Division of a Decimal by the Power of 10
Step 1: Shift the point in the decimal left as many places as the number of zeroes the power of 10 contains.
Step 2: If there are short of digits left to the point in the decimal, add zeroes left to it and follow the step 1.
Divide 256.52 by 100.
Explanation
100 contains two zeroes, therefore, shift the point two digit left in the decimal
Thus \[256.52\div 100=2.5652\text{ }.\]
Divide 3.25 by 10000.
Explanation
10000 contains 4 zeroes and 3.25 has only 1 digit left to the point so add ; zeroes left to it and shift the decimal 4 places left.
Thus \[3.25\div 10000=0.000325.\]
Division of a Decimal by a Whole Number
Step 1: Remove the point from the decimal and divide it by the whole number.
Step 2: Shift the point left in the quotient as many places as the given decimal has total number of digits right to the point.
Step 3: If the quotient is a whole number then insert a point in it so that it has equal number of decimal places as the given 'decimal has.
or
Step 1: Convert the decimal into fraction.
Step 2: Divide the fraction by the whole number.
Divide 45.25 by 8
Solution:
Write the decimal without point and divide it by the whole number
Thus \[4525\div 88=565.625\]
The given decimal 45.25 has two digit right to the point, therefore, shift the decimal two digit left.
Thus \[45.25\div 8=5.65625.\]
Or
\[45.25=\frac{4525}{100}\]
Thus \[45.25\div 8=\frac{4525}{100}\div =\frac{4525}{100}\times \frac{1}{8}\]
Or \[\frac{4525}{8}\times \frac{1}{100}=565.625\times \frac{1}{100}=5.65625\]
Division of a Whole Number by a Decimal
Step 1: Remove the point from the decimal.
Step 2: Divide the whole number by the obtained number in the first step.
Step 3: Multiply the quotient by 10 if there are one digit right to the point in the given decimal, by 100 if there are two digit right to the point in the given decimal and so on.
or
Step 1: Convert the decimal into fraction.
Step 2: Divide the whole number by the fraction.
Divide 910 by 3.5.
Explanation
Remove the point from the decimal and divide the whole number by it
Thus\[910\div 35=26\]
The decimal 3.5 contains only one digit right to the point, so multiply the quotient by 10
Thus \[910\div 3.5=260.\]
Or
\[3.5=\frac{35}{10}\]
Thus \[910\div 3.5=910\div more... 
Multiplication of Decimals by Power of 10
Step1: Shift the point in the decimal right as many places as there are zeroes the power of 10 contains. .
Step 2 : If there are short of digits to the right of the point in the decimal, add zeroes right to it and follow the first step.
Multiply 23.256 by 100.
Explanation
Power of ten (here 100) contains two zeroes, so point will shift two digit right.
Thus \[23.256\times 100\text{ }=2325.6.\]
Multiply 27.5 and 1000.
Here power of 10 contains three zeroes, therefore, decimal point will shift three digit right but the decimal contains only one digit right to the point so add two zeroes right to it.
Thus \[27.500\times 1000=27500\]
Multiplication of a Decimal by a Whole Number
Step 1: Remove the point from the decimal and multiply with the whole number simply.
Step 2: Insert a point in the product so that the given decimal and the product have same number of decimal places.
or
Step 1: Convert the decimal into fraction.
Step 2: Now multiply the fraction and the whole number.
Find the product of 45 and 78.63.
Solution:
Multiply 45 and 7863
Thus \[45\times 7863\text{ }=353835\]
Now place a point in 353835 so that 353835 and 78.63 have same decimal places.
Thus the product \[45\times 78.63=3538.35\]
Or
\[78.63=\frac{7863}{100}\]
Thus \[45\times 78.63=45\times \frac{7863}{100}=\frac{7863\times 45}{100}=3538.35\]
Multiplication of Decimals
Step 1: Remove the point from the decimals and multiply them like whole numbers.
Step 2: Insert a point in the product so that number of decimal places of the product is equal to the sum of the number of decimals places of the multiplier and the multiplicand.
or
Step 1: Convert the decimals into fractions.
Step 2: Multiply numerator with numerator and denominator with denominator of the fractions.
Multiply 3.625 and 4.6.
Solution:
Multiply 3625 and 46
Thus \[3625\times 46=166750\]
Now 3.625 has three decimal places and 4.6 has one decimal place so place a point in 166750 so that it has four decimal places.
Thus \[3.625\times 4.6=16.6750.\]
Or
\[3.625=\frac{3625}{1000}\]
\[4.6=\frac{46}{10}\]
Thus \[3.625\times 4.6=\frac{3625}{1000}\times \frac{46}{10}=\frac{3625\times 46}{1000\times 10}=\frac{166750}{10000}=16.6750\] 
Subtraction of Decimals
Step 1: Convert the minuend and subtrahend into like decimals.
Step 2: Arrange the digits of minuend and subtrahend column wise one below other so that decimal points come in the same column.
Step 3: Now subtract the digits column wise from right to left, write the difference directly below the respective digits and place a point in the point column.
Subtract: 83.455 and 23.201
Explanation
Arrange the digits of 83.455 and 23.201 column wise and subtract.
\[\begin{align} & \,\,\,\,83.478 \\ & \underline{\frac{-23.201}{60.254}} \\ \end{align}\]
Subtraction of a Decimal From a Whole Number
Write the whole number in the decimal form then follow the steps given for subtraction of decimals.
Subtract 12. 32 from 57.
Explanation
Write 57 in the decimal form = 57.00
Now subtract 12.32 from 57.00
\[57.00-12.32=44.68.\]
Subtraction of a Whole Number From a Decimal
Write the whole number in the decimal form and follow the steps given for subtraction of decimals.
Subtract 25 from 30.213 .
Explanation
Write 25 in the decimal form = 25.000
Now subtract 25.000 from 30.213
\[30.213-25.000=5.213.\] 
Addition of Decimals
Step 1: Convert the addends into like decimals.
Step 2: Arrange the addends one below other in columns so that decimal points come in the same column.
Step 3: Now add the digits which are in the same column.
Add 73.478 and 45.02.
Explanation
45.02 can be written as 45.020
Now 73.478 and 45.020 are like decimals. Arrange their digits column wise and add.
Addition of a Decimal and a Whole Number
Write the whole number in the decimal form and follow the steps given for the addition of decimals.
Add 45 and 82.12.
Decimal form of 45 = 45.00 (to write a whole number in decimal form, a decimal point is placed in the extreme right of the number and then zeroes are added as per the requirement).
Now add 45.00 and 82.12
45.00+82.12=127.12. 
Introduction
In the previous chapter we have studied about the decimals. In this chapter we will study how to add two or more than two decimals, how to subtract a decimal from other decimal, how to multiply decimals, and how to divide a decimal by other decimal. 
Comparison of Decimals
Step 1: Compare the integral parts of the decimals, the decimal having greater integral part is greater.
Step 2: If the integral parts are equal, compare the digits at tenth place in thedecimals. The decimal having greater digit at tenth place is greater.
Step 3: If the digits at tenth place are equal, compare the digits at hundredth place and so on.
Compare 217.15 and 217.26.
Explanation
Integral part in 217.15 = 217
Integral part in 217.26 = 217
Thus both the decimals have same integral part. Therefore, compare the digits at tenths place.
Digit at the tenth place in 217.15 = 1
Digit at the tenth place in 217.26 = 2
2 is greater than 1. Therefore, 217.26 > 217.15.
5 is at the ....... place in the decimal 21.456. Choose the correct option to fill in the blank.
(a) Tenths
(b) Hundredths
(c) Thousandths
(d) Ten-thousandths
(e) None of these
Answer: (b)
Explanation
Place value of 5 in the decimal Thus 5 is at the hundredths place.
Which one of the following is the expanded form of 740.023?
(a)\[700+40+\frac{2}{10}+\frac{3}{100}\]
(b) \[700+4+\frac{2}{10}+\frac{3}{1000}\]
(c) \[700+40+\frac{2}{10}+\frac{3}{1000}\]
(d) \[700+40+\frac{2}{10}+\frac{3}{1000}\]
(e) None of these
Answer: (c)
Which one of the following options contains the decimal indicated in the following place value chart?
Hundreds Tens Ones Tenths Hundredths
| Hundreds | Tens | Ones | more...
 Conversion of Decimals
 Conversion of Unlike
Decimals into Like Decimals and Vice-Versa
Step 1: Select the decimal which has the highest number of decimal places.
Step 2: Now place the zeroes in the extreme right side in the other decimals so that they have equal number of digits right to the decimal point.
Convert 4.5, 9.03, 7.551, 2.1 into like decimals.
Explanation
The decimal 7.551 has the highest number of decimal places among the decimals 4.5, 9.03, 7.551, and 2.1.
The decimal 4.5 has only one decimal place, thus put 2 zeroes in the extreme right side =4.500.
The decimal 9.03 has only two decimal places, thus put 1 zeroes in the extreme right side = 9.030
The decimal 2.1 has only one decimal place, thus put 2 zeroes in the extreme right side = 2.100.
Now 4.500, 9.030, 7.551, and 2.100 are like decimals.
Note: In the same way you can convert like decimals into unlike decimals.
Conversion of a Decimal into a Fraction
Step 1: Remove the point from the decimal and write the obtained number as the numerator.
Step 2: Write 1 as denominator and put zeroes right to it so that the number of zeroes is equal to the number of digits right to the point in the given decimal.
Convert 23.56 into a fraction.
Explanation
On removing the point from the decimal 23.56 we get the number 2356.
Thus 2356 becomes numerator for the required fraction. There are two digits right to the point in the decimal 23.56 thus the required denominator will be 100 as 100 contains two zeroes.
Thus the required fraction for \[23.56=\frac{2356}{100}\]
Conversion of a Fraction into a Decimal
If denominator of the fraction is power of 10, count the digits of the numerator from right and put a decimal point in the numerator so that the number of digits right to the decimal point is equal to the number of zeroes in the denominator.
Convert \[\frac{12562}{100}\] into a decimal.
Explanation
The denominator 100 contains two zeroes, therefore, put a point after two digits counting from right.
Thus the required decimal\[\frac{12562}{100}=125.62\].
Division Method
Step 1: Insert a point in the extreme right to the dividend, and add zeroes right to the point (Note: you may increase the number of zeroes as per the requirements).
Step 2: Now divide the numerator by the denominator.
Step 3: Insert a point extreme right more...  Expanded Form of Decimals
Expanded form of a decimal represents the addition of place values of the digits respected to their position in the decimal. For the example: Expanded form of 315.162 is \[300+10+5+\frac{1}{10}+\frac{6}{100}+\frac{2}{1000}.\]
Write the expanded form of the decimal 0.956
Explanation
\[0\frac{9}{10}+\frac{5}{100}+\frac{6}{1000}.\]
 Decimal Places
The number of digits right to the point in a decimal is called decimal places of that decimal For example: In the decimal 26.345, there are three digits right to the point, therefore, the decimal 26.345 has three decimal places.
How many decimal places does the decimal 25.26 has?
Explanation
There are two digits right to the point in the decimal 25.26, thus it has two decimal places.
Like Decimals
The decimal having same number of digits right to the point are called like decimals.
In other words like decimals have same decimal places.
For example: 2.56, 5.48, 0.25, etc. are like decimals as they have the same number of decimal places.
"4.56 and 256.35 are like decimals". Justify the statement.
Explanation
Both the decimals 4.56 and 256.35 have two decimal places and the decimals have same number of decimal places are like decimals.
Unlike Decimals
Decimal numbers of different decimal places are called unlike decimals. In other word unlike decimals have different decimals places.
For example: 0.2, 1.23, 2.236 etc. are unlike decimals as they have different decimal places.
Check, 25.36 and 5.256 like decimals or unlike decimals.
Explanation
25.36 has two decimal places whereas 2.256 has three decimal places. They have the different decimal places, therefore, they are unlike decimals.
Equivalent Decimals
The decimals which have same value are called equivalent decimals.
For example: 2.5, 2.50, 2.500, are equivalent decimals as they have the same value.
Write two equivalent decimals of 2.57
Explanation
Two equivalent decimals of 2.57 are 2.570 and 2.5700. You may find many other equivalent decimals of 2.57 by just adding zeroes in the extreme right side of the decimal.
Note: If the number of zeroes is increased in the extreme right side of a decimal, the value of the decimal remains constant.  Introduction
A fraction with the denominator power of 10 (like 10, 100, 1000 etc.) is called decimal. It is expressed as a number using a point called decimal point. Decimal consist of two parts which are separated by a decimal point.
 Integral Part
The part which is left to the decimal point is called integral part or whole number part. For example, in the decimal 896.3, 896 is the integral part.
Decimal Part
The part which is right to the decimal point is called fractional part or decimal part. For example: 45.683 is a decimal number in which 683 is fractional part or decimal part.
Note: Decimal part read as separately one by one like 35.721 is read as thirty five point seven, two, one not as thirty five point seven hundred twenty one.
Decimal Place Value Chart
|
Operation on the Fraction
Addition and Subtraction of Like Fractions
Like fractions have same denominator. In the operation of addition, numerators of the like fractions are added and their sum become the numerator for the required fraction and their common denominator becomes denominator. For the example:
\[\frac{P}{Q}+\frac{R}{Q}+\frac{P+R}{Q}=\frac{S}{Q}\] (Where\[S=P+R\]). In the operation of subtraction, difference of numerators is found
Ex: \[\frac{P}{Q}-\frac{R}{Q}=\frac{P-R}{Q}=\frac{S}{Q}\] (Where \[S=P-R\])
Add \[\frac{15}{7}\] and \[\frac{9}{7}\]
Explanation
Addition of \[\frac{15}{7}\] and \[\frac{9}{7}=\frac{15}{7}+\frac{9}{7}\]
\[=\frac{9+15}{7}=\frac{24}{7}.\]
Find the product of \[\frac{5}{17}\] and \[\frac{32}{85}.\] 
