**Decimal Fractions :**Fractions in which denominators are powers of 10 are known as decimal fractions. Thus , 1/10 = 1 tenth =.1; 1/100 = 1 hundredth = .01**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=25/100=1/4;2.008=2008/1000=251/125.**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, 1.84/2.99 = 184/299 = 8/13; 0.365/0.584 = 365/584 = 5**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.**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 = 0.0730 x 10000 = 730.**Multiplication of Decimal Fractions :**Multiply the given numbers considering them without the 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 .02 x .002). Now, 2x2x2 = 8. Sum of decimal places = (1 + 2 + 3) = 6. .2 x .02 x .002 = .000008**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.**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, 0.00066/0.11 = (0.00066*100)/(0.11*100) = (0.066/11) = 0.006**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.**Pure Recurring Decimal :**A decimal fraction in which all the figures after the decimal point are repeated, is called a pure recurring decimal.

**Convert a Pure Recurring Decimal into Vulgar Fraction**

Remove the number left to the decimal point, if any.

Write the repeated figures only once in the numerator without the decimal point

Write as many nines in the denominator as the number of repeating figures.

Add the number removed in step 1(if any) with the fraction obtained in the above steps.

\( 0.\bar{3} = \frac{3}{9} = \frac{1}{3} \)

\( 0.\bar{7} = \frac{7}{9} \)

\( 0.\overline{36} = \frac{36}{99} = \frac{4}{11} \)

\( 0.\overline{067} = \frac{067}{999} = \frac{67}{999} \)

\( 1.\overline{3} = 1 + \frac{3}{9} = 1 + \frac{1}{3} = \frac{4}{3} \)

\( 5.\overline{36} = 5 + \frac{36}{99} = 5 + \frac{4}{11} = \frac{59}{11} \)

**Convert a Mixed Recurring Decimal into Vulgar Fraction**

Remove the number left to the decimal point, if any.

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

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.

Add the number removed in step 1(if any) with the fraction obtained in the above steps.

\( 0.54\overline{29} = \frac{(5429-54)}{9900} \\~\\= \frac{5375}{9900} = \frac{215}{396} \)

\( 0.1\bar{6} = \frac{(16-1)}{90} \\~\\= \frac{15}{90} = \frac{1}{6} \)

\( 0.58\overline{3} = \frac{(583-58)}{900} \\~\\= \frac{525}{900} = \frac{21}{36} = \frac{7}{12} \)

\( 1.3\overline{18} = 1 + \frac{(318-3)}{990} \\~\\= 1 + \frac{315}{990} = 1 + \frac{63}{198} \\~\\= 1 + \frac{21}{66} = 1 + \frac{7}{22} = \frac{29}{22} \)

**Ans . **

3/4

**Explanation :**0.75 = 75/100 = 3/4

**Ans . **

751/250

**Explanation :**3.004 = 3004/1000 = 751/250

**Ans . **

7/1250

**Explanation :**0.0056 = 56/10000 = 7/1250

**Ans . **

16/29 < 7/12 < 5/8 < 3/4 < 13/16

**Explanation :**Converting each of the given fractions into decimal form, we get : 5/8 = 0.624, 7/12 = 0.8125, 16/29 = 0.5517, and 3/4 = 0.75 Now, 0.5517 < 0.5833 < 0.625 < 0.75 < 0.8125. So 16/29 < 7/12 < 5/8 < 3/4 < 13/16

**Ans . **

8/9 > 9/11 > 3/4 > 13/16

**Explanation :**Clearly, 3/5 = 0.6, 4/7 = 0.571, 8/9 = 0.88, 9/111 = 0.818. Now, 0.88 > 0.818 > 0.6 > 0.571 . and so 8/9 > 9/11 > 3/4 > 13/16

**Ans . **

6891.59775

**Explanation :**6202.5 + 620.25 + 62.025 + 6.2025 + 0.62025 ------------- = 6891.59775

**Ans . **

19.6476

**Explanation :**5.064 + 3.98 + 0.7036 + 7.6 + 0.3 + 2 ------------- = 19.6476

**Ans . **

13.7654

**Explanation :**31.0040 – 17.2368 ----------- = 13.7654

**Ans . **

7.8033

**Explanation :**13.0000 – 5.1967 ----------- = 7.8033

**Ans . **

3767.836

**Explanation :**Let 5172.49 + 378.352 + x = 9318.678 Then , x = 9318.678 – (5172.49 + 378.352) = 9318.678 – 5550.842 = 3767.836

**Ans . **

12498.34

**Explanation :**Let x – 7328.96 = 5169.38. Then, x = 5169.38 + 7328.96 = 12498.34

**Ans . **

632.04

**Explanation :**6.3204 * 1000 = 632.04

**Ans . **

690

**Explanation :**0.069 * 10000 = 0.0690 * 10000 = 690

**Ans . **

3.393

**Explanation :**261 * 13 = 3393. Sum of decimal places of given numbers = (2+1) = 3. 2.61 * 1.3 = 3.393.

**Ans . **

3.03702

**Explanation :**21693 * 14 = 303702. Sum of decimal places = (4+1) = 5 2.1693 * 1.4 = 3.03702.

**Ans . **

0.002560

**Explanation :**4 * 4 * 4 * 40 = 2560. Sum of decimal places = (1 + 2+ 3) = 6 0.4 * 0.04 * 0.004 * 40 = 0.002560

**Ans . **

1.9832

**Explanation :**Sum of decimal places = (2 + 2) = 4 2.68 * 0.74 = 1.9832

**Ans . **

0.7

**Explanation :**63 / 9 = 7. Dividend contains 2 places decimal. 0.63 / 9 = 0.7

**Ans . **

0.0012

**Explanation :**204 / 17 = 12. Dividend contains 4 places of decimal. 0.2040 / 17 = 0.0012

**Ans . **

0.2431

**Explanation :**31603 / 13 = 2431. Dividend contains 4 places of decimal. 3.1603 / 13 = 0.2431

**Ans . **

0.01

**Explanation :**Let 0.006 / x = 0.6, Then, x = (0.006 / 0.6) = (0.006*10) / (0.6*10) = 0.06/6 = 0.01

**Ans . **

2689

**Explanation :**(1 / 0.0003718 ) = ( 10000 / 3.718 ) = 10000 * (1 / 3.718) = 10000 * 0.2689 = 2689

**Ans . **

37/99

**Explanation :**`0.37 = 37/99`

**Ans . **

53 / 999

**Explanation :**53 / 999

**Ans . **

142857/333333

**Explanation :**3.142857 = 3 + 0.142857 = 3 + (142857 / 999999)

**Ans . **

8/45

**Explanation :**= (17 – 1)/90 = 16 / 90 = 8/45

**Ans . **

69 / 550

**Explanation :**= (1254 – 12 )/ 9900 = 1242 / 9900 = 69 / 550

**Ans . **

2 + (161/300)

**Explanation :**= 2 + 0.536 = 2 + (536 – 53)/900 = 2 + (483/900) = 2 + (161/300)

**Ans . **

0.09

**Explanation :**Given the expression : = \( \frac{a^3 + b^3}{a^2 - ab + b^2}\) , where a = 0.05 , b = 0.04 = (a +b ) = (0.05 +0.04 ) = 0.09

Real numbers are of two types: Decimal and Integers. The Integers have no decimal values like 0.33, 0.45; Decimals can be 0.23, 0.333... etc

Integers are of three types negative numbers like -1, -2 .. , zero and positive numbers like 1, 2, 3...

The negative numbers and zero are called non positive and zero and positive numbers are called non negative numbers.

Decimal numbers are of finite or terminating decimal types or infinite decimal types. The infinite decimal type is classified as rational numbers if they can be expressed in the form p / q or irrational number if they can't be expressed in the form p / q.

**Ans . **

61

**Explanation :**Let the number be x. Then, x - 36 = 86 - x => 2x = 86 + 36 = 122 => x = 61. Hence, the required number is 61.

**Ans . **

5

**Explanation :**Let the number be x. Then, 7x - 15 = 2x + 10 => 5x = 25 =>x = 5. Hence, the required number is 5

**Ans . **

2/3 or 3/2

**Explanation :**Let the number be x. Then, x + (1/x) = 13/6 => (x

^{2}+ 1)/x = 13/6 => 6x^{2}– 13x + 6 = 0 => 6x^{2}- 9x – 4x + 6 = 0 => (3x – 2) (2x – 3) = 0 So x = 2/3 or x = 3/2 Hence the required number is 2/3 or 3/2.

**Ans . **

72

**Explanation :**Let the numbers be x and (184 - x). Then, (X/3) - ((184 – x)/7) = 8 => 7x – 3(184 – x) = 168 => 10x = 720 => x = 72. So, the numbers are 72 and 112. Hence, smaller number = 72.

**Ans . **

28 and 17

**Explanation :**Let the number be x and y. Then, x – y = 11 ----(i) and 1/5 (x + y) = 9 => x + y = 45 ----(ii) Adding (i) and (ii), we get: 2x = 56 or x = 28. Putting x = 28 in (i), we get: y = 17. Hence, the numbers are 28 and 17.

**Ans . **

4

**Explanation :**Let the numbers be x and y. Then, x + y = 42 and xy = 437 x - y = sqrt[(x + y)

^{2}- 4xy] = sqrt[(42)^{2}- 4 x 437 ] = sqrt[1764 – 1748] = sqrt[16] = 4. Required difference = 4.

**Ans . **

7 and 8

**Explanation :**. Let the numbers be x and (15 - x). Then, x

^{2}+ (15 - x)^{2}= 113 => x^{2}+ 225 + X^{2}- 30x = 113 => 2x^{2}- 30x + 112 = 0 => x^{2}- 15x + 56 = 0 => (x - 7) (x - 8) = 0 => x = 7 or x = 8. So, the numbers are 7 and 8

**Ans . **

30

**Explanation :**Let the four consecutive even numbers be x, x + 2, x + 4 and x + 6. Then, sum of these numbers = (27 x 4) = 108. So, x + (x + 2) + (x + 4) + (x + 6) = 108 or 4x = 96 or x = 24. :. Largest number = (x + 6) = 30.

**Ans . **

27, 29 and 31

**Explanation :**Let the numbers be x, x + 2 and x + 4. Then, X

^{2}+ (x + 2)^{2}+ (x + 4)^{2}= 2531 => 3x^{2}+ 12x - 2511 = 0 => X^{2}+ 4x - 837 = 0 => (x - 27) (x + 31) = 0 => x = 27. Hence, the required numbers are 27, 29 and 31.

**Ans . **

59 and 43

**Explanation :**Let the numbers be x and y, such that x > y Then, 3x - 4y = 5 ...(i) and (x + y) - 6 (x - y) = 6 => -5x + 7y = 6 …(ii) Solving (i) and (ii), we get: x = 59 and y = 43. Hence, the required numbers are 59 and 43

**Ans . **

36

**Explanation :**Let the ten's digit be x. Then, unit's digit = (x + 3). Sum of the digits = x + (x + 3) = 2x + 3. Number = l0x + (x + 3) = llx + 3. 11x+3 / 2x + 3 = 4 / 1 => 1lx + 3 = 4 (2x + 3) => 3x = 9 => x = 3. Hence, required number = 11x + 3 = 36.

**Ans . **

81

**Explanation :**Let the ten's digit be x. Then, unit's digit = (9 - x). Number = l0x + (9 - x) = 9x + 9. Number obtained by reversing the digits = 10 (9 - x) + x = 90 - 9x. therefore, (9x + 9) - 63 = 90 - 9x => 18x = 144 => x = 8. So, ten's digit = 8 and unit's digit = 1. Hence, the required number is 81.

**Ans . **

3 / 5

**Explanation :**Let the required fraction be x/y. Then, x+1 / y+1 = 2 / 3 => 3x – 2y = - 1 …(i) and x – 1 / y – 1 = 1 / 2 2x – y = 1 …(ii) Solving (i) and (ii), we get : x = 3 , y = 5 therefore, Required fraction= 3 / 5.

**Ans . **

30 and 20

**Explanation :**Let the two parts be x and (50 - x). Then, 1 / x + 1 / (50 – x) = 1 / 12 => (50 – x + x) / x ( 50 – x) = 1 / 12 => x

^{2}– 50x + 600 = 0 => (x – 30) ( x – 20) = 0 => x = 30 or x = 20. So, the parts are 30 and 20.

**Ans . **

6, 4 and 15

**Explanation :**Let the numbers be x, y and z. Then, x+ y = 10 ...(i) y + z = 19 ...(ii) x + z = 21 …(iii) Adding (i) ,(ii) and (iii), we get: 2 (x + y + z ) = 50 or (x + y + z) = 25. Thus, x= (25 - 19) = 6; y = (25 - 21) = 4; z = (25 - 10) = 15. Hence, the required numbers are 6, 4 and 15.