Tiêu đề 02-Fraction and Decimals(1).pdf ✅

FRACTIONS & DECIMALS 2 CHAPTER CONTENTS • Definition : Fraction • Pictures form • Types of Fraction • Simplest form of Fractions • Addition & Subtraction of Fractions • Multiplication of Fractions • Division of Fractional Numbers • Simplifying Brackets in Fractions • Decimals • Definition : Decimals • Comparing Decimals • Addition & Subtraction of Decimals • Conversion of a Decimal Number into a Decimal Fraction • Multiplication of Decimal Numbers • Division of Decimal Numbers • Conversion of Units ➢ DEFINITION : FRACTION A fraction is a number which can be written in the form b a , where both a and b are natural numbers and the number 'a' is called numerator and 'b' is called the denominator of the fraction b a , b  0. For example, 15 7 , 5 0 , 3 1 , 5 2 , are fractions. ➢ PICTURES FORM A fraction represents a part of a whole, where the denominator of the fraction represents the number in which equal parts the whole is divided and the numerator shows the number of equal parts taken. 8 3 For example, the shaded part of the figure represents the fraction 8 3 . ➢ TYPES OF FRACTION  Proper Fraction : A proper fraction is a fraction in which the numerator is smaller than the denominator. For example, 29 12 , 7 3 , 9 2 ,..., etc. are proper fractions.  Improper Fraction : An improper fractions is a fraction in which the numerator is greater than the denominator.
For example, 13 17 , 17 29 , 5 7 , ...... , etc. are improper fractions.  Like Fractions : The fractions with the same denominator are called like fractions. For example, 12 11 , 12 5 , 12 7 , ..... , etc. are like fractions.  Unlike Fractions : The fractions with different denominators are called unlike fractions. For example, 8 7 , 13 11 , 5 4 , 3 2 , .... , etc. are unlike fractions.  Unit Fractions : The fraction with numerator 1 are called unit fractions. For example, 7 1 , 3 1 , 4 1 , 2 1 , ...., etc. are unit fractions.  Mixed Numerals : Mixed numerals are combination of a whole number and a proper fraction. For example, fractions 4 1 , 8 3 1 , 5 2 1 3 , etc. are mixed numerals or mixed fractions.  Equivalent Fractions : If m b m a d c   = , then the fractions b a and d c are called equivalent fractions because they represent the same portion of the whole. For example, 3 2 2 2 6 4   = ; 16 3 5 3 48 15   = For example, the shaded parts of each of the following figures are same but they are represented by different fractional numbers. 2 1 4 2 8 4 They are called equivalent fractions. So we write 2 1 = 4 2 = 8 4 , etc.  Decimal fractions : A fraction whose denominator is any of the number 10,100,1000 etc. is called a decimal fraction. For example : 1000 17 , 100 11 , 10 8 etc. are decimal fractions.  Vulgar fractions : A fraction whose denominator is a whole number, other than 10,100,1000 etc. is called a vulgar fractions. For example , 17 11 , 8 3 , 7 2 etc. are vulgar fractions. ➢ SIMPLEST FORM OF FRACTIONS If numerator and denominator of a fraction have no common factor other than 1, then the fraction is said to be in its simplest form i.e. HCF of both is 1. For example, 13 12 , 7 3 , 5 4 , 5 3 , etc. are the fractions in simplest form. ➢ ADDITION AND SUBTRACTION OF FRACTIONS There are two case of adding and subtracting fractions : 1. Fractions with Similar Denominators. (Like fractions) 2. Fractions with Different Denominators (Unlike fractions)  Fractions with Similar Denominators : For example : Ex.1 Solve the following : (i) 5 3 5 2 + (ii) 7 3 7 4 − Sol.(i) 5 3 5 2 + = 5 2 + 3 = 5 5 = 1 (ii) 7 3 7 4 − = 7 4 −3 = 7 1  Fractions with Different Denominators : Use of L.C.M. of denominators. For example : Ex.2 Solve the following : (i) 3 4 5 2 + (ii) 8 1 9 3 −
Sol. (i) 3 4 5 2 + [L.C.M. of 5 and 3 = 15] = 15 23+ 45 = 15 6 + 20 = 15 26 = 1 15 11 (ii) 8 1 9 3 − [L.C.M. of 8 and 9 = 72] = 72 38−91 = 72 24 −9 = 72 15 = 24 5 ❖ EXAMPLES ❖ Ex.3 Solve the following : (i) 5 3 2 − (ii) 8 7 4 + (iii) 15 4 11 9 − (iv) 8 5 3 2 1 8 − (v) 2 1 3 3 2 2 + (vi) 2 3 5 2 10 7 + + Sol. (i) 5 3 2 − = 5 3 1 2 − [L.C.M. of 1 and 5 = 5] = 5 25−31 = 5 10 − 3 = 5 7 = 5 2 1 (ii) 8 7 4 + = 8 7 1 4 + = 8 48+ 71 = 8 32 + 7 = 8 39 = 8 7 4 (iii) 15 4 11 9 − [L.C.M. of 11 and 15 = 165] = 165 159 −114 = 165 135 − 44 = 165 91 (iv) 8 5 3 2 1 8 − = 2 17 – 8 29 [L.C.M. of 2 and 8 = 8] = 8 174 − 291 = 8 68− 29 = 8 39 = 4 8 7 (v) 2 1 3 3 2 2 + = 2 7 3 8 + [L.C.M. of 3 and 2 = 6] = 6 16 + 21 = 6 37 = 6 6 1 . (vi) 2 3 5 2 10 7 + + [L.C.M. of 2, 5 and 10 = 10] = 10 71+ 22 + 35 = 10 7 + 4 +15 = 10 11+15 = 10 26 = 5 13 = 5 3 2 Ex.4 Arrange the following in descending order : 10 7 , 7 3 , 5 1 Sol. 10 7 , 7 3 , 5 1 [L. C. M of 5, 7 and 10 = 70] = 5 14 1 14   , 7 10 3 10   , 10 7 7 7   = 70 49 , 70 30 , 70 14 Descending order is 70 14 70 30 70 49   i.e., 5 1 7 3 10 7    5 1 , 7 3 , 10 7 are in descending order. Ex.5 A rectangular sheet of paper is 12 2 1 cm long and 10 3 2 cm wide. Find its perimeter. Sol. Length of paper = 12 2 1 cm = 2 25 cm Breadth of paper = 10 3 2 = 3 32 cm Perimeter of rectangular paper sheet = 2(length + breadth) =       + 3 32 2 25 2 =        +  6 25 3 32 2 2 =       + 6 75 64 2 = 6 2139 = 3 139 = 46 3 1 cm.
Ex.6 Ritu ate 5 3 part of an apple and the remaining apple was eaten by her brother Somu. How much part of the apple did Somu eat ? Who had the larger share ? By how much ? Sol. Ritu ate = 5 3 part of apple Somu ate =       − 5 3 1 part of apple So, Somu ate = 5 2 part of apple Ritu ate more apple than Somu.        5 2 5 3  Ritu ate       − 5 2 5 3 more share. i..e., Ritu ate       = − 5 1 5 3 2 more apple than Somu. Ex.7 Michael finished colouring a picture in 12 7 hr. Vaibhav finished colouring the same picture in 4 3 hr. Who worked longer ? By that fraction was it longer ? Sol. Michael finished colouring picture in = 12 7 hr. Vaibhav finished colouring picture in = 4 3 hr. i.e., Vaibhav finished colouring picture in = 4 3 3 3   = 12 9 hr. Vaibhav worked longer, because 12 7 12 9  Vaibhav worked longer by =       − 12 7 12 9 hr. = 12 9 − 7 = 12 2 = 6 1 hr. Ex.8 Write three equivalent fractions of 7 2 . Sol. 14 4 7 2 2 2 7 2 =   = 21 6 7 3 2 3 7 2 =   =  28 8 7 4 2 4 7 2 =   = So, three equivalent fractions of 7 2 are 14 4 , 21 6 and 28 8 . Ex.9 Identify proper, improper and mixed fractions from the following numbers : 9 4 , 1 13 37 , 4 1 , 3 100 11 , 7 4 , 3 7 , 3 1 2 Sol. Proper fractions are = 100 11 , 7 4 Improper fractions are = 13 37 , 3 7 Mixed fractions are = 9 4 , 1 4 1 , 3 3 1 2 ➢ MULTIPLICATION OF FRACTIONS Rule : Product of fractions = Product of their Denomin ators Product of their Numerators (i) Whole number by a fraction (ii) Fraction by a fraction (iii) Whole number by a mixed fraction (iv) Multiplication of two mixed fractions  Whole number by a fraction : To multiply a whole number by a fraction, we simply multiply the numerator of the fraction by the whole number, keeping the denominator same. ❖ EXAMPLES ❖ Ex.10 Find the product (i) 3 × 7 2 (ii) 3 × 8 1 (iii) 9 7 × 6 Sol. (i) 3 × 7 2 = 7 2 1 3  = 1 7 3 2   = 7 6 (ii) 3 × 8 1 = 8 1 1 3  = 1 8 3 1   = 8 3 (iii) 9 7 × 6 = 1 6 9 7  = 3 14 = 3 2 4