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  1. Decimal Fractions:

    Fractions in which denominators are powers of 10 are known as decimal fractions.

     

    Thus,1= 1 tenth = .1;        1= 1 hundredth = .01;
    10100

     

    99= 99 hundredths = .99;  7= 7 thousandths = .007, etc.;
    1001000
  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, 0.25 =25=1;       2.008 =2008=251.
    10041000125
  3. Annexing Zeros and Removing Decimal Signs:

    Annexing zeros to the extreme right of a decimal fraction does not change its value. Thus, 0.8 = 0.80 = 0.800, etc.

    If numerator and denominator of a fraction contain the same number of decimal places, then we remove the decimal sign.

     

    Thus,1.84=184=8.
    2.9929913
  4. Operations on Decimal Fractions:

    1. Addition and Subtraction of Decimal Fractions: The given numbers are so placed under each other that the decimal points lie in one column. The numbers so arranged can now be added or subtracted in the usual way.

    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 x 100 = 596.32;   0.073 x 10000 = 730.

    3. Multiplication of Decimal Fractions: Multiply the given numbers considering them without decimal point. Now, in the product, the decimal point is marked off to obtain as many places of decimal as is the sum of the number of decimal places in the given numbers.

      Suppose we have to find the product (.2 x 0.02 x .002).

      Now, 2 x 2 x 2 = 8. Sum of decimal places = (1 + 2 + 3) = 6.

      Decimal Fractions: Notes and Important Formulas | CSAT Preparation - UPSC .2 x .02 x .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. Dividing a Decimal Fraction By a Decimal Fraction: Multiply both the dividend and the divisor by a suitable power of 10 to make divisor a whole number.

      Now, proceed as above.

       

      Thus,0.00066=0.00066 x 100=0.066= .006
      0.110.11 x 10011
  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.

     

    Let us to arrange the fractions3,6and7in descending order.
    579

     

    Now,3= 0.6,  6= 0.857,  7= 0.777...
    579

     

    Since, 0.857 > 0.777... > 0.6. So,6>7>3.
    795
  6. Recurring Decimal:

    If in a decimal fraction, a figure or a set of figures is repeated continuously, then such a number is called a recurring decimal.

    n 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.

     

    Thus,1= 0.333... = 0.3;22= 3.142857142857.... = 3.142857.
    37

    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; 0.53 =53; 0.067 =67, etc.
    999999

    Mixed Recurring Decimal: A decimal fraction in which some figures do not repeat and some of them are repeated, is called a mixed recurring decimal.

    Eg. 0.1733333.. = 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 many nines as there are repeating digits followed by as many zeros as is the number of non-repeating digits.

     

    Thus, 0.16 =16 - 1=15=1;   0.2273 =2273 - 22=2251.
    9090699009900
  7. Some Basic Formulae:

    1. (ab)(ab) = (a2b2)

    2. (ab)2 = (a2 + b2 + 2ab)

    3. (ab)2 = (a2 + b2 - 2ab)

    4. (abc)2a2 + b2c2 + 2(abbcca)

    5. (a3b3) = (ab)(a2abb2)

    6. (a3b3) = (ab)(a2abb2)

    7. (a3b3c3 - 3abc) = (abc)(a2b2c2abbcac)

    8. When abc = 0, then a3b3c3 = 3abc.

    9. Decimal Fractions:

      Fractions in which denominators are powers of 10 are known as decimal fractions.

       

      Thus,1= 1 tenth = .1;        1= 1 hundredth = .01;
      10100

       

      99= 99 hundredths = .99;  7= 7 thousandths = .007, etc.;
      1001000
    10. 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, 0.25 =25=1;       2.008 =2008=251.
      10041000125
    11. Annexing Zeros and Removing Decimal Signs:

      Annexing zeros to the extreme right of a decimal fraction does not change its value. Thus, 0.8 = 0.80 = 0.800, etc.

      If numerator and denominator of a fraction contain the same number of decimal places, then we remove the decimal sign.

       

      Thus,1.84=184=8.
      2.9929913
    12. Operations on Decimal Fractions:

      1. Addition and Subtraction of Decimal Fractions: The given numbers are so placed under each other that the decimal points lie in one column. The numbers so arranged can now be added or subtracted in the usual way.

      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 x 100 = 596.32;   0.073 x 10000 = 730.

      3. Multiplication of Decimal Fractions: Multiply the given numbers considering them without decimal point. Now, in the product, the decimal point is marked off to obtain as many places of decimal as is the sum of the number of decimal places in the given numbers.

        Suppose we have to find the product (.2 x 0.02 x .002).

        Now, 2 x 2 x 2 = 8. Sum of decimal places = (1 + 2 + 3) = 6.

        Decimal Fractions: Notes and Important Formulas | CSAT Preparation - UPSC .2 x .02 x .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. Dividing a Decimal Fraction By a Decimal Fraction: Multiply both the dividend and the divisor by a suitable power of 10 to make divisor a whole number.

        Now, proceed as above.

         

        Thus,0.00066=0.00066 x 100=0.066= .006
        0.110.11 x 10011
    13. 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.

       

      Let us to arrange the fractions3,6and7in descending order.
      579

       

      Now,3= 0.6,  6= 0.857,  7= 0.777...
      579

       

      Since, 0.857 > 0.777... > 0.6. So,6>7>3.
      795
    14. Recurring Decimal:

      If in a decimal fraction, a figure or a set of figures is repeated continuously, then such a number is called a recurring decimal.

      n 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.

       

      Thus,1= 0.333... = 0.3;22= 3.142857142857.... = 3.142857.
      37

      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; 0.53 =53; 0.067 =67, etc.
      999999

      Mixed Recurring Decimal: A decimal fraction in which some figures do not repeat and some of them are repeated, is called a mixed recurring decimal.

      Eg. 0.1733333.. = 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 many nines as there are repeating digits followed by as many zeros as is the number of non-repeating digits.

       

      Thus, 0.16 =16 - 1=15=1;   0.2273 =2273 - 22=2251.
      9090699009900
    15. Some Basic Formulae:

      1. (ab)(ab) = (a2b2)

      2. (ab)2 = (a2 + b2 + 2ab)

      3. (ab)2 = (a2 + b2 - 2ab)

      4. (abc)2a2 + b2c2 + 2(abbcca)

      5. (a3b3) = (ab)(a2abb2)

      6. (a3b3) = (ab)(a2abb2)

      7. (a3b3c3 - 3abc) = (abc)(a2b2c2abbcac)

      8. When abc = 0, then a3b3c3 = 3abc.

The document Decimal Fractions: Notes and Important Formulas | CSAT Preparation - UPSC is a part of the UPSC Course CSAT Preparation.
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FAQs on Decimal Fractions: Notes and Important Formulas - CSAT Preparation - UPSC

1. What are decimal fractions?
Ans. Decimal fractions are numbers that have a decimal point and digits after it. They represent a part of a whole number and are commonly used for precise measurements and calculations.
2. How do you convert a decimal fraction to a common fraction?
Ans. To convert a decimal fraction to a common fraction, we count the number of decimal places and write the decimal as the numerator over a denominator of 1 followed by the appropriate number of zeros. Simplify the fraction if possible.
3. How do you convert a common fraction to a decimal fraction?
Ans. To convert a common fraction to a decimal fraction, divide the numerator by the denominator using long division or a calculator. If the division terminates, the result is a terminating decimal. If the division repeats, the result is a repeating or recurring decimal.
4. How do you compare decimal fractions?
Ans. To compare decimal fractions, line up the decimal points and compare the digits from left to right. If the digits are equal, move to the next digit until a difference is found. The decimal fraction with the larger digit in the first differing place is greater.
5. How do you add or subtract decimal fractions?
Ans. To add or subtract decimal fractions, line up the decimal points and perform the addition or subtraction as usual. Be careful to align the digits correctly and carry or borrow as needed.
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