Fractions and Decimals

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1 Chapter Fractions and Decimals FRACTION: A fraction is a number representing a part of a whole. The whole may be a single object or a group of objects. 5 is a fraction with numerator and denominator 5. PROPER FRACTION: A fraction whose numerator is less than the denominator, is called a proper fraction. 7,, 9 5 is a proper fractions. IMPROPER FRACTION: A fraction whose numerator is more than or equal to the denominator, is called an improper fraction. 95 is a improper fractions. MIXED FRACTION: A combination of a whole number and a proper fraction is called a mixed fraction. EQUIVALENT FRACTIONS: A given fraction and various fractions obtained by multiplying (or dividing) its numerator and denominator by the same non-zero number, are called equivalent fractions. 6 9,, 8 6 fraction. If a b and c d are two equivalent fractions, then etc., are equivalent fractions equivalent to the c ad b c i.e., a c a d b c b d b d LIKE FRACTIONS: Fractions having the same denominators are called like fractions. 7,, etc., are like fractions. UNLIKE FRACTIONS: Fractions with different denominators are called unlike fractions. 7 9,, 5 etc., are unlike fractions.

2 FRACTION IN LOWEST TERMS: A fraction is in its lowest terms of its numerator and denominator have no common factor other than. COMPARING FRACTIONS In order to compare fractions, we may use the following steps: I II Find the LCM of the denominators of the given fractions. Convert each fraction to its equivalent fraction with denominator equal to the LCM obtained in step I. Arrange the fractions in ascending or descending order by arranging numerators in ascending or descending order. CONVERSION OF UNLIKE FRACTIONS TO LIKE FRACTIONS In order to convert unlike fractions to like fractions, we follow the following steps: I Find the LCM of the denominators of the given fractions. Convert each of the given fractions into an equivalent fraction having denominator equal to the LCM obtained in step I. ADDITION AND SUBTRACTION OF FRACTIONS In order to add and subtract unlike fractions, we follow the following steps: I II V Obtain the fractions and their denominators. Find the LCM of the denominators. Convert each fraction into an equivalent fraction having its denominator equal to the LCM obtained in step II. Add or subtract like fractions obtained in step III LCM of and is 6. So, convert each fraction to an equivalent fraction with denominator 6

3 MULTIPLICATION OF FRACTIONS We define the multiplication of fractions as follows: Product of their numerators Fractions of two fractions = Product of their denominators i.e., a c ( a c) b d ( b d) Division of Fraction: The division of a fraction a b DECIMALS DECIMALS: Decimals are in extension of our number system. Each decimal number or decimal has two parts, namely (i) Whole-number part, (ii) Decimal part. These parts are separated by a dot (.), called the decimal point. DECIMAL PLACES The number of digits contained in the decimal part of a decimal number is known as the number of decimal places. COMPARING DECIMALS In order to compare decimal numbers, we may follow the following steps: I II Obtain the decimal numbers. Compare the whole parts of the numbers. The number with greater whole part will be greater. If the whole are equal, go to next step. Compare the extreme left digits of the decimal parts of two numbers. The number with greater extreme left digit will be greater. If the extreme left digits of decimal parts are equal, then compare the next digits and so on. Illustration Which is greater of 8. and 9.5? Solution The given decimals have distinct whole number parts, so we compare whole number parts only. In 8., the whole number part is 8. In 9.5, the whole number part is 9. 8 > 9 8. > 9.5 ADDITION AND SUBTRACTION OF DECIMALS In order to add or subtract decimals, we may use the following steps: Convert the given decimals to like decimals.

4 I II V Write the decimals in columns with their decimal points directly below each other so that tenths come under tenths, hundredths come under hundredths and so on. Add or subtract as we add or subtract whole numbers. Place the decimal point, in the answer, directly below the other decimal points. Illustration : Simplify: Solution: We have Converting the given decimals into = like decimals = ( ) ( ) = = MULTIPLICATION OF A DECIMAL BY 0, 00, 000, etc. We follow the following rules to multiply a decimal by 0, 00, 000, etc. I II On multiplying a decimal by 0, the decimal point is shifted to the right by one place. On multiplying a decimal by 00, the decimal point is shifted to the right by two places. On multiplying a decimal by 000, the decimal point is shifted to the right by three places, and so on. MULTIPLICATION OF A DECIMAL BY ANOTHER DECIMAL In order to multiply a decimal by another decimal, we follow the following steps: I Multiply the two decimals without decimal point just like whole numbers. Insert the decimal point in the product by counting as many places from the right to left as the sum of the number of decimal places of the given decimals. Illustration : Find the product of.5 and 6.5? Solution: We have,

5 Since the sum of the decimal places in the given decimals is + =. So, the product must contain places of decimals. Hence, = 5.65 DIVIDING A DECIMAL BY 0, 00, 000 etc. In order to divide a decimal by 0, 00, 000 etc., we follow the following rules: Rule I: Rule : Rule : When a decimal is divided by 0, the decimal point is shifted to the left by one place. When a decimal is divided by 00, the decimal point is shifted to the left by two places. When a decimal is divided by 000, the decimal point is shifted to the left by three places. DIVIDING A DECIMAL BY A DECIMAL In order to divide a decimal by another decimal, we follow the following steps: I Multiple the dividend and divisor by 0 or 00 or 000 etc., to convert the divisor into a whole number. Divide the new dividend by the whole number obtained in step I. Following examples will illustrate the above procedure. Illustration : Divide 0.06 by 0.6 We have, Hence, = 0.06 SOLVED PROBLEMS Problem : Solution: Simplify: We have,

6 [ LCM of 6,8, is = ] 6 8 Problem : Divide: 5 by / Solution: We have, THINGS TO REMEMBER. A fraction is a number representing a part of a whole.. A fraction can be expressed in the form a, where a, b are whole numbers and 0 b. A fraction whose numerator is less than the denominator is called a proper fraction.. Product of their numerator Product of two fraction = Product of their denominators 5. Two fractions are said to be reciprocal of each other, if their product is. The reciprocal of a non-zero fraction a b is equal to. b a 6. The division of a fraction a b by a non-zero fraction c d is the product of a b of c d. with the reciprocal TIPS. Comparison of fractions can be done in the following ways: If ad bc, then a c b d (b) If ad bc, then a c b d If ad bc, then a c b d. To add or subtract find equivalent fractions that have the same denominator. a c a c. To multiply:, then simplify. b d b d

7 . a b Reciprocal of a fraction ( a 0, b 0). b a 5. To divide a c a d a d. b d b c b c 6. If a b and c d ( a N, b N, c N, d N) a b are two fractions, then c d is a fraction between a b and c d. 7. A fraction p is non-terminating repeating or recurring decimal if a digit a block of digits are q repeated in the decimal part Solve: PART I: MISCELLANEOUS DOMAIN (b) Sahil solved 5 part of an exercise while Rahim solved 7 much? of it. Who solved lesser part? By how. A cyclist covers. Find: km in one hour. How far does he go in 8 (b) 5 5. Which is greater:.05 or.50? hours? (d) (e) Express as rupees using decimals: 9 paise (b) 8 rupees 75 paise. 7. Multiply: 8 8 ; (b) 6 ; (d) ; (e) A rectangular table top is m its long and m wide. Find its perimeter? 9. Calculate 7 0. Write the place value of 5 in the following decimal numbers:.5 (b) (d) 8.50 (e) Find (b) (d) 0. 0 (e). (f)

8 . The side of an equilateral triangle is 5.5 cm. Find its perimeter?. If a car covers a distance of 9.5 km in one litre of petrol. How much distance will it cover in 9.5 litres of petrol?. Find: (b) (d) (e) (f) (g) Find the average of 8., 7.5,.9 and 0.6? 6. A train covers.50 km in.5 hours. What is the distance covered by it in hour? 7. Production of wheat is times that of rice, but the cost of rice is times that of wheat. If a farmer produces wheat in place of rice, then what is his income in terms of the previous income? 8. Simplify (0. 0.) 9. Eight people are planning in share equally the cost of a rental car. If one person withdraws from arrangement and the others share equally the entire cost of the car, then by how much is the share of a?? of the remaining persons is increased in terms of original share? 0. Mohan ate half a pizza on Monday. He ate half of what was left to Tuesday and so on. He followed this pattern for one week. How much of the pizza would he have eaten during the week?. A lamp post has half of its length in mud, of its length in water and m above the water, but in the mud. Find the total length of the post?. Chandran gave one-fourth of his money to Suresh. Suresh in turn gave one-third of what he received to Jayesh. If the difference between the amount of Suresh and Jayesh is `00, how much did Chandran have?. Convert (i) 0.89 (ii).7 (ii) to vulgar fraction.. Which of the following numbers 0., 0., (0.), is the greatest? 5. Simplify: In the expression: [.6 [. {.. x)}] 0.65, find the value of x? 7. Simplify: 8. Evaluate: What is equal to?

9 VALUE BASED QUESTIONS. Find the value of. If N, what is the value of N? What is the value of ? A woman sells to the first customer half her stock of apples and half an apple, to the second customer sells half her remaining stock and half an apple, and so on to the third and to the fourth customer. Such that she has now 5 apples left. How many apples did she have before she started selling? 5. Find x if x HIGHER ORDER THINKING SKILLS (HOTS). Simplify: of. 6. Find the value of of of Calculate: A student was asked to solve the fraction his answer wrong? 7 5 of and is answers was. By how much was 5. In the expression [ 6 {. (.. x)}] 0.65, find the value of x?

10 PART II: MULTIPLE CHOICE QUESTIONS Find the value of? (b) (d) is equal to (b) 0 0 (d) None of these. The value of is: (b) (d). When 0.55 is converted into a fraction the result is: 6 (b) Which of the following fractions is the largest? (b) Which of the following fractions is less than 7 8 and greater than? (b) 0 (d) (d) 6 80 (d) 7 7. At the first stop on his route, a driver unloaded 5 of the packages in his van. After he unloaded another packages at his next step, of the original number of packages remained. How many packages were in the van before the first delivery? 5 (b) 0 0 (d) 6 8. What fraction of 7 must be added to itself to make the sum? (b) (d) 5 9. x Find the value of x of the following: The value of upto places of decimal is (b).97.6 (d).65. The product of two fractions is 5 7 (b) and their q is. The greater fraction is: 7 (d) 5

11 . Which of the following sets of fractions is in the correct sequence of ascending order of their values? ,, (b),,,, (d),, The GCD of 5 7,, 6 8 is: 05 8 (b) 8 (d) None of these. Suppose a b, b c, and (b) c d, what was be the value of b as a fraction of d? 8 (d) [( ) ( ) is equal to:.0 (b).0.9 (d) Simplify: [0.9 {.. ( )}] 0.8 (b).8 0 (d) Simplify: (b) (d) 80, ,, is same as (b) (d) is equal to:.0 (b).50.0 (d) expressed as a rational number is: (b) (d) divided by 0.0 is equal to: 56 (b) (d) is the same as: (b) (d) None of these. The value of ( ) is: 0.59 (b) (d) None of these

12 . Consider the following statements:. cannot be written as a terminating decimal.. can be written as a terminating decimal. 5. can be written as a terminating decimal. 6 Which of the statements given above is/are correct? only (b) only and (d) and p q 5. If.5555 (in the lowest form), then what is the value of? q p 0. (b) (d) If A C 6 E, then the value of 5A B 6C D E is: B D (b) (d) Which pair of operations will make the equation below true when inserted into the blank spaces in the shown? and + (b) and + + and (d) and 8. 5 of 5 7 of a number is greater than 9 of of the same number by 8. What is half of the 5 number? 60 (b) 5 0 (d) If two-third of three-fourth of a number added to three-fourth of the fourth-fifth of the number is x times the number, the value of x is (b) 0 9 (d) 0 0. What would be the reciprocal of the sum reciprocals of the numbers 5 and 7? (b) 5 (d) 6 55

DECIMAL FRACTIONS. Thus, 0.25=25/100=1/4;2.008=2008/1000=251/125.

DECIMAL FRACTIONS. Thus, 0.25=25/100=1/4;2.008=2008/1000=251/125. DECIMAL FRACTIONS I. 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; 99/100=99 hundreths=.99;7/1000=7 thousandths=.007,etc