Chapter Fractions IMPORTANT DEFINITIONS. Fractions A fraction is a number representing a part of a whole. The whole may be a single object is a group of objects. A fraction is part of an entire object. One fourth is white Two fourths are white. One half is black. Three fourths are white Four fourths are white. Fractions on Number Line Suppose we want to represent the fraction on the number line. Take a line segment OA of the unit length. Divide OA into equal points and take parts out of it to reach the point P. Then the point P represents the number. O / A
. Types of Fraction We can define the three types of fractions like this: i) Proper Fractions The numerator is less than the denominator Examples :,, ii) Improper Fractions iii) Mixed Fractions: The numerator is greater than (or equal to) the denominator Examples:,, A whole number and proper fraction together Examples:,, 6. Equivalent Fraction Equivalent fractions are fractions that have the same value or represent the same part of an object. If a pie is cut into two pieces, each piece is also one-half of the pie. If a pie is cut into pieces, then two pieces represent the same amount of pie that / did. We say that / is equivalent to /. Rule: To get a fraction equivalent to a given fraction. We multiplies divide the numerator and the denominator of gives fraction, by the same non-zero number. To compare / and / we would multiply / by / to produce /6. Since /6 is not the same as /, the fractions are not equivalent. Fractions equivalent to / are /, /6, /8, /0, 6/... Fractions equivalent to / are /6, /9, /, /,... Fractions equivalent to / are /8, /, /6, /0,... Fractions equivalent to / are /0, /, /0, /,... Fractions equivalent to / are /0, 6/, 8/0, 0/,.... Like and Unlike Fraction Like Fractions: Fractions having the same denominator are called like fraction. E.g.,, etc., the like fraction. Unlike fractions: Fraction with different denominators are called unlike fractions for e.g.,, etc. the unlike fraction. 6. Comparing Fractions Comparing Fractions with the Same Denominator Rule : Among the fraction with the same denominator, the one with the greater numerator is the greater of the two. Example: 8 (ii) 6 (iii) 9 9 9 0 0
Comparing Fractions with the Same Numerator Rule : Among the fractions with the same numerator, the one with the smaller denominator is the greater of two. Example: (ii) (iii) 9 9 6 8 0 General Method of Comparing Two Fractions i) Method of Cross Multiplication Let a b and c d be two given fractions Cross multiply, as shown b Find cross products ad and bc. (a) if ad > bc then a c b d (b) if ad < bc then a c b d (c) ad = bc then a c b d c d Example: Compare the fractions 9 and 6? By multiplying, we get = and 9 6 = 9 6 Clearly, > Hence, 6 9 ii) Method of converting the given fractions into like fractions Rule : Change each one of the given fraction into an equivalent fraction with the denominator equal to the cm of the given fraction. Now, the new fraction are like fraction which may be compared to Rule. Example Compare the fractions 6 and 8 9? Less of 6 and 9 = ( ) = 8 Now, we convert each one of 6 and 8 9 into an equivalent fraction having 8 as denominator.
= 6 6 8 6 Clearly, 8 8 Hence, 8 6 9 and 8 8 6 9 9 8. Addition of Fractions TIPS FOR COMPETITIVE LEVEL i) Adding Fractions with the Same Denominator Fractions consist of two numbers. The top number is called the numerator. The bottom number is called the denominator. numerator denominator To add two fractions with the same denominator, add the numerators and place that sum over the common denominator. ii) Adding Fractions with Different Denominators How to Add Fractions with different denominators: Find the Least Common Multiple (LCM) of the denominators of the fractions Rename the fractions to have the LCM Add the numerators of the fractions Simplify the Fraction Example: Find the Sum of /9 and / Determine the Least Common Multiple of 9 and which is 6 Rename the fractions to use the Least Common Multiple (/9 = 8/6, / = 9/6) The result is 8/6 + 9/6 Add the numerators and put the sum over the LCM = /6 Simplify the fraction if possible. In this case it is not possible iii) Adding Mixed Numbers with the Same Denominator Mixed numbers consist of an integer followed by a fraction. How to add two mixed numbers whose fractions have the same denominator: Add the numerators of the two fractions Place that sum over the common denominator. If this fraction is improper (numerator larger than or equal to the denominator) then convert it to a mixed number
Add the integer portions of the two mixed numbers If adding the fractional parts created a mixed number then add its integer portion to the sum. Example: Add the fractional part of the mixed numbers / + / = / Convert / to a mixed number / = Add the integer portions of the mixed numbers + = 8 Add the integer from the sum of the fractions 8 + = 9 State the final answer: 9 How to add two mixed numbers whose fractions have different denominator: Add the fractional part and whole part of the two fractions separately. If the addition of fractional part is improper (numerator larger than or equal to the denominator) then convert it to a mixed number Add the integer portions of the two mixed numbers If adding the fractional parts created a mixed number then add its integer portion to the sum. Example: Add the fractional part of the mixed numbers /+ / = /6 + /6 = /6 Convert / to a mixed number /6 = 6 Add the integer portions of the mixed numbers + = 8 Add the integer from the sum of the fractions 8 + = 9 State the final answer: 9 6. Subtraction of Fractions i) Subtracting Fractions with the Same Denominator Fractions consist of two numbers. The top number is called the numerator. The bottom number is called the denominator. numerator denominator To subtract two fractions with the same denominator, subtract the numerators and place that difference over the common denominator.
ii) Subtracting Fractions with Different Denominators To Subtract Fractions with different denominators: Find the Least Common Multiple (LCM) of the denominators of the fractions Rename the fractions to have the LCM Subtract the numerators of the fractions The difference will be the numerator and the LCD will be the denominator of the answer. Simplify the Fraction Example: Find the difference between / and /9. Determine the Least Common Multiple of 9 and which is 6 Rename the fractions to use the Least Common Multiple (/9 = 8/6, / = 9/6) The result is 9/6-8/6 Subtract the numerators and put the difference over the LCM = /6 Simplify the fraction if possible. In this case it is not possible iii) Mixed Numbers Consist of an Integer Followed by a Fraction. How to subtract mixed numbers having the same denominator: Make the first numerator larger than the second if it is not. Subtract the second numerator from the first Place that difference over the common denominator. Subtract the integer portions of the two mixed numbers State the answer Example: Make the first numerator larger than the second Subtract the fractional parts of the mixed numbers / - / = / Subtract the integer portions of the mixed numbers - = State the final answer: How to subtract two mixed numbers whose fractions have different denominator: Subtract the fractional part and whole part of the two fractions separately. If the subtraction of fractional part is improper (numerator larger than or equal to the denominator) then convert it to a mixed number Subtract the integer portions of the two mixed numbers If subtracting the fractional parts created a mixed number then subtract its integer portion to the sum. Example:
How to subtract two mixed numbers whose fractions have different denominator: Subtract the fractional part and whole part of the two fractions separately. If the subtraction of fractional part is improper (numerator larger than or equal to the denominator) then convert it to a mixed number Subtract the integer portions of the two mixed numbers If subtracting the fractional parts created a mixed number then subtract its integer portion to the sum. Example: Subtract the fractional part of the mixed numbers / - / = /6 - /6 = -/6 Subtract the integer portions of the mixed numbers - = - State the final answer: 6 Problem : Add: 9 (ii) 0 SOLVED PROBLEMS LCM of 0 and is ( ) = 0 So, we convert the given fractions into equivalent fractions with denominator at or 0. We have, and 0 0 0 0 0 0 0 0 0 6 (ii) We have, 8 8 LCM of and is 6. So, convert each fraction to an equivaelnt fraction with denominator 6 6 6 6 6 6 6 Example : Simplify: 6 8
Problem : We have 6 8 6 6 8 9 9 6 8 9 9 6 8 [ LCM of 6, 8, is ] 6 86 6 86 0 Arrange the following fractions in descending order: 8,, 9 (ii),,, 0 8 6 8 6 First we convert the given fractions into the fractions i.e. fractions having common denominator. For this, we first find the LCM of the denominators of the given fractions. Denominators are 9,,. LCM of (9,, ) = = 6 Now, we convert each fraction into equivalent fractions With 6 as its denominator We have, 9 9 6 8 8 6 We know that [ 6 9 ] > > 8 6 6 6 9 [ 6 ] [ 6 ] 9 (ii) Denominators of the given fractions are:,, 0, 8 LCM of denominators = = 0 We now convert each fraction into an equivalent fraction with 0 as its denominator. 8 8 0 8 [ 0 8] 8 0 8 0 60 [ 0 0] 0 0
98 [ 0 0 ] 0 0 0 6 [ 0 8 ] 8 8 0 98 6 60 8 98 6 60 8 0 0 0 0 0 8 Problem : Simplify: (ii) (iii) 9 (iv) 8 8 8 We have, [ LCM of and is ] 0 0 (ii) We have, 8 8 8 [ LCM of and 8 of 8] 8 8 9 8 8 8 8 (iii) The LCM of and is = 6. 9 9 9 6 6 6 6 (iv) We have 8 8 9 8 9 [ LCM of and 8 is 8] 8 68 9 68 9 9 8 8 8 8 8 Alter We have 8 8 8 8 8
Problem : Problem 6: 8 8 8 8 8 8 8 ( ) 8 8 8 0 0 9 8 9 8 8 8 8 8 8 Simplify: 6 We have, 6 6 6 [ LCM of, and 6 is, so we convert each fraction into an equivalent fraction with denominator ] 6 9 6 6 9 6 8 9 Sameera purchased kg apples and kg oranges. What is the total weight of fruits purchased by her? Total weight of the fruits purchased by Sameera is kg 9 Now, 9 9 9 8 Hence, total weight 8 kg. Multiplication of Fractions Let there be a rectangle of length on and breadth cm. In the previous section, we have learnt how to find the perimeter of the rectangle by using addition of fractions. If we want to calculate the area of the rectangle, we will have to find the product of its length and breadth i.e., or,. This can be calculated if we know how to multiply two fractions. So, we define the multiplication of fractions as follows:
Product of their numerations Product of two fractions = Product of their denominations i.e. a c ( a c) b d ( b d) For example, (ii) 6 6 (iii) (iv) 9 60 0 9 9 9 SOLVED PROBLEMS Problem : Multiply by 9 (ii) by (iii) by (iv) by Problem : We have, 8 9 9 6 (ii) (iii) 9 (iv) 8 8 Multiply and reduce to lowest form (if possible): (ii) (iii) 8 (iv) 6 0 6 We have, 6 (ii) 8 8 8 8 8 (iii) 0 0 (iv) 6 6 8
Problem : Simplify: (ii) 0 We have, 9 (ii) 0 0 0 0 9 9 Problem : Which is greater? of or, of 8 of = and of 8 8 8 8 8 In order to compare these fractions, we convert them into equivalent fractions having some denominator equal to the LCM of and 8. LCM of and 8 = = 6 6 and 8 8 6 Clearly, > 6 6 8 Alter: We know that of numerators of two fractions are same, then the fraction having smaller denominator is greater. Problem : Find: of a rupee (ii) of a year (iii) of a day (iv) 8 of a kilogram (v) of an hour (vi) of a litre We have, rupee 00 paise of a rupee Now, of 00 paise 00 00 0 of 00 00 60 of a rupee = 60 paise
(ii) We have, year = months of a year = of months Now, of = 9 of a year = 9 months (iii) We have, day = hours of a day = of hours Now, of 8 6 (iv) We have, kilogram = 000 gram of a kilogram = 8 8 of 000 grams 000 8 grams 000 000 Now, 000 6 8 8 8 of a kilogram = 6 grams 8 (v) We have hour = 6 minutes of an hour 60 minutes 60 60 0 Now, 60 0 of a hour = 0 minutes (vi) We have, litre = 000 ml of a litre 000 ml 000 000 Now, 000 0 80 Problem 6: Sugar is sold at ` per kg. Find the cost of 8 kg of a sugar. We have, Cost of kg of sugar = ` = ` Cost of 8 of sugar = ` 8
` 8 = ` = 0 8 = ` 0 8 Hence, the cost of 8 kg of sugar is `0 8 Problem : A car runs 6 km using litre of petrol. How much distance will it cover using litre of petrol. In litre petrol, car runs 6 km. 6 In litres of petrol car will travel 6km km = ( ) km, = km Hence, car travels km in litres of petrol.. Division of Fractions Reciprocal of Fraction: Two fractions are said to be the reciprocal or multiplicative inverse of each other, if their product is. For example: and are the reciprocals of each other, because. (ii) The reciprocal of is i.e., because (iii) The reciprocal of (iv) The reciprocal of i.e. is, because i.e.,. is, because. Reciprocal of 0 does not exist because division by zero is not possible. Clearly, the reciprocal of a non-zero fraction a b is the fraction b a. Division of Fractions: The division of a fraction a b by a non-zero fraction c d is defined as the product of a b with the multiplicative inverse or reciprocal of c d. i.e., a c a d b d b c
For example, 9 9 9 6 6 (iii) 6 6 (ii) (iv) 8 8 8 Problem : Divide by 9 SOLVED PROBLEMS (ii) 8 by (iii) 6 by 6 (iv) by 9 Problem : Problem : We have, 9 9 9 6 8 8 8 6 (ii) 8 0 6 0 6 (iii) 6 6 6 0 (iv) 9 9 9 9 99 Simplify: (ii) 9 (iii) 8 (iv) We have, 9 9 9 (ii) 8 8 (iii) 8 8 08 08 9 08 9 (iv) 9 9 Simplify: 6 8 0 We have, (ii) 6 8 6 0 0 8 6 0 8 8 9 0 9
(ii) 9 0 9 0 90 6 68 8 8 6 8 8 8 8 8 8 8 8 Problem : The cost of kg of sugar is `0, find its cost per kg. We have, Cost of kg of sugar `0 Cost of of sugar 0 Cost of kg of sugar 0 0 ` ` 0 ` ` 8 PART I: MISCELLANEOUS DOMAIN. Compare the following fractions by using the symbol < or > = : 9 and 8 (ii) 9 and (iii) and 9 9 0 (iv) 9 and 0. Arrange the following fraction is in descending order: 9,,, (ii),,, 0 0 8. Find the sum: 8 0 (ii) (v) 0 (iii) 9 9 (iv) 6. Find the difference of and 6 (v) and 0 (ii) 6 and (iii) (vi) 8 and 8 and 0 (iv) and 0. Simplify: (ii) 6 9 (iii) 6 8
6. Suman studies for hours daily. She devotes much time does she devote for other subjects?. A piece of wire is of length m, what is the length of the other piece. hours of her time for Science and Mathematics. How m. If it is cut into two pieces in such a way that the length of one piece of 8. In a magic square, the sum of the numbers in each row, in each column and along the diagonal is the same is this a magic square. 9. Which is greater? 6 of or or 8 9 6 0. Shikha plants saplings in a row in a her garden. The distance between two adjacent saplings is m. Find the distance between the first and the last sapling.. Lipika reacts a book for hours everyday. She reacts the entire book in 6 days. How many hours in all were required by her to read the book?. Find the area of a rectangular park which is m long and 8 m broad.. Sharda can walk 8 km in one hour. How much distance will she cover in hours?. Each side of a square is 6 m long. Find its area.. There are students in a class and of them are boys. How many girls are there in the class? HIGHER ORDER THINKING SKILLS (HOTS). Copy and complete the following table: (a) Symbol Words Numerator Denominator Meaning Number Line One half One whole divided into two equal parts and one is being considered. one half
(b) (c) (d) (e) Symbol Words Numerator Denominator Meaning Number Line Three quarters Two sevenths One whole divided into four equal parts and three are being considered three quarters two thirds One whole divided into nine equal parts and seven are being considered. Using identical square pieces of paper make copies of this diagram. Number the pieces on both sheets. Cut one of the slices into its seven pieces. Use the pieces to help you work out the following: (a) How many triangles like piece would fit into the largest square? (b) What fraction of the largest square is piece? (c) What function f piece is piece? (d) What fraction of the largest square in each tan gram piece? 6. Use grid paper to construct 6 identical squares with sides cm long, or click on the icon to obtain a template. Use the grid lines on the paper to guide you. (a) Divide the first square into equal parts. Each part is one half. One half has been shaded. Divide the second square into quarters. Each half is now equivalent to two quarters or. Shade in the same half as you did in the first square. (b) Divide the third square into eighths. Shade in the one half of the big square. (c) Divide the fourth square into sixteenths. Shade in the one half of the big square. (d) In the fifth square show that one half equals. 6 (e) Copy and complete:...... 6.
. (a) Use a protractor to outline identical circles. (b) From the centre in the first circle measure and rule lines, 0 apart. Since 0 = 60, you have divided the circle into thirds. Shade. (c) In the second circle draw lines 60 apart. Since 6 60 = 60, you have divided the circle into sixth. Shade 6. (d) In the third circle draw lines 0 degrees part. Shade the appropriate equal area. (e) Continue the pattern in the fourth circle. (f) Copy and complete:...... 6. Gordon spent $ on a drink and $ on chocolates. What fraction of $0 did he spend? 6. Jeny scored correct answer in test of 0 questions. What fraction of her answers were incorrect?. Linda had a bag of 9 apples. She ate and the fed others to her horse. What fraction of her apples remain? 8. Ram started his homework at 8: p.m. and completed it at 9:08 p.m. If he had allowed on how to do his homework, what fraction of that time did he use? 9. Vijay had 9 cm of rope. He cut pieces from it, each cm long. When fractionof the rope remained? 0. There are 60 in full revolution or turn. (a) Find the number of degrees in I. One quarter turn II a half turn III. those quarter of a turn (b) What fraction of a resolution is: 0 (ii) 60 8 (iii) 0? PART II: MULTIPLE CHOICE QUESTIONS. A fraction equivalent to 8 is (a) 8 (b) 8 (c) 8 (d) None of these. If is equivalent to x 0 then the value of x is (a) (b) 8 (c) (d) None of these
. Which of the statement is correct? (a) (b) (c) and (d) Cannot be compared. The largest of the fractions,,, 9 is. (a) (b) 9? 6 (a) (b) (c) 6 (c) (d) 9 (d) 9 8 6. Which is greater: (a) or? 0 (b) 0 (c) Both are equal (d) None of these. is an example of 8 (a) a proper fraction (b) an improper fraction (c) a mixed fraction (d) None of these 8. The largest of the fraction,, 9 and is (a) (b) 9 (c) (d) 9. 8 8 (a) 0 (b) (c) 8 (d) 6 0. Write T for true and F; for false for each of the statement given below: (a) (b) 8 0 6 6 (c), and are like fraction (d) lies between and. (e) Among,,,, the largest fraction is