568.9
Example 2:
Multiply 21.167 x 1000
Solution:
Following the method of multiplication of a decimal by powers of 10, the decimal point will be shifted three places towards the right
Hence, 21.167 X 1000 = 21167
→ Read MoreThere are four different methods involved in dividing decimals:
 Division of a decimal number by a whole number
 Division of a decimal number by powers of 10
 Division of a decimal number by another decimal number
Division of a decimal number by a whole number
While dividing a decimal number, we can follow the long division method used in division of whole numbers. The only difference is that a decimal point is placed in the quotient such that it lines up with the decimal point in the dividend.
Examples on Dividing Decimals by a Whole Number
Given below are some examples that explains how to divide a decimal number by a whole number.
Example 1:
Divide 8.241 by 3
Step 1: Write the decimal number as a whole number, without considering the decimal point.
Step 2: Place the decimal point in the quotient such that it lines up with the given decimal number (dividend).
We can see that $\frac{8241}{3}$ = 2747
Hence, we get, $\frac{8.241}{3}$ = 2.747
And so, $\frac{82.41}{3}$ = 27.47
And, $\frac{824.1}{3}$ = 274.7
The decimal point of the quotient has to line up with that of the divisor.
Example 2:
Divide 9.346 by 7.
In this example, the remainder does not become zero. This is a case of nonterminating decimal. We may stop division at any stage depending on the accuracy required.
Division of a decimal number by powers of 10
Consider the example 42.5 divided by 10.
42.5 = $\frac{425}{10}$
$\frac{42.5}{10}$ = $\frac{425}{10}$ / $\frac{10}{1}$
= $ \frac{425}{10}$ x $\frac{1}{10}$ (Multiplying by the reciprocal of $\frac{10}{1}$)
= $\frac{425}{100}$
= 4.25
We see that the digits of the divisor and the quotient are the same here, but the decimal point is shifted one place towards the left.
When we divide by 10, the decimal point will shift one place to the left. When we divide by 100, the decimal point will shift 2 places towards the left.
While dividing a decimal number by powers of 10, the digits of the decimal number and the quotient will be same but the decimal point in the quotient will be shifted to the left by as many places as the power of 10 ( the number of zeros in the power of 10)Examples on Dividing Decimals by Powers of 10:
Given below are examples that explain division of decimal numbers by powers of 10.
Example 1:
Divide 89.998 by 1000
Solution:
Following the method of division of a decimal number by powers of 10, we will get the same quotient as the dividend. But, the decimal point will be shifted to the left by as many places as the number of zeros in the divisor.
$\frac{89.998}{1000}$ = 0.089998, because the decimal point should be shifted 3 places to the left.
$\frac{89.998}{1000}$ = 0.089998
Example 2:
Find the quotient of $\frac{203967}{100000}$
Solution:
Following the method of division, we get,
$\frac{203967}{100000}$ = 2.03967
Dividing a decimal number by another decimal number
We can divide a decimal number by another decimal number through the following steps:
Step 1: Convert the divisor to a whole number by multiplying it by suitable powers of 10.
Step 2: While multiplying the divisor by a power of ten, we have to multiply the dividend also by the same power of 10.
Step 3: Now, we have to divide a decimal number by a whole number. Use the steps to divide a decimal number by a whole number.
Examples on Dividing Decimals by another Decimal Number
Given below are some examples on dividing a decimal number by a decimal number.
Example 1:
Divide $\frac{20.6816}{8.992}$
Solution:
Step 1: Convert the divisor to a whole number by multiplying it by a suitable power of 10.
The divisor is 8.992. We have to convert this as 8.992 x 1000 = 8992
Step 2: While multiplying the divisor by a power of ten, we have to multiply the dividend also by the same power of 10.
20.6816 x 1000 = 20681.6
Step 3: Now, we have to divide a decimal number by a whole number. Use the steps to divide a decimal number by a whole number.
Now, we have to find $\frac{20681.6}{8992}$
We know that, $\frac{206816}{8992}$ = 23
Hence, $\frac{20681.6}{8992}$ = 2.3
Example 2:
Find the answer to $\frac{35.568}{7.8}$
Solution:
Following the steps of division, we get
$\frac{35.568}{7.8} = \frac{(35.568\times10)}{(7.8)\times10}$
$ = \frac{355.68}{78}$
We have to divide 35568 by 78, and put the decimal point in the quotient such that it lines up with that of the dividend.
$\frac{35568}{78}$ = 456
Hence, $\frac{355.68}{78}$ = 4.56
→ Read MoreThere are a number of conversions that involve decimal numbers. Some of them are:
 Decimal to Fraction
 Decimal to Percent
 Decimal to Binary Number
 Decimal to Hexadecimal Number
To convert a decimal into fraction we follow these steps:
Steps to Convert Decimal to Fraction
 Write down the decimal divided by 1
 Multiply both the numerator and the denominator by a number that is a power of ten based on the number of digits after the decimal point.
 Simplify the fraction.
Examples to Convert Decimals to Fractions
Given below are some examples that explain how to convert decimals to fractions
Example 1:
Convert the following decimals into fractions (a) 0.85 (b) 0.231
(a) 0.85
Solution:
Step 1: Write down the decimal divided by 1
$\frac{0.85}{1}$
Step 2: Multiply both numerator and denominator by 100 (Since, there are two digits after the decimal point).
$\frac{0.85}{1} = \frac{(0.85 \times 100)}{(1 \times 100)} = \frac{85}{100}$
Step 3: We will simplify the fraction so obtained
$\frac{85}{100} = \frac{17}{20}$
(b) 0.231
Solution:
Step 1: Write down the decimal divided by 1
$\frac{0.231}{1}$
Step 2: Multiply both the numerator and denominator by 1000 ( Since, there are three digits after the decimal point)
$\frac{0.231}{1} = \frac{(0.231\times1000)}{(1\times1000)} = \frac{231}{1000}$
Step 3: Since greatest common factor of 231 and 1000 is 1, it cannot be reduced. Therefore, the answer is $\frac{231}{1000}$
The Dewey decimal system of classification is a numerical method that libraries apply to classify non fiction publications into groups based on respective subjects. This was invented by American librarian Melville Louise Kossuth Dewey in the 19th century as a system for small sized libraries.
The Dewey number is a code for a specific subject. This system can play an important role in facilitating access to children’s material both on the shelves of libraries and if required in online systems. It can help provide multilingual access.
Decimal to Inches:
When units smaller than a sixtyfourth of an inch are required, the decimal inches system is used.
The inch is divided into ten equal parts each having a value of one hundredths and each of these is again divided into ten equal parts and so on until the inch is divided into tenths, hundredths, thousandths and ten thousandths of an inch is obtained.
In the inch system, it is not necessary to place a zero to left of the decimal point for dimension less than one inch.
Example: a measurement of eight hundred and forty three thousandths of an inch could be written as ".843"
The most common decimal inch fraction used is one – thousandth of an inch, or 0.001 inch.
Whole inches are represented by numbers to the left of the decimal point.
Let us try one example:
A dimension of 3.413 inches means 3 inches plus four hundred and thirteen thousandths of an inch.
Inch metric equivalents:
The inch system is the standard of measurement in North American industries.
In this system, the base unit of length is the inch and further units are related to the base by odd unusual factors.
The inch is commonly divided into fractions such as halves $\frac{1}{2}$, quarters $\frac{1}{4}$, eights $\frac{1}{8}$, sixteenths $\frac{1}{16}$ etc.
Inch to metric conversion:
Inch  Metric 
.001 inch  0.025 mm 
1 inch  25.4 mm 
1 foot  0.3048 m 
1 yard 
0.9134 m 
1 ounce 
28.35 g 
1 pound  0.4536 Kg 
Metric to inch conversion:
Metric 
Inch 
1 mm 
.039 inches 
1 m 
39.37 inches 
1 Km 
.6213 miles 
1 g 
15.432 grains 
1 Kg 
2.204 pounds 
1 liter 
1.0567 quarts 
When decimal fractions are divided, the divisor is placed to the left of the dividend as we do in the division of whole numbers.
When dividing decimal fractions, the divisor must be a whole number and not a fraction.
The divisor could be made a whole number by moving the decimal point to the right of the number and after doing this the decimal of the dividend is moved the same number of places to the right as well.
The decimal of the dividend is then placed above the division bracket and hereafter the numbers are then divided in same manner as whole numbers.
Example for Decimal Division:
Divide 19.44 by 3.6
Step 1: As we could not have a divisor with decimal fraction so we need to convert this into a whole number.
Step 2: The decimal is moved right for both the divisor (3.6) and the dividend (19.44) by same places.
Step 3: after converting the divisor into a whole number (36) the quotient decimal is placed exactly on top of the decimal of dividend (194.4) decimal.
Percent literally means "out of 100". A percent can be represented as a fraction and as a decimal number.
We can convert a percent to a decimal number and we can write a decimal number as a percent.
Consider the shaded region in the 10 x 10 grid sheet. We see that half of the grid sheet is shaded. We can say that 50% of the grid is shaded.
We know $\frac{1}{2}$ = 0.5, and 0.5 =50%
We can convert a decimal number to a percent by multiplying the decimal number by 100, and placing the percent symbol with the number. When we multiply a decimal number by 100, we are shifting the decimal point two places towards right.
We can represent a decimal number as a percent by shifting the decimal point two places towards right and place the percent symbol, %, with the number.Examples to Convert Decimals to Percent
Given below are some of the examples that explains how to convert decimals to percent.
Example 1:
Write the decimal number, 0.95 as percent
Solution:
Multiplying the given decimal number by 100,
0.95 x 100 = 95.0, by shifting the decimal point two places towards right
We have to place the percent symbol with the number.
Hence, the required percent is 95%
0.95 = 95%
Example 2:
Convert the decimal number 0.234 to percent
Solution:
We can multiply the given decimal number by 100 and place the percent symbol with the number.
Therefore, 0.234 = 0.234 x 100 = 23.4%, by shifting the decimal point two places towards right.
Decimal fraction can also be rounded off to the nearest one or to one place of decimal or to two places of decimals, as required.
Rounding Decimals to the Nearest 1:
While rounding off a decimal fraction to the nearest one, if the digit in the tenths place is < 5, then the tenths place and all the following places are replaced by 0. If the digit in the tenths place is >= 5, then the digit in the ones place is increased by 1 and all digits after the decimal are reduced to 0.
For example, 49.37 = 49.00 [Since 3 < 5, it will be 49.00 i.e. 49].
96.872 = 97.000 [Since 8 > 5, it will be 97.00 i.e. 97].
Rounding Decimals to One Decimal Place:
1) When the digit in the hundredths place is < 5, the digit in the hundredths place and following digits are replaced by 0.
For example, 14.732 = 14.700 [Since 3 < 5, the hundredths place and following digits are replaced by 0].
2) When the digit in the hundredths place is >= 5, the digit in the tenths places is increased by 1 and the following digits become 0.
For example, 56.897 = 56.900 [Since 9 > 5, the digit in the tenth places is increased by 1 and the following digits become 0].
Rounding Decimals to Two Decimal Places:
1) When the digit in the thousandths place is < 5, the thousandths place and the following digits are replaced by 0.
For example, 189.434 = 186.430 [Since 4 < 5, the thousandths place and the following digits are replaced by 0].
2) When the digit in the thousandths place is >= 5, the digit in the hundredths places is increased by 1 and the following digits become 0.
For example, 27.987 = 27.990 [Since 7 > 5, the digit in the hundredths place is increased by 1 and the following digits become 0].
Let us try to represent 0.9, 1.3 and 1.7 on a number line.
We know that, 0.9 is greater than 0 and less than 1. Also, there are nine tenths in it.
From 0 to 1, we divide the line into ten parts. Each part will represent the tenths.
So, to represent 0.9, we choose the ninth place from 0 reading in tenths.
Hence, P represents 0.9.
We know that 1.3 is greater than 1 and less than 2. Also, there are one and three tenth in it.
Here, we can see that 1.3 is third tenth after the number 1.
We know that 1.7 is greater than 1 and less than 2. Also, there are one and seven tenth in it.
See that 1.7 is the seventh point from 1.
A decimal number which has a decimal part with repeating digits is known as repeating decimals.
These are decimals of the form,
i. 0.333333......
ii. 5.161616........
Representing repeating decimals:
The common practice to represent repeating decimals is to put a dot or a bar above the digit which is repeated.
0.3 = 0.33333 (The digit is repeated after one digit)
0.09 = 0.090909 (The digit is repeated after two digits)
0.428571 = 0.428571428571428571(The digit is repeated after 6 digits)
Converting repeating decimals to fractions:
There is a standard method for converting repeating decimal numbers into fractions.
Steps to Convert Repeating Decimals to Fractions
1. Name the given fraction as x2. Multiply this fraction by a power of 10, such that the power is equal to the number of repeating digits in the decimal.
3. Subtract the equation in step 1 from the equation in step 2.
4. Represent x as a fraction.
Examples to convert repeating decimals to fractions:
Given below are some of examples that explain how to convert repeating decimals to fractions.
Example 1:
Let us convert 0.33333.... to a fraction.
Solution:
Take x = 0.3333..... (1)
Only one digit is repeated. So, multiply equation (1) by 10
We get, 10x = 3.3333.... (2)
Subtracting equation (1) from (2), 9x = 3
x = $\frac{3}{9}$
= $\frac{1}{3}$ (Writing in the reduced form)
Hence, 0.3333.... = $\frac{1}{3}$
Example 2:
Find the equivalent fraction of the decimal number 5.161616........
Solution:
Let, x = 5.16161616...... (1)
Here, two digits are repeating. So, we multiply x by 100.
100 x = 516.161616........ (2)
Subtracting equation (1) from equation (2), we get,
99 x = 511
Dividing both sides of the above equation by 99, x = $\frac{511}{99}$ , which is an improper fraction.
Hence, 5.161616.... = $\frac{511}{99}$
Any decimal that has only a finite number of non zero digits is called terminating decimals. 0.25, 0.682, 0.04 etc are some of the examples of terminating decimals.
Terminating decimal involves those fractions whose denominator is 2 or 5 , or a combination of 2 and 5.Examples of Terminating Decimals:
$\frac{1}{2}$ = 0.5 , $\frac{1}{4}$ = 0.25 , $\frac{1}{8}$ = 0.125 , $\frac{1}{16}$ = 0.0625 etc.$\frac{1}{5}$ = 0.2 , $\frac{1}{10}$ = 0.2 , $\frac{1}{25}$ = 0.04 , $\frac{1}{125}$ = 0.008$
In all the examples shown above, the denominators are positive exponents of 2 or 5 or a combination of both.
4 = 2^{2} , 8 = 2^{3} , 16 = 2^{4} ,10 = 2 x 5 , 25 = 5^{2} , 125 = 5^{3}
In all other cases, decimals are non terminating.
Examples of Non Terminating Decimals:
$\frac{1}{3}$ = 0.3333 = 0.3 , $\frac{1}{7}$ = 0.1428571 etc.
Basic Operation on terminating Decimals:
Addition, subtraction, multiplication and division are some of the basic operations that can be performed on terminating decimals.
Addition of Terminating Decimals:
For this operation, the numbers are written one above another lining up the decimal point.Add 2.19 + 3.25
2.19
+3.25

5.44
Subtraction of Terminating Decimals:
Subtract 6.31  4.87
6.31
4.87

1.44
Multiplication of Terminating Decimals:
4.25 x 2.63
Write the numbers one above another. Multiply without considering decimal point and place decimal point after counting total decimal places.
4.25
X 2.63

1275
2550x
850xx

11.1775
Division of Terminating Decimals:
All decimal fractions can be represented in it's equivalent decimals form. In a decimal fraction, the denominator is always a power of 10. The equivalent decimal form of $\frac{5}{10}$ is 0.5. The equivalent decimal form of $\frac{25}{100}$ is 0.25.
Given below are some of the fractions and it's equivalent decimals:
Fractions  Equivalent Decimals 
$\frac{1}{10}$  0.1 
$\frac{1}{5}$  0.2 
$\frac{1}{2}$  0.5 
$\frac{1}{4}$  0.25 
$\frac{3}{4}$  0.75 
$\frac{5}{10}$  0.5 
Given below is the decimal chart for fractions:
Given below are some of the decimal word problems.
Example 1:
Mary's school is 6.45 miles away from her house. She travels 5. 35 miles by bus and the rest by walk. How much distance does she walk daily?
Solution:
Following the steps of subtraction, we get
6.45 
5.35

1.10
Mary has to walk 1.10 miles to her school.
Example 2:
If the cost of one chocolate candy is $2.30, find the cost of 8 candies.
Solution:
Cost of 8 chocolate candies = 8 X cost of one candy
= 8 X 2.30
Following the method of multiplication of a whole number and a decimal number,
We get, 8 X 2.30 = 18.40
Thus, the cost of 8 chocolate candies = $ 18. 40
Decimal notation is defined as a way of writing a number in the base  10 numerical system. A fraction or a real number when represented in the base  10 numerical system having any of the digits from 0 to 9 and a decimal point is called as it's decimal notation.
In the expanded notation of decimals, each value in the place value chart has a value ten times the value of the next place on its right.
For example, 100 + 10 + 1 + $\frac{1}{10}$ + $\frac{1}{100}$ + $\frac{1}{1000}$ ...
Examples for Expanded Notation of Decimals:
Given below are some of the examples that explains the expanded notation of decimals.
Example 1:
If we have to represent the number 257+ $\frac{3}{10}$ + $\frac{2}{100}$ , then 2 goes to hundreds place, 5 goes to tens place, 7 goes to ones place, 3 goes to tenths place and 2 goes to hundredths place.
The number 257.32 is called a decimal or a decimal number. It is read as two hundred fifty seven point three two.
The number is expanded in the form $2 \times 100 + 5 \times 10 + 7 \times 1+ 3 \times$ $\frac{1}{10}$ + 2 $\times$ $\frac{1}{100}$
(or)
$2 \times 100 + 5 \times 10 + 7 \times 1 + 3 \times 0.1 + 2 \times 0.01$
Example 2:
In 10.456, 10 goes to tens place, 0 goes to ones place, 4 goes to tenths place, 5 goes to hundredths place and 6 goes to thousandths place. 10.456 is expanded in the form of,
$1\times 10 + 0 \times 1+ 4 \times$ $\frac{1}{10}$ + $5 \times$ $\frac{1}{100}$ + $6 \times$ $\frac{1}{1000}$
(or)
$1 \times 10 + 0 \times 1 + 4 \times 0.1 + 5 \times 0.01 + 6 \times 0.001$
Example 3:
10.05 can be expanded as follows:
$1 \times 10 + 0 \times 1+ 0 \times$ $\frac{1}{10}$ + 5 $\times$ $\frac{1}{100}$
(or)
$1 \times 10 + 0 \times 1 + 0 \times 0.1 + 5 \times 0.01$
Example 4:
0.755 can be expanded as follows:
$0 \times 1+ 7 \times $ $\frac{1}{10}$ + 5 $\times$ $\frac{1}{100}$ + 5 $\times$ $\frac{1}{1000}$
(or)
$0 \times 1 + 7 \times 0.1 + 5 \times 0.01 + 5 \times 0.001$