1. Decimal Fraction : Fractions in which denominators are powers of 10 are known as decimal fractions. A fraction written as a decimal. A decimal point separates whole numbers from decimal fractions
for example, 1/10 = 1 tenth = .1
1/100 = hundredth = .01
99/100 = 99 hundredths = .99
7/1000 = 7 thousandths = .007 , etc.
2. Conversion of a Decimal Into Vulgar Fraction : Put 1 in the denominator under the decimal point and annex with it as many zeros as is the number of digits after the decimal point. Now, remove the decimal point and reduce the fraction to its lowest terms.
Thus, 0.20 = 20/100 = 1/5
2.008 = 251/125
3. numerator and denominator of a decimal fraction :
a. Annexing zeros to the extreme right of a decimal fraction does not change its value. Thus, 0.8 = 0.80 = 0.800, etc.
b. If numerator and denominator of a decimal fraction contain the same number of decimal places, then we remove the decimal sign
for example: 1.84/2.99 = 184/299 = 8/13
4. Operation on Decimal Fractions :
1. Addition and Subtraction of Decimal Fractions:
To add/subtract decimals,
(i) Write down the numbers, one under the other, with the decimal points lie in one column.
(ii) Now the numbers can be added normally (remember to put the decimal point in the answer).
Examples : 1.3 + 0.24 = 1 . 5 4
2. Multiplication of a decimal Fraction By a power of 10 :
Shift the decimal point to the right by as many places as is the power of 10.
Thus, 5.9632 × 100 = 596.32; 0.073 × 10000 = 0.073 × 10000 = 730
3. Multiplication of Decimal Fractions:
To multiply decimals,
(i) Multiply the given numbers considering them without decimal point.
(ii) In the product, the decimal points is marked off to obtain as many places of decimal as is the sum of the number of decimal places on the given numbers.
Suppose we have to find the product (.2 × .2 × .002).
Now, 2 × 2 × 2 = 8. Sum of decimal places = (1 + 2 + 3) = 6.
∴ .2 × .02 × .002 = .000008.
4. Dividing a Decimal Fraction By a Counting Number :
Divide the given number without considering the decimal point, by the given counting number. Now, in the quotient, put the decimal point to give as many places of decimal as there are in the dividend.
Suppose we have to find the quotient (0.0204 ÷ 17). Now, 204 ÷ 17 = 12.
Dividend contains 4 places of decimal. So, 0.0204 ÷ 17 0.0012.
5. Division of Decimal Fraction/Counting Number by a Decimal Fraction:
To divide a decimal/counting number by a decimal,
(i) Multiply both the dividend and the divisor by a suitable power of 10 to make divisor a whole number.
(ii) Now divide the numbers normally.
Examples 0.00066/0.11 = 0.00066 x 100 /0.11 x 100 = 0.066/11 = .006
5. Comparison of Fractions : Suppose some fractions are to be arranged in ascending or descending order of magnitude. Then, convert each one of the given fractions in the decimal form, and arrange them accordingly.
Suppose, we have to arrange the fractions 3/5, 6/7 and 7/9 in descending order.
Now, 3/5 = 0.6, 6/7 = 0.857, 7/9 = 0.77……
Since 0.857 > 0.777…… > 0.6, so 6/7 > 7/9 > 3/5.
6. Recurring Decimal : A repeating decimal, also called a recurring decimal, is a number whose decimal representation eventually becomes periodic (i.e., the same sequence of digits repeats indefinitely). The repeating portion of a decimal expansion is conventionally denoted with a vinculum so, for example,
1/3 = 0.333... (the 3 repeats forever)
1/7 = 0.142857142857... ( the "142857" repeats forever)
77/600 = 0.128333... (the 3 repeats forever)
In a recurring decimal, if a single figure is repeated, then it is expressed by putting a dot on it. If a Set of figures is repeated, it is expressed by putting a bar on the set.
Pure Recurring Decimal : A decimal fraction in which all the figures after the decimal point are repeated, is called a pure recurring decimal
Converting a Pure Recurring Decimal Into Vulgar Fraction : Write the repeated figures only once in the numerator and take as many nines in the denominator as is the number of repeating figures.
Thus, 0.5 = 5/9; 0.53 = 53/99
Mixed Recurring Decimal : A decimal fraction in which some figures do not repeat and some of the are repeated, is called a mixed recurring decimal.
e.g., 0.17333…… = 0.173
Converting a Mixed Recurring Decimal Into Vulgar Fraction : In the numerator, take the difference between the number formed by all the digits after decimal point. (taking repeated digits only once) and that formed by the digits which are not repeated. In the denominator, take the number formed by as there are repeating digits followed by as many zeros as is the number of non-repeating digits.
7. Basic Formulae :
1. (a + b) (a – b) = (a2 – b2).
2. (a + b)2 = (a2 + b2 +2ab).
3. (a – b)2 = (a2 + b2 – 2ab).
4. (a + b + c)2 = a2 + b2 + c2 + 2 ( ab + bc + ca).
5. (a3 + b3) = (a + b) (a2 - ab + b2)
6. (a3 – b3) = (a – b) (a2 + ab + b2).
7. (a3 + b3 + c3 – 3abc) = ( a + b + c) (a2 + b2 + c2 – ab - bc – ac).
When a + b + c = 0, then a3 + b3 + c3 = 3abc.