Decimal Fractions

                                                      

Questions from the topic of Decimal  Fractions are being frequently asked in the exams and there are strong chances that the question from this topic will be asked again in the competitive exams like SSC, BANK PO, IB, PLACEMENT EXAMS, GATE and many more. The concepts of this chapter help students in simplifying the advance calculations that seems to be CUMBERSOME and TIME TAKING. With the help of these concepts, students can easily untangle such wide calculations

Students who are  preparing for any exam must learn the rules of DECIMALS  FRACTIONS , and ace themselves in this topic. Having a sound knowledge of this topic and put it in an application, increases their chances of being selected.

Decimal Fractions :

Fractions in which denominator is power of 10 are known as Decimal fractions.

Example :  .063

                                 63             63      

                 .063 =  ━━━   =   ━━━ = 63 x 10⁻³

                              1000            10³

How to convert Decimals into Fractions :

•  Putting 1 in the denominator and put as many Zeros after 1 as is the number of digits after the decimal point. 

• Then remove the Decimal and convert the fraction into the simplest form. 

Example :                 625          5

                    .625 = ━━━  =  ━━

                                 1000          8

KEY HIGHLIGHTS : 

• Putting any number of zeros to the Extreme Right of the decimals does not change the value.

Example :  0.63 = 0.630 = 0.6300 = 0.63000

• If the numerator and denominator of the fraction has the same number of Decimal Places, the Decimals can be removed directly without putting any number of Zeros after it.

Simplifying Decimals :

• Addition & Subtraction : Numbers should be placed so that all the Decimals lie in the same column.

Example :  Adding  2.345 + 0.2345 + 23.45 + 234.5 + 0.02345

                   2.345              (3 Places after Decimal)

                               0.2345            (4 Places after Decimal)

                             23.45                (2 Places after Decimal)

                           234.5                  (1 Place after Decimal)

                     ╀      0.02345           (5 Places after Decimal)

                      ━━━━━━━

                           260.55295

            To ease the calculation of decimals, the blank places can be filled with the Zeros.

• Multiplication : Multiply the number without considering the Decimals from the digits and mark the Decimals in the Final Answer accordingly.

Example :  Multiply 0.14 x 0.26

Sol :          Simply multiply 14 x 26 = 364

Now put the Decimal as in Sum of the Number of Decimal Places in the question.

Here Number of Decimal Places = 2 + 2 = 4

Hence, the required answer is 0.0364

• Multiplying Decimal Fraction by a Power of 10 :

• Shift the Decimals to the Right Hand Side Of the number , if Power of 10 is Positive

• Shift the Decimal to the Left Hand Side of the Number, if the Power of 10 is Negative.

Example : 0.92348 x 10⁶ 

Sol : Shifting Decimal to the Right Place; Since Power of 10 is Positive. 

        Shifting decimal to 6 Places Right Hand Side.

        Hence, 0.92348 x 10⁶  = 923480

Example : 6238.58739 x 10⁻³

Sol : Shifting Decimal to the Left Side Place; Since Power of 10 is Negative.

        Shifting decimal to 3 Places Left Hand Side.

        Hence, 6238.58739 x 10⁻³ = 6.23858739

• Dividing Decimal Fraction by Natural Number : Dividing the numbers without considering the Decimals of the digits. Finally, put the Decimal in the Quotient to give as many places of Decimal as there are in the Dividend. 

Example : Divide 0.0343  by 7.

Sol :                              343

                  0.0343 =   ━━━ = 49 = 0.0049

                                       7

• Dividing Decimal Fraction by another Decimal Fraction : Divide both Dividend and Divisor by putting a suitable power of 10 making them into a Natural Number. 

Example : Divide 0.289 by 0.17

Sol :        0.289          289 x 10²        17

             ━━━━ =  ━━━━━ =  ━━  =  17 x 10⁻¹ = 1.7

                0.17            17 x 10³         10¹

Comparison of Fractions :

While rearranging Fractions either in the Ascending or the Descending Order, converting the Fraction into the Decimals makes the task easy.

Another Approach : 

• Taking the L.C.M. of the Denominator  

• Convert each of the given Fraction in the same Denominator form 

• Arrange the Numerators accordingly in the required pattern of Ascending or Descending.

Example :   Arrange these  in Ascending Order. 5/6, ¾ , ⅔ , ⅛, 2/9.

Sol : Taking L.C.M. of 6, 4, 3, 8, 9

        L.C.M. of 6, 4, 3, 8, 9 = 72

        Respective Numerators  = (5 x 12), (3 x 18), (2 x 24) , (1 x 9), (2 x 8)

                                               = 60, 54, 48, 9 , 16

       Ascending Order =  ⅛ < 2/9 < ⅔ < ¾ <  ⅚.

Recurring Decimal : 

• Pure Recurring Decimals : Decimal  Fractions in which all the figures after the decimal are repeated , such decimal fractions are Pure Recurring Decimals.

Example : ⅔ = 0.6666…….. , 

                

                 22   

             ━━━   = 3.142857142857…..

                 7

• Mixed Recurring Decimals : Decimal Fractions in which some values do not repeat themselves after the decimal, such decimal fractions are Mixed  Recurring Decimals.

Example : 0.1733333

Converting a Recurring Decimal into Fraction :

• Pure Recurring Decimals : Writing repeated numbers only once in the Numerator and take as many 9’s in Denominator as is the number of repeated digits of Numerator

Example :    

                    ━        59

                 0.59 =  ━━─

                               99

• Mixed Recurring Decimals : In Numerator, take the difference between  the number formed by all the digits after Decimal Point  and the non- repeating digits. In Denominator, put as many numbers of 9’s as of repeating digit  and as many numbers of 0’s for the non- repeating digits. 

Example :    

                       ━         1729 - 17         1712          428

                0.1729  =   ━━━━━  = ━━━━ = ━━━

                                        9900        9900 2475

Basic Formulas : These formulae are also used while solving the questions.

• (a + b)² = a² + b² + 2 ab

• (a - b)² = a² + b² - 2 ab

• (a + b)² + (a - b)² = 2 ( a² + b² )

• (a + b)² - (a - b)² = 4 ab

• a² - b² = (a + b) (a - b)

• (a + b + c)² = a² + b² + c² + 2 ( ab + bc + ca)

• (a + b)³ = a³ + b³ + 3ab (a + b) = a³ + b³ + 3a²b + 3ab²

• (a - b)³ = a³ - b³ - 3ab (a - b) = a³ - b³ - 3a²b + 3ab²

• a³ + b³ = ( a + b ) ( a² + b² - ab )

• a³ - b³ = ( a - b ) ( a² + b² + ab )

• (a³ + b³+ c³ - 3 abc)  = (a + b+ c) (a² + b² + c² - ab - bc - ca)

In the above identity ;

CASE 1 :  if (a + b+ c) = 0 

(a³ + b³+ c³ - 3 abc)  = 0

Hence, a³ + b³+ c³ = 3abc

CASE 2 : (a³ + b³+ c³ - 3 abc)  = ½ { 2(a + b+ c) (a² + b² + c² - ab - bc - ca)} 

Or,  (a³ + b³+ c³ - 3 abc) = ½ {(a + b+ c) (2a² + 2b² + 2c² - 2ab - 2bc - 2ca)

Or,  (a³ + b³+ c³ - 3 abc) = ½ {(a + b+ c) ( a - b )² ( b - c )² ( c - a )² }

Practice Question :

Q 1. Which of the following is the fraction for 0.36?

1. 9/25

2. 51/25

3. 3/400

4. 2081/250

Sol :                36            9

          0.36 = ━━━ = ━━

                       100         25

Q2. What decimal of an hour is a second ?

1. .0025

2. .0256

3. .00027

4. .000126

Sol :    1 Hour = 60 Minutes = 60 x 60 Seconds

           1 Hour = 3600 Seconds 

                                     1

           1 Second =  ━━━━ Hr.  = 0.00027

                                  3600

Q3. 

The value of	(0.96)3 - (0.1)3	:
	(0.96)2 + 0.096 + (0.1)2

1. 0.86

2. 0.95

3. 0.97

4. 1.06

Sol : Using the identity ; a³ - b³ = ( a - b ) ( a² + b² + ab )

Q4. 

The value of	0.1 x 0.1 x 0.1 + 0.02 x 0.02 x 0.02	:
	0.2 x 0.2 x 0.2 + 0.04 x 0.04 x 0.04

1. 0.0125

2. 0.125

3. 0.25

4. 0.5

Q5. When 0.5656565656……. is converted into fraction, then the result is : 

1. 56/100

2. 56/99

3. 5656/10000

4. 5656/9999

Sol : Convert Pure Recurring Decimal into the Fraction.

Q6. The expression (11.98 x 11.98 + 11.98 x x + 0.02 x 0.02) will be a perfect square for x equal to :

1. 0.02

2. 0.4

3. 0.2

4. 0.04

Sol : Using the identity; (a + b)² = a² + b² + 2 ab

Q7. 

(0.1667)(0.8333)(0.3333)	is approximately equal to:
(0.2222)(0.6667)(0.1250)

1. 2.0

2. 2.4

3. 2.5

4. 2.9

Q8. 0.04 x 0.0162 is equal to :

1. 6.48 x 10⁻³

2. 6.48 x 10⁻⁴

3. 6.48 x 10⁻⁵

4. 6.48 x 10⁻⁶

Sol : Multiply 4 x 162 = 648

        No. of Decimal Places = 2 + 4 = 6

        According to this, required answer = 6.48 x 10⁻⁴

Q 9. 

4.2 x 4.2 - 1.9 x 1.9 is equal to:         2.3 x 6.1 0.5 1.0 20 22 Sol : Using the identity; a² - b² = (a + b) (a - b) Q 10. The price of commodity X increases by 40 paise every year, while the price of commodity Y increases by 15 paise every year. If in 2001, the price of commodity X was Rs. 4.20 and that of Y was Rs. 6.30, in which year commodity X will cost 40 paise more than the commodity Y ? 2010 2011 2012 2013 Sol :    According to Question,                               40                            15               40              4.20  +  ━━─ X   -   6.30  +  ━━ Y  =  ━━━

                              100                          100             100

Ask me in the comment section, if you face any problem while solving questions of Decimal Fractions . Visit next page for more concepts and problems.