Chapter - 3
1) The fractions in which the denominators are 10, 100, 1000 etc. are known as decimal fractions.
2) Numbers written in decimal form are called decimals.
3) A decimal has two parts, namely, the whole number part and the decimal part.
4) The number of digits contained in the decimal part of a decimal is called the number of its decimal places.
5) We have 0.1 = 0.10 = 0.100 etc
Also 0.2 = 0.20 = 0.200 etc.
6) Unlike decimals may be converted into like decimals by annexing the requisite number of zeros at the end of the decimal part.
7) Addition of decimals.
Step 1 Convert the given decimals into like decimals.
Step 2 Write the numbers to be added, one below the other so that the decimal points of all these decimals are in the same column.
Step 3 Add as in case of whole numbers
Step 4 Put the decimal point in the sum under the decimal points in the addends.
8) Subtraction of decimals.
Step 1 Convert the given decimals into like decimal
Step 2 Write the smaller number below the larger one so that their decimal points are in the same column
Step 3 Subtract as in the case of whole numbers
Step 4 In the difference, put the decimal point directly under the decimal pts. of the given numbers.
9) Converting a decimal into fraction.
Step 1 Write the given decimal without the decimal point as the numerator of the fraction.
Step 2 In the denominator write 1 followed by as many zeros as there are decimal places in the given decimal.
Step 3 Convert the above fraction into its simplest form.
10) Converting a fraction into a decimal
a) If the denominator of the fraction is 10 or a power of 10.
b) If the denominator is other than a power of 10
Step I Divide the num. by the den. till a non-zero remainder is obtained
Step II Put a decimal point in the dividend as well as in the quotient
Step III Put a zero on the right of the decimal point as well as on the right of the remainder
Step IV Divide again as we do in whole numbers
Step V Repeat step IV till the remainder is zero
Example 1 Convert 29/4 into a decimal fraction
Solution: On dividing we get
Example 2 Convert into a decimal fraction.
We find that in the above two examples, the process of division has terminated after a few decimal places i.e there are finite places after the decimal point. Such fractions are said to be terminating decimals.
In the case of the fractions the process of division is un ending and there are infinite number of places after the decimal point. Such fractions are said to be non-terminating decimals.
Decimal representation of is 0.888 ------ or is represented as 0.
So every positive rational number is either a terminating or a non-terminating repeating decimal.
To search whether a rational number is a terminating or non-terminating repeating decimal.
If the denominator of a rational number (which is in its lowest terms) as no prime factor other than 2 or 5 or both, the rational number is a terminating decimal. Otherwise the rational number is a non-terminating repeating decimal.
Example: etc are terminating
decimals where as are non-terminating repeating decimals.
Multiplication of decimals.
a) Multiplication of a decimal by a whole number.
To multiply a decimal by a whole number, we multiply the two numbers as if they are whole numbers. We put the decimal point in the product so that there are as many places of decimal in it as there are in the given decimal.
b) Multiplication of a decimal by 10, 100, 1000 ----- etc.
To multiply a decimal by 10, 100, 1000 etc, we have to give a jump to the decimal by one. Two or three etc places to the right
For example 2.96 x 10 = 29.6
2.96 x 100 = 296
2.96 x 1000 = 2960 etc
c) Multiplication of a decimal by a decimal
To multiply two decimals, we first multiply them as if we are multiplying two whole numbers
(ignoring the decimal points). Then the decimal is placed in the product such that it has as many decimal places as there are in multiplicand and the multiplier taken together.
For example: To find the product of 34.17 x 3.2 we first multiply 3417 by 32 and get 109344
The multiplicand and the multiplier taken together have 3 decimal places, the product should also have 3 decimal places. Hence 34.17 x 3.2 = 109.344
Division of Decimal
Therefore 442.4 � 7 = 63.2
b) Division of a decimal by 10,100,1000 etc.
To divide a decimal by 10,100,1000 etc we have to give a jump to the decimal by one, two, three etc places to the left.
For example :
c) Division of a decimal by a decimal
If we want to divide 6.25 by 2.5 we write
Since the divisor has one decimal place we multiply both numerator and denominator by 10 and get
Now the problem reduces to the division of a decimal by a whole number.
d) Division of a whole number by a decimal Convert the decimal of divisor to a whole number and divide.