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Fractions A fraction is a part of the whole (object, thing, region). It forms the part of basic aptitude of a person to have and idea of the parts of a population, group or territory. Civil servants must have a feel of ‘fractional’ thinking. eg, 5 , here ‘12’ is the number of equal part into which the whole has been divided, is called denominator and ‘5’ is 12 the number of equal parts which have been taken out, is called numerator. Example1: Name the numerator of Solution: Numerator of Denominator of 3 5 . and denominator of 13 7 3 is 3. 7 5 is 13. 13 Lowest Term of a Fr action Dividing the numerator and denominator by the highest common element (or number) in them, we get the fraction in its lowest form. eg, To find the fraction 6 in lowest form Since ‘2’ is highest common element in numerator 6 and denominator 14 14 so dividing them by 2, we get Equiv alent 3 6 . Which is the lowest form of . 7 14 Frac tions If numerator and denominator of any fraction are multiplied by the same number then all resulting fractions are called equivalent fractions. eg, 1 2 3 4 1 , , , all are equivalent fractions but is the lowest form. 2 4 6 8 2 Example 2: Find the equivalent fractions of 2 having numerator 6. 5 Solution: We know that 2 × 3 = 6. This means we need to multiply both the numerator and denominator by 3 to get the equivalent fraction. Hence, required equivalent fraction 2 2 3 6 5 5 3 15 Additio n and Subtraction of Fractions Here two cases arise as denominators of the fraction are same or not. Case I: When denominators of the two fractions are same then we write denominator once and add (or subtract) the numerators. 2 3 5 = 7 7 7 eg, Case II: If denominators are different, we need to find a common denominator that both denominators will divide into. 1 3 6 8 eg, 1 2 3 4 , = = 6 12 18 24 We can write, 3 6 9 = = 8 16 24 13 1 3 4 9 = = 24 6 8 24 24 Example 3: Calculate Solution. 1 3 2 7 1 1 7 7 3 3 2 6 = = and = = 2 2 7 14 7 7 2 14 1 3 7 6 1 = = 2 7 14 14 14 Multipl ication and Division of Fr actions To multiply fractions, the numerators are multiplied together and denominators are multiplied together. eg, 1 3 1 3 3 1 = = = 6 8 6 8 48 16 In division of fraction, the numerator of first fraction is multiplied by the denominator of second fraction and gives the numerators. Also denominator of first fraction is multiplied by the nemerator of second fraction and gives the denominator. eg, 1 3 becomes 16 8 1 8 8 4 = = 6 3 18 9 P ro pe r a n d Im pro per F ra c t io ns The fractions in which the number in numerator is less than that of denominator, are called proper fractions. Also if the number in numerator is greater than that of denominator, then the fractions are called improper frations. eg, 4 3 is an improper fraction while is a proper fraction. 3 4 Mix e d Nu m be rs A mixed number is that, which contains both a whole number and a fraction. eg, 4 7 1 5 ,3 ,6 are mixed numbers. 12 4 8 Example 4: Which of the following are proper and improper fractions? 7 6 12 5 (b) (c) (d) 9 5 7 13 Solution. (a) and (d) are proper fractions as numerator is less than denominator. Also, (b) and (c) are improper fractions as numerator is greater than denominator. (a) Illusration 5: Solution. Are 7 5 and 4 mixed number? 13 6 7 5 is only a proper fraction as it does not contain any whole number, while 4 is a mixed number 13 6 as it contains ‘4’ as a whole number and D ec im al 5 is a fraction. 6 F ra c t io ns The fractions in which denominators has the power of 10 are called decimal fractions. eg, 0.25 = nought point two five = 0.1 = nought point one = 25 1 = = one quarter 100 4 1 = one-tenth. 10 For converting a decimal fraction into simple fraction, we write the numerator without point and in the denominator, we write ‘1’ and put the number of zeros as many times as number of digits after the point in the given decimal fraction eg, 0.037 = 37 1257 ,0.1257 = 1000 10000 Example 6: Convert the each of the following decimal fractions into simple fractions. (a) 5.76 (b) 0.023 (c) 257.5 Solution. (a) 5.76 = 576 288 144 23 = = (b) 0.023 = 100 50 25 1000 (c) 257.7 = 2575 515 = 10 2 Addit io n or Subt rac t ion o f D e cim a l F rac t ions In the addition or subtraction of decimal fractions, we write the decimal fractions in such a way that all the decimal points are in the same straight line then these numbers can be add or subtract in simple manner. Example 7: Solve 0.68 + 0.062 + 0.20 Solution. 0.680 +0.062 +0.200 0.942 Multip lication of Decimal Fractions To multiply by multiplies (powers) of 10 the decimal point is moved to the right by the respective number of zero. Example 8: 0.75 × 10 = ? Solution. 0.75 × 10 = 7.5 (The decimal point is shifted to right by one place) To multiply decimals by number other than 10. We ignore the decimal point and multiply them in simple manner and at last put the points after the number of digits (from right) corresponding to the given problem. Example 9: Multiply 8 and 10.24 Solution. First we multiply 8 and 1024 8 × 1024 = 8192 Now, 8 × 10.24 = 81.92 (We put decimal points after two digits from right as in given question). Example 10: 12.4 × 1.62 Solution. We know that 124 × 162 = 20088 12.4 × 1.62 = 20.088 Division of Decimal Fr actions Division of decimal numbers is the reverse of the multiplication case ie, we move the decimal point to the left while dividing by multiplies of 10. Example 11: 25.75 ÷ 10 = ? Solution. 25.75 ÷ 10 = 2.575 When a decimal number is divided by an integer, then at first divide the number ignoring the decimal point and at last put the decimal after the number of digits (from right) according at in the given problem. Example 12: Divide 0.0221 by 17 Solution. First we divide 221 by 17 ie, without decimal 221 ÷ 17 = 13 Now, we put decimal according as in the given problem 0.0221 ÷ 17 = 0.0013 If divisor and dividend both are decimal numbers then first we convert them in simple fraction by putting number of zero in the denominator of both. Then divide by the above manner. Example 13: Divide by 0.0256 by 0.016 Solution. 0.0256 256 1000 256 25.6 = = = =1.6 0.016 16 10000 160 16 Example 14: Divide 70.5 by 0.25 Solution. 70.5 705 100 7050 = = 282 = 0.25 25 10 25 To Find HCF and LCM of Decimal Fr actions First we make the decimal digits of the given decimal numbers, same by putting some number of zero if necessary. Then find HCF or LCM ignoring decimals. And at last put the decimal according as the given numbers. Example 15: Determine HCF and LCM of 0.27, 1.8 and 0.036. Solution. Given numbers are 0.27, 1.8 and 0.036. or 0.270, 1.800 and 0.036. These numbers without decimals are 270, 1800 and 36. Now, HCF of 270, 1800 and 36 = 18 HCF of 0.270, 1.800 and 0.036 = 0.018 LCM of 270, 1800 and 36 = 18 × 5 × 2 × 3 × 10 = 5400 18 270 1800 36 5 15 100 2 2 3 20 2 3 10 1 LCM of 0.270, 1.800 and 0.36 = 5.400 or 5.4 Terminati ng and Non-Termi nating Recurring Decimals If decimal expression of any fraction be terminated then fraction is called terminating. as 5 = 0.3125 16 But if we take example 33 ÷ 26, then 26) 33 (1.2692307 26 70 52 180 A 156 240 234 60 52 80 78 200 182 180 B In this division, we see that remainder at the stages A and B are same. In the continued process of division by 26, the digits 6, 9, 2, 3, 0, 7 in the quotient will repeat onwards. 33 = 1.2692307692307... 26 This process of division is non terminating. Therefore, such decimal expressions are called non terminating repeating (recurring) decimals. In repeating digit, we put (–) bar. 33 = 1.2692307 26 ie, Example 16: Write the following fractions in decimal form and till that these are terminating or non terminating recurring. 2 3 Solution. (b) (a) 4 5 (c) (a) 2 = 0.6666.... = 0.6 non terminating recurring 3 (b) 4 = 0.8 terminating 5 (c) 3 = 0.272727... = 0.27 non terminating recurring 11 (d) 17 = 0.1888... 0.18 non terminating recurring 90 No n- Te rm in at ing, Non- R e curring 3 11 (d) 17 90 De c ima ls Every fraction can be put in the form of terminating or non-terminating recurring decimals ie, these decimal p numbers can be put in the form of q . These are called rational numbers. But some decimals numbers are there that can’t be put in the form of p , these are non-terminating, non-recurring decimals. Also these are called q irrational numbers. eg, 0.101001000100001... To c o n ve rt n o n-t e rmina ting re c urring dec im als into sim ple fra c tio n First write the non-terminating recurring decimal in bar notation. Then write the digit ‘a’ in the denominators as many times as number of digits recurring in the numerator. Also don’t put decimal in the numerator. Example 17 Convert the following in simple fraction (i) 0.33333... (ii) 0.181818... Solution. 3 1 (i) 0.33333... = 0.3 = = 9 3 18 2 (ii) 0.181818... = 0.18 = = 99 11 Mixed Recurring Decimals A decimal fraction in which some digits are not repeated and some are repeated, called mixed recurring decimal. Ho w to C on ve rt Mix ed R ec urring D ec im al int o a Simple F ra ct io n? First we subtract non repeated part from the number (without decimal) and put number 9 as many times as number of recurring digits and also put the number ‘0’ as many times odd number of non-recurring digits. Example 18: Convert the following in simple fraction (i) (ii) 3.0072 0.18 Solution. (i) 0.18 = 18 1 17 = 90 90 2 72 2 = 3 =3 (ii) 3.0072 = 3 + 0.0072 = 3 275 9900 275 Example 19: Arrange 2 13 4 15 , , and in ascending order, 3 15 5 16 Solution. We know that 2 = 0.67 3 13 = 0.86 15 4 = 0.80 5 15 = 0.94 16 Here, it is clear that 0.67 < 0.80 < 0.86 < 0.94 2 4 13 15 3 5 15 16 7 8 5 2 Example 20: Arrange , , and is descending order. 9 11 13 7 Solution. We know that 7 = 0.77 9 8 = 0.72 11 5 = 0.38 13 2 = 0.28 7 Here it is clear that 0.77 0.72 0.38 0.28 7 8 5 2 9 11 13 7 EXERCISE 1. If 2025 = 45, then the value of 0.00002025 0.002025 20.25 = (a) 49.95 (c) 4.9995 2. If 2 3 1 7 of 3 5 of is equal to: 9 11 7 9 (a) 8 (b) 9 (c) 7/9 (d) 5/4 11. 9 1 2025 (b) 49.5495 (d) 499.95 15 = 3.88, the the value of 12. 5 is: 3 (a) 1.39 (b) 1.29 (c) 1.89 (d) 1.63 3. If 2805 ÷ 2.55 = 1100, then 280.5 ÷ 25.5 is: (a) 111 (b) 1.1 (c) 0.11 (d) 11 4. The value of 213 + 2.013 + 0.213 + 2.0013 is: (a) 217.2273 (b) 21.8893 (c) 217.32 (d) 3.217.32 0.05 0.05 0.05 0.04 0.04 0.04 ? 0.05 0.05 0.05 0.04 0.04 0.04 (a) 0.09 (b) 0.9 (c) 0.009 (d) 0.001 6. The numerator of a non-zero rational number is five less than the denominator. If the denominator is increased by eight and the numerator is doubled, then again we get the same rational number. The required rational number is: (a) 1/8 (b) 4/9 (c) 2/8 (d) 3/8 5. 7. The simplification of 0.6 0.7 0.8 0.13 yields: (d) 1.46 (b) 2.46 (c) 3.46 (a) 2.46 8. What will be the approximate value of 779.5 × 5 – 46.5 × 19 – 9? (a) 10800 (b) 18008 (c) 10080 (d) 10008 9. A fraction becomes 2 when 1 is added to both the numerator and the denominator and it becomes 3 when 1 is subtracted from both numerator and denominator. The numerator of the given fraction is: (a) 7 (b) 5 (c) 3 (d) 8 10. The simplification of 3.36 2.05 1.33 equals: (a) 2.46 (b) 2.61 (c) 2.64 (d) 2.64 1 1 1 1 1 1 1 1 20 30 42 56 72 90 110 132 is equal to: (a) 3/6 (b) 1/7 (c) 1/6 (d) 1/10 13. The value of (0.3467 0.1333) is (a) 4.38 (b) 0.4801 (c) 0.48 (d) 4.481 3 7 1 1 5 3 3 14. Simplify: 4 of 5 8 3 4 7 4 7 1 6 2 (a) 2/9 (b) 56/77 (c) 50/73 (d) 3 2 9 15. What approximate value should come in place of z in the following question? 242 + 2.42 + 0.242 + 0.0242 = z (a) 670 (b) 575 (c) 580 (d) 680 16. What should come in place of the question mark (?) in the following question? 248.8 – 190.48 + 68.08 = ? (a) 126.4 (b) 127.4 (c) 128.4 (d) 124.6 17. Find a fraction such that if 1 is added to the denominator it reduces to ½ and reduces to 3/5 on adding 2 to the numerator. (a) 12/25 (b) 13/25 (c) 11/13 (d) 11/25 5.32 56 5.32 44 is: 7.662 2.342 (a) 14 (b) 13 (c) 10 (d) 12 19. The rational number, which equals the number 18. The value of 2.357 with recurring decimal is: (a) 235/101 (b) 235/997 (c) 2335/1001 (d) 2379/997 © 2011 www.upscportal.com 24 Fractions 20. xy is a number that is divided by ab where xy < ab and gives a result 0.xyxyxy... then ab equals: (a) 77 (b) 33 (c) 99 (d) 66 21. Express (a) 6 20 as mixed fraction. 3 2 3 22. Express 7 30 (a) 4 (b) 5 2 3 (c) 4 2 3 (d) 3 33 (c) 4 2 (b) 5 2 (c) 6 (a) 18 27 (b) 180 5 and 360 8 (c) 180 2 and 200 5 (d) 220 2 and 550 5 (a) 34 (d) 4 1 2 (b) 3 2 (d) 9 6025 241 (b) 10000 400 29. Solve (a) 2 (d) 7 to 25. Simplest form of (b) (c) 2 3 (d) 4 3 (d) 605 844 (d) 5 8 28. Simplest fraction form of 0.6025 is 24. The numerator of the fraction which is equivalent 15 with denominator 7 is 35 (a) 5 (b) 3 (c) 7 250 2 and 400 3 (a) 1 1 23. Which one of the following is equivalent to ? 3 1 (a) 2 (a) 27. Which of the following is proper fraction? 2 3 3 as improper fraction. 4 31 (b) 4 26. Which of the following is proper fraction? 30. (c) 525 700 1.15 to its simplest form. 1.84 115 184 (b) 23 184 (c) 115 23 Solve. 0.635 + 1.8 + 2.9 + 18.358 + 0.02345 (a) 23.78645 (b) 22.87654 (c) 23.88456 (d) 24.78645 36 is 54 6 9 (c) 2 3 (d) 1 3 ANSWERS 1. (b) 11. (a) 21. (a) 2. (b) 12. (c) 22. (b) 3. (d) 13. (b) 23. (c) 4. (a) 14. (d) 24. (b) 5. (a) 15. (d) 25. (c) 6. (d) 16. (a) 26. (d) 7. (a) 17. (b) 27. (c) 8. (a) 18. (c) 28. (b) 9. (a) 19. (d) 29. (d) 10. (d) 20. (c) 30. (a) EXPLANATIONS 1. 2025 = 45 = 3 2 1 0.00002025 0.002025 = 0.0045 + 0.045 + 45 + 4.5 = 49.5495 2025 20.25 11. 5 = 3 3. 1100 1 100 = 11 0.05 3 0.04 3 0.05 2 0.05 0.04 0.04 2 5. if 0.05 = a, 0.04 = b a3 b3 =a+b a ab b2 = 0.05 + 0.04 = 0.09 6. Let the denominator be x, then by the given x 5 . Also, condition the rational number is x 2 x 5 x 5 2 1 = = or or 2x = x + 8 or x = 8 x 8 x x x 8 then, 11 36 36 7 =9–4÷4 9 11 7 9 =9–1=8 9 5 3 = 3 3 280.5 ÷ 25.5 3.88 15 = 1.29 = 3 3 2805 = 2.55 10 10 The rational number is 3 . 8 7. 0.6 0.7 0.8 0.13 6 7 8 13 246 = = 2.46 9 9 9 99 99 8. 779.5 × 15 – 46.5 × 19 – 9 = ? = 11692.5 – 883.5 – 9 = 11692.5 – 892.5 = 10800 = 10. 3.36 2.05 1.33 2 3 1 5 of 3 5 of 7 9 11 7 Using BODMAS rule, 15 = 3.88 2. 9 1 36 5 33 64 = 2 = 2.64 99 99 99 99 3467 34 3433 , = 9900 9900 1333 13 1320 = 0.1333 = 9900 9900 3433 1320 .03467 0.1333 = 9900 9900 4753 = = 0.4801. 9900 3 2 4 7 1 1 5 3 of 3 14. 5 8 3 4 7 4 7 1 6 Using BODMAS rule, 13. 0.3467 = 11 4 = 11 6 6 = 4 6 = 4 12 = 7 7 7 5 3 3 8 12 7 4 7 7 7 5 9 8 12 7 28 8 7 5 28 7 12 7 9 7 20 20 = 1 12 9 9 29 2 = 3 . = 9 9 357 999 2 299 357 2355 = = 999 999 xy . 20. 0.xy = 99 21. 3) 20 (6 18 2 19. 2.357 = 2 0.357 = 2 20 2 = 6 3 3 22. 7 3 31 = 4 4 23. 1 1 2 2 = = 3 3 3 6 24. 15 15 5 2 = = 35 35 5 3 25. 36 36 18 2 = = 54 54 18 3 26. 220 22 2 = = 550 55 5 27. 2 is proper fraction as 2 is less than 3. 3 28. 0.6025 = 6025 241 = 10000 400 1.15 115 115 23 5 = = = 1.84 184 184 23 8 30. 0.635 1.87 2.9 18.358 + 0.02345 23.78645 29.