This document discusses decimal fractions and various operations involving them such as addition, subtraction, multiplication, division, comparison of fractions, and conversion between decimal and vulgar fractions. Some key points: - Decimal fractions are fractions where the denominator is a power of ten, and can be written using a decimal point. - To add or subtract decimal fractions, they are written under each other so the decimal points line up, then added or subtracted as usual. - To multiply decimal fractions, the numbers are multiplied without regard to the decimal point, and the decimal point is placed to make the total number of decimal places equal the sum of the individual numbers' decimal places. - To divide decimal fractions, the numbers are

1. Decimal Fraction
- Govinda Darade

2. Decimal Fraction
 A fraction where the denominator (the bottom number)
is a power of ten (such as 10, 100, 1000, etc ).
 You can write decimal fractions with a decimal point
(and no denominator), which make it easier to do
calculations like addition and multiplication on fractions.
 Example::
1. 7/10=0.7
2. 43/100=0.43
3. 51/1000=0.051

3. Decimal to Vulgar Fraction
 Put 1 in denominator under the decimal point and an
next 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 term.
 Example::
1. 0.25= 25/100
=1/4
2. 2.008=2008/1000
=251/125.
Que:: 0.75

4. Addition & Subtraction
• The Numbers are so placed under each other that
the decimal point lie in one column. The Numbers so
arranged can now be added or subtracted in the
usual way.
E.g.:: 6202.5+620.25+62.025+6.2025+0.62025

5.  Example::
1. 6202.5+620.25+62.025+6.2025+0.62025
6202.5
+ 620.25
+ 62.025
+ 6.2025
+ 0.62025
6891.59775

6. • Example::
5172.49 + 378.352 + ? =9318.678
Sol:
let,
5172.49 + 378.352 + x =9318.678
x = 9318.678 – (5172.49 + 378.352)
x = 3767.836
Que::
31.004 – 17.2386

7. Multiplication
• Multiply the given numbers considering them
without the decimal point. Now, in the product, the
decimal point narked off to obtain as many places of
decimal as is the sum of the number of decimal
places in the given numbers.
• E.g.::
0.2 * 0.02* 0.002
= (2*2*2) and Addition of Decimal Place(1+2+3=6)
= 8 so, final Ans = 0.000008

8. Division
 By 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 in the dividend.
E.g.::
0.0204 / 17
=204/17
=12 now, dividend contain 4 places of decimal so,
=0.0012

9.  By Decimal Fraction:
 Multiply both the dividend and divisor by suitable power
of 10 to make divisor a whole number then it is similar to
previous one.
E.g.::
0.00066 / 0.11
= (0.00066 * 100)/ (0.11 * 100)
= 0.066 /11=66/11=6
= 0.006

10. • Que::
1. 2.61 * 1.3
2. 0.63 / 9
3. 2.5/0.0005

11. Comparison of Fraction
 Suppose some fractions are to be arrange in
ascending or descending order of magnitude.
 Then convert, each of the given fractions in the
decimal form and arrange them accordingly.
E.g.::
3/5 , 6/7 , 7/9 arrange then in descending order.
Now, first convert to decimal form.
3/5 = 0.6 6/7 = 0.857 7/9 = 0.777
Since 0.857 > 0.777 > 0.6 so,
6/7 > 7/9 > 3/5.

12. Recurring Decimal
 If in a decimal fraction, a figure or a set of figures is
repeated continuously, then such a number is called
recurring decimal.
 If single figure is repeated we put dot on it.
 If set of figures is repeated we put bar on that set.
E.g.::
1/3 =0.3 single figure repeats i.e., 0.33333…
22/7 = 3.142857 set of figures repeated.

13. Pure Recurring Decimal
 A decimal fraction in which all the figures after the
decimal point are repeated, is called pure recurring
decimal.
 Converting pure recurring decimal to vulgar fraction::
1. Remove the number left to the decimal point, if any.
2. Write the repeated figures only once in the numerator
without the decimal point.
3. Write as many nines in the denominator as the number
of repeating figures.
4. Add the number removed in step 1(if any) with the
fraction obtained in the above
E.g.:: 0.53

14. E.g.::
1. 0.53
= 53/99
2. 0.067
= 67/999
3. 5.36
= 5 + (36/99)
= 5+ (4/11)
=(55+4)/11
=59/11

15. Mixed Recurring Decimal
 A decimal fraction in which some figures after the
decimal point are not repeated and some of them are
repeated, is called mixed recurring decimal.
 Converting mixed recurring decimal to vulgar fraction::
1. Remove the number left to the decimal point, if any.
2. Numerator is the difference between the number formed by
all the digits (taking repeated digits only once) and that
formed by the digits which are not repeated.
3. Denominator is the number formed by taking as many nines as
the number of repeating figures followed by as many zeros as
the number of non-repeating digits.
4. Add the number removed in step 1(if any) with the fraction
obtained in the above steps.
E.g.:: 0.583

16. • E.g.::
1. 0.583
= (583 - 58)/900
= 525/900
= 7/ 12
2. 1.318
= 1 + ((318-3)/990)
= 1 +(315/990) = 1 +(21/66)
= 1 + (7/22)
= (22+7) / 22
= 29 /22

17. • Que::
1. 0.053
2. 2.536

18. Basic Rules
1. (a + b)2 = a2 + 2ab + b2
2. (a – b)2 = a2 – 2ab + b2
3. a2 – b2 = (a – b)(a + b)
4. a2 + b2 = (a + b)2 – 2ab
5. (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
6. (a – b – c)2 = a2 + b2 + c2 – 2ab + 2bc – 2ca
7. (a + b)3 = a3 + 3a2b + 3ab2 + b3
8. (a + b)3 = a3 + b3 + 3ab(a + b)
9. (a – b)3 = a3 – 3a2b + 3ab2 – b3 = a3 – b3 – 3ab(a – b)
10.a3 – b3 = (a – b)(a2 + ab + b2)
11.a3 + b3 = (a + b)(a2 – ab + b2)

19. E.g.::
0.05 * 0.05 *0.05 + 0.04 * 0.04 *0.04
0.05 * 0.05 – 0.05 * 0.04 + 0.04 *0.04
given expression = a3 + b3 /(a2 – ab + b2)
here a=0.05 & b=0.04
so, ans = (a+b) (a2 – ab + b2)/(a2 – ab + b2)
=(a+b)
=0.05+0.04
=0.09

20. • Que::
(0.1*0.1*0.1+0.02*0.02*0.02)/(0.2*0.2*0.2+0.04*0.04*0.04)