202110795-TRAVELLER_PREMIUM-STUDENT-TEXTBOOK-MATHEMATICS-G05-PART2

MATHEMATICS 5 TEXTBOOK – 2 Name: ___________________________________ Section: ________________ Roll No.: _________ School: __________________________________

Preface ClassKlap partners with schools, supporting them with learning materials and processes that are all crafted to work together as an interconnected system to drive learning. Our books strive to ensure inclusiveness in terms of gender and diversity in representation, catering to the heterogeneous Indian classroom. ClassKlap presents the Traveller series, designed specifically to meet the requirements of the new curriculum released in November 2016 by the Council for the Indian School Certificate Examinations (CISCE). Guiding principles: The 2016 CISCE curriculum states the following as a few of its guiding principles for Mathematics teaching:  D  evelop mathematical thinking and problem-solving skills and apply these skills to formulate and solve problems.  A  cquire the necessary mathematical concepts and skills for everyday life and for continuous learning in Mathematics and related disciplines.  R  ecognise and use connections among mathematical ideas and between Mathematics and other disciplines.   R eason logically, communicate mathematically and learn cooperatively and independently. Each of these principles resonates with the spirit in which the ClassKlap textbooks, workbooks and teacher companion books have been designed. The ClassKlap team of pedagogy experts has carried out an intensive mapping exercise to create a framework based on the CISCE curriculum document. Key features of ClassKlap Traveller series:  Theme-based content that holistically addresses all the learning outcomes specified by the CISCE curriculum.  T he textbooks and workbooks are structured as per Bloom’s taxonomy to help organise the learning process according to the different levels involved.  Student engagement through simple, age-appropriate content with detailed explanation of steps.  Learning is supported through visually appealing images, especially for Grades 1 and 2.  Increasing difficulty level in sub-questions for every question.  Multiplication tables provided as per CISCE requirement. All in all, the Traveller Mathematics books aim to develop problem-solving and reasoning skills in the learners’ everyday lives while becoming adept at mathematical skills as appropriate to the primary level. – The Authors

Textbook Features I Will Learn About I Think Contains the list of concepts to be covered Arouses the student’s in the chapter along with the learning curiosity before objectives introducing the concept I Recall I RUenmdeermsbtearndand Pin-Up-Note Recapitulates the Elucidates the basic Highlights the key points or prerequisite knowledge for elements that form the definitions the concept learnt previously basis of the concept ? Train My Brain I Apply I Explore(H.O.T.S.) C hecks for learning to gauge Connects the concept E ncourages the student to the understanding level of the to real-life situations by extend the concept learnt student providing an opportunity to more complex scenarios to apply what the student has learnt Maths Munchies Connect the Dots Drill Time Aims at improving speed of Aims at integrating Revises the concepts with calculation and problem Mathematical concepts practice questions at the solving with interesting facts, with other subjects end of the chapter tips or tricks A Note to Parent E ngages the parent in the out-of- classroom learning of their child

Contents 7 Integers 7.1 Introduction to Negative Numbers���������������������������������������������������������������������������� 1 8 Fractions 8.1 Classification of Fractions���������������������������������������������������������������������������������������� 14 8.2 Comparison of Fractions������������������������������������������������������������������������������������������ 20 9 Fraction Operations 9.1 A dd and Subtract Fractions������������������������������������������������������������������������������������ 28 9.2 Multiply and Divide Fractions���������������������������������������������������������������������������������� 34 10 Money 10.1 U nitary Method in Money��������������������������������������������������������������������������������������� 46 11 Decimals 11.1 Introduction to Decimals���������������������������������������������������������������������������������������� 52 11.2 Compare and Order Decimals������������������������������������������������������������������������������� 61 12 Decimal Operations 12.1 A dd and Subtract Decimals����������������������������������������������������������������������������������� 71 12.2 Multiply Decimals���������������������������������������������������������������������������������������������������� 75 13 Percentages 13.1 Introduction to Percentage������������������������������������������������������������������������������������ 82 14 Measurement 14.1 Conversion of Units of Measurement��������������������������������������������������������������������� 90 14.2 Area and Perimeter������������������������������������������������������������������������������������������������� 94 15 Data Handling 15.1 L ine Graphs and Pie Charts���������������������������������������������������������������������������������� 104

Integers7Chapter I Will Learn About • negative numbers. • comparing integers using a number line. • ordering of integers. • rules of addition and subtraction of integers. 7.1 Introduction to Negative Numbers I Think Pooja’s family visited Kuttanad, a region in Kerala. It is well known for rice farming that is carried out around 1 m to 3 m below sea level. She wanted to represent this numerically in her diary. How do you think Pooja can write it? I Recall We have already learnt: • Roman numerals • Indian and international systems of numeration • Representing and comparing numbers on the number line 1

As we move from left to right on the number line, the numbers increase. As we move from right to left on the number line, the numbers decrease. Let us revise this concept by answering the following. 1) Arrange the given numbers in ascending and descending orders. a) 1436, 7001, 5998, 3291 b) 56891, 80149, 32748, 82013 c) 31752, 37251, 35721, 31572 2) Fill in the blanks and complete the sequences given: a) 238, __________, 240, 241, 242, ______________, 244 b) 8997, 8998, ___________, 9000, ______________, 9002 I Remember and Understand We always start counting as 1, 2, 3, 4 and so on. Counting from 1 comes naturally to us. That is why the collection of numbers starting from 1 are called natural numbers. For example, 1, 2, 3, 6, 100, 4536 and so on. The natural numbers along with ‘0’ form the collection of numbers called the whole numbers. Therefore, the smallest whole number is ‘0’ whereas the smallest natural number is ‘1’. On a number line, as we move to the right by one unit, the value increases by 1. Similarly, moving to the left by one unit decreases the value by 1. When we move by one unit to the left of 0, we The numbers +1, +2, + 3 and so denote it as –1, read as negative 1. We write on are usually written as 1, 2, negative numbers with a minus sign to show that 3… These are called positive it is in the opposite direction from 0 to 1. Similarly, numbers. The numbers -1, -2, moving left by 2, 3, 4 and so on, units are denoted -3, …. are called negative as – 2, – 3, – 4 and so on. numbers. 0 is neither positive We extend the number line to the left of 0 and nor negative. write the negative numbers as shown. For each positive number to the right of 0, there is a corresponding negative number to the left of 0. Negative numbers Positive numbers 2

Positive numbers (1, 2, 3, ….), negative numbers (–1, –2, –3, ….) and 0 together form a set of numbers called integers. In other words, the set of whole numbers and negative numbers is called the set of integers. It is denoted by the letter Z. Hence, we write Z = {…, –3, –2, –1, 0, 1, 2, 3, …}. The dots before –3 and after 3 indicate that there are more numbers on either side. Every number to the left of a given number is less than the given number. For example, 5 < 6, 2 < 3, 0 < 1, –1 < 0, –3 < –2 and so on. So, all negative numbers are less than 0 and all positive numbers are greater than 0. In the same way, 6 > 5 whereas –6 < –5. So, the greater a positive integer, the smaller is its corresponding negative integer. Predecessor and successor of a given integer Every integer has an integer that comes before it. This integer is called the predecessor of the given integer. On a number line, the integer just before the given integer (that is, to its left) is called its predecessor. Similarly, every integer has an integer that comes after it. This integer is called the successor of the given integer. On a number line, the integer just after the given integer (that is, to its right) is called its successor. Every number has a predecessor obtained by subtracting 1 from it and a successor obtained by adding 1 to it. So, the smallest negative integer and the largest positive integer cannot be determined. Predecessor Predecessor of -3 of 2 Successor of Successor -3 of 2 Uses of integers The concept of opposites is indicated by the positive and negative numbers. So, we use integers to represent opposites such as the following. a) Moving forward by 5 steps: +5 or 5 Moving backward by 5 steps: –5 b) Profit of ` 10: ` 10 Integers 3

Loss of ` 5: – ` 5 c) Saving of ` 50: ` 50 Expenditure of ` 12: – ` 12 d) Height of 1200 m above sea level: 1200 m Depth of 708 m below sea level: – 708 m e) 10° C above 0° C: +10° C 7° C below 0° C: –7° C Example 1: Represent the following on a number line. a) –4 b) –2 c) 5 d) 3 Solution: Draw a number line with markings at equal intervals. Number them a) according to their distances from the starting point 0. Circle the given number. b) c) d ) Example 2: Write the following using integers. a) A deposit of ` 215 into a savings account. b) A withdrawal of ` 108 from a savings account. c) A mountain is 4326 m above sea level. d) An octopus in water is at a depth of 573 m below sea level. Solution: a) ` 215 b) – ` 108 c) 4326 m d) – 573 m Example 3: Write the predecessors and successors of the following integers. a) 65 b) 308 c) – 4937 d) –2086 4

Solution: Given Predecessor Successor integer (Just before the given (Just after the given a) b) 65 integer) integer) c) 308 64 66 d) -4937 307 309 -2086 -4938 -4936 -2087 -2085 ? Train My Brain Write the predecessor and successor of each of these integers. a) 549 b) - 356 c) -333 d) -286 I Apply Compare and order integers In previous classes, we have learnt to compare and order whole numbers. Similarly, we can compare and order integers. We know that, • an integer to the left of a given integer is smaller than the given integer. • an integer to the right of a given integer is greater than the given integer. • all positive integers are greater than all negative integers. • 0 is less than all positive integers and greater than all negative integers. Consider the following examples. Example 4: Write the greater of the given integers in each of the following pairs. a) 615, 378 b) 173, – 508 c) – 479, – 124 Solution: a) C omparing two positive integers is like comparing two whole numbers. So, the greater of 615 and 378 is 615. b) A positive integer is greater than a negative integer. So, the greater of 173 and –508 is 173. c) A greater number with a minus sign lies to the left of a smaller number with a minus sign. So, it is smaller than the smaller number with a minus sign. So, the greater of –479 and –124 is –124. Integers 5

Example 5: Write the least of the given integers in each of the following. a) 135, 683, 738 b) 173, –508, 418 c) – 547, – 935, – 827 Solution: a) C omparing three positive integers is like comparing three whole numbers. So, the smallest of 135, 683, 738 is 135. b) A negative integer is smaller than a positive integer. So, the least of 173, – 508, 418 is – 508. c) A larger number with a minus sign lies to the left of a smaller number with a minus sign. So, it is smaller than the smaller number. So, the least of – 547, – 935, – 827 is – 935. Example 6: Write the given integers in the ascending and descending orders. a) 315, 638, 378 b) 731, – 580, 841 c) – 457, – 953, –782 Solution: a) Comparing the given integers, we get 315 < 378 < 638. Ascending order: 315, 378, 638 Descending order: 638, 378, 315 b) Comparing the given integers, we get – 580 < 731< 841. Ascending order: – 580, 731, 841 Descending order: 841, 731, – 580 c) Comparing the given integers, we get – 953 < –782 < – 457. Ascending order: – 953, –782, – 457 Descending order: – 457, –782, – 953 Example 7: The temperatures recorded in Shimla on three days are as given: Monday: –3 °C; Tuesday: –2 °C; Wednesday: –4°C. Which day was the coldest? Solution: The temperatures of Shimla on Monday, Tuesday and Wednesday were –3 °C, –2 °C and –4 °C respectively. Comparing the three temperatures, we have –4 °C < –3 °C < –2 °C. Since , –4 °C is on Wednesday, Wednesday was the coldest. Absolute value of an integer We have learnt that every positive integer to the right of 0 has a corresponding negative integer to its left at the same distance from 0. Observe the number line of integers. 6

While four units to the right of 0 is positive 4, four units to the left of 0 is negative 4. 4 units 4 units Similarly, two units to the right of 0 is 2 and two units to the left of 0 is –2 and so on. The distance of an integer from 0, without considering its direction (right or left) is called its absolute value. So, the absolute value of an integer is just its numerical value irrespective of its sign (plus or minus). Thus, the absolute value of a positive integer or a negative integer or 0 is positive. Note: The absolute value of a number is never negative. The absolute value of a number, 32 is written as |32| and is read as the absolute value of 32. For example, the absolute value of –93 is written as |–93| and is equal to 93. The absolute value of 0 is written as |0| and is equal to 0. The absolute value of 68 is written as |68| and is equal to 68. Example 8: Write the absolute values of the given integers. a) 878 b) – 743 Solution: The absolute value of a positive integer or a negative integer is always positive. So, the required absolute values are: a) |878| = 878 b) |–743| = 743 I Explore (H.O.T.S.) Addition and subtraction of integers We already know that to add a whole number to another, we move as many places to the right of the first number on the number line as the number to be added. For example, to find 4 + 5, we move 5 places to the right of 4 on the number line to get 9 as the sum. Integers 7

Similarly, to subtract a whole number from another, we move as many places to the left on the number line as the number to be subtracted. For example, to find 4 – 2, we move 2 places to the left of 4 on the number line to get 2 as the difference. Similarly, we can add and subtract integers. Same is true in the case of addition and subtraction of a positive integer to any integer. Whereas, it is exactly opposite in case of addition and subtraction of a negative integer to any integer. Rules of addition and subtraction Rule 1: Addition and subtraction of ‘0’ to and from an integer results in the same integer. For example, 6 + 0 = 6 as shown on the following number line. Rule 2: To add a positive integer we move as many places to the right of the first integer as the integer to be added. For example, 4 + 5 = 9 as shown on the number line. Rule 3: To add a negative integer we move as many places to the left of the first integer as the integer to be added. For example, 8

a) 3 + (–4) = (–1), as shown. b) (–3) + (–4) = (–7), as shown on the following number line. Rule 4: To subtract a positive integer we move as many places to the left of the first integer as the integer to be subtracted. For example, 5 – 4 = 1 as shown on the number line. Rule 5: To subtract a negative integer we move as many places to the right of the first integer as the integer to be subtracted. For example, a) 4 – (–3) = 7 as shown on the following number line. b) (–4) – (–3) = (–4) + (+3) = (–1) as shown. Integers 9

Let us now solve a few examples based on these rules. Example 9: Solve using a number line: a) (–3) + 5 b) 3 + (–2) Solution: a) (–3) + 5 = 2 From rule 2, b) 3 + (–2) = 1 From rule 3, Example 10: Solve using a number line: a) 5 – 4 b) (–6) – (–3) Solution: a) 5 – 4 = 1 From rule 4, Train My Brain b) (–6) – (–3) = –3 From rule 5, 10

Maths Munchies Brahmagupta, an Indian astronomer and mathematician introduced the concept of negative numbers. During his time (7th century), Indians used negative integers to represent debts. He called positive numbers as fortunes and negative numbers as debts. He also gave rules for operations of negative numbers and zero. Connect the Dots Science Fun The body temperature of a polar bear is around 37°C. It is maintained through a thick layer of fur, a tough hide, and a thick fat layer (up to 11 cm or around 5 in. thick). This excellent insulation keeps a polar bear warm even when the air temperatures drop to -37°C. Social Studies Fun About one-third of Netherlands lies below sea level, with the lowest point being 22 feet (around 6.7 metres) below sea level. Meanwhile, the highest point is about a thousand feet above sea level. Integers 11

Drill Time 7.1 Introduction to Negative Numbers 1) Represent the following on a number line. a) 7 b) –5 c) 11 d) –6 2) Write the following using integers. a) A withdrawal of ` 521 from a savings account. b) A deposit of ` 186 into a savings account. c) An aeroplane flying at a height of 1565 m above sea level. d) A submarine at a depth of 531 m below sea level. e) The night temperature of a place is 5° C below 0° C. 3) Write the predecessors and successors of the following integers. a) –396 b) 830 c) – 350 d) –8260 4) Write the smaller of the given integers in each of the following pairs. a) 615, 378 b) 173, –508 c) – 479, – 124 5) Write the greatest of the given integers in each of the following. a) 423, 389, 156 b) 702, –723, 682 c) – 418, – 365, – 903 6) Write the given integers in the ascending and descending orders. a) 426, 719, 754 b) 302, – 108, – 634 c) 853, – 486, –758 7) Write the absolute values of the given integers. a) –482 b) 690 c) –136 8) Add the following using a number line. a) 2 + 3 b) 4 + (–2) c) (–4) + 6 d) (–5) + (–3) 9) Subtract the following using a number line. a) 4 – 1 b) 3 – (–3) c) (–2) – 4 d) (–3) – (–6) 12

10) Word problem a) The temperatures recorded in Manali on three days are as given. Monday: –4 °C; Tuesday: –1 °C; Wednesday: –7 °C Which day was the hottest? 11) Write the following using integers. a) The night temperature of city A is 7 °C below 0 °C. b) The depth of a man-made tank is 743 m below the ground level. c) A bird is flying at a height of 38 m above sea level. d) A diver dived into a river to a depth of 59 m below sea level. 12) The following are the temperatures recorded in four places on a particular day of a year. Place Temperature Darjeeling 15 °C below 0 °C Nainital 6 °C above 0 °C Ooty 23 °C below 0 °C Munnar 7 °C above 0 °C From the given table, answer the following questions. a) Write the temperatures of the given places using integers. b) Which place was the hottest on that day? c) Which place was the coldest on that day? d) Write the names of the places in the increasing order of their temperatures. A Note to Parent Show your child your savings bank passbook. Ask them to observe the deposits and withdrawals. Then ask them to find the overall balance at the end of a month using integers. Integers 13

8Chapter Fractions I Will Learn About • proper, improper and mixed fractions. • comparison of three or more fractions. 8.1 Classification of Fractions I Think Pooja’s father told her that he spends two-thirds of his monthly salary and saves the rest. Pooja calculated the amount her father saves from his salary of ` 25,000 per month. How do you think Pooja could calculate her father’s savings per month? I Recall In class 4, we have learnt how to find the fraction of a collection. Let us answer these to recall the concept. a) A half of a dozen bananas = _______________ bananas b) A quarter of 16 books = _______________ books c) A third of 9 balloons = _______________ balloons 14

d) A half of 20 apples = _______________ apples e) A quarter of 8 pencils = _______________ pencils I Remember and Understand We have learnt about like and unlike fractions. Let us now learn about the other types of fractions. Proper, Improper and Mixed Fractions Consider 1+ 5 = 6 sum of the two like fractions is a like fraction with its 8 8 8 . Here, the numerator less than its denominator. Such fractions are called proper fractions. Sometimes it is possible that we get the sum with its numerator greater than the denominator. 7 5 12 For example, 8 + 8 = 8 . Here, the sum of the two like fractions is a like fraction with its numerator greater than its denominator. Such fractions are called improper fractions. Note: In some cases, the sum of the numerators of the like fractions may be equal to the denominator. Then, the fraction is said to be an improper fraction. For example, 3 + 4 = 7 , 3 + 5 = 8 and so on. 7 7 7 8 8 8 78 Fractions such as 7 , 8 can also be written as a whole, that is 1. 12 8 4 8 We can write 8 as the sum of like fractions as 8 + 8 . This has a whole  8  and a proper fraction  4  . That is, 12 =1+ 4 = 14 . Such fractions are called mixed fractions.  8  8 8 8 A mixed fraction is also called a mixed number. For example, in the mixed fraction 12 3 , 12 is the whole and 3 is the proper fraction. 8 8 Example 1: List out the proper fractions, improper fractions and mixed fractions from the following: 13 ,15 7 , 11 , 37 , 9 , 65 13 , 143 , 75 3 ,107 27 , 72 , 68 2 , 29 , 50 23 , 69 , 53 18 9 34 6 14 17 98 4 49 59 5 32 35 32 30 Fractions 15

Solution: From the given fractions, 13 11 9 29 Proper fractions: 18 , 34 , 14 , 32 Improper fractions: 37 143 72 69 53 6 , 98 , 59 , 32 , 30 Mixed fractions: 15 7 , 65 13 , 75 3 , 107 27 , 68 2 , 50 23 9 17 4 49 5 35 We usually write fractions as proper or mixed fractions. So, we need to learn to convert improper fractions to mixed fractions and mixed fractions to improper fractions. Conversion of improper fractions to mixed fractions Consider an example. Example 2: Convert 37 to its mixed fraction form. 6 Solution: To convert improper fractions into mixed fractions, follow these steps. Steps Solved 143 Solve these 53 98 30 Step 1: Divide the 37 72 69 numerator by the 6 59 32 denominator. )6 37(6 Step 2: Write the quotient and the − 36 remainder. 1 Step 3: Write the Quotient = 6 quotient as the whole. Remainder = 1 Write the remainder as the numerator The mixed and the divisor as the denominator of the fraction form of proper fraction. This gives the required 37 is 6 1 . mixed fraction. 6 6 Conversion of mixed fractions to improper fractions Consider an example. Example 3: Convert 15 7 into an improper fraction. 9 Solution: To convert mixed fractions into improper fractions, follow these steps. 16

Steps Solved 65 13 Solve these 107 27 15 7 17 75 3 49 Step 1: Multiply the whole by the 9 4 denominator. 15 × 9 = 135 Step 2: Add the numerator of the 135 + 7 = 142 proper fraction to the product obtained in step 1. Step 3: Write the sum as the numerator of the mixed fraction. The improper The denominator does not change. fraction form of This gives the required improper 15 7 is 142 . fraction. 9 9 ? Train My Brain Classify following into proper, improper and mixed fractions: 5 b) 6 5 a) 6 5 c) 1 6 I Apply Let us now see some real-life examples in which we find the fraction of a number. Example 4: Rohan wants to arrange 60 books on his shelf. If only 13 books can be put in a rack, how many racks will be filled with the books? Give your answer as a mixed fraction and as an improper fraction. Solution: Number of books Rohan wants to arrange = 60 Number of books that can be arranged on each rack = 13 488 Number of racks that are filled = 60 ÷ 13 = 4 13 Improper fraction equivalent to 4 488 = 60 Example 5: For a science fair, a group of 13 13 418 students prepared 12 123 litres of orange juice. Express the quantity of orange juice as an improper fraction. Fractions 17

Solution: Quantity of orange juice prepared = 12 1 litres 2 Improper fraction equivalent to 12 1 = 12×2+1 = 25 2 2 2 Example 6: A school auditorium has 2500 chairs. On the annual day, 41 of the 150 auditorium was occupied. How many chairs were occupied? Solution: Total number of chairs in the auditorium = 2500 41 Fraction of chairs occupied = 150 Number of chairs occupied = 41 × 2500 = 4× 2500 150 5 = 10000 = 2000 5 Therefore, 2000 chairs in the auditorium were occupied. I Explore (H.O.T.S.) Conversion of fractions is done when we need to add and subtract fractions. We have already learnt the addition and subtraction of like (proper) fractions. Let us see some real-life examples of addition and subtraction of improper and mixed fractions. Example 7: Solve: 5 + 11 12 12 Solution: W e can add the fractions with the following representations. + 5 + 11 12 12 We see that there are 16 coloured fractions in all. Replace the one uncoloured 18

fraction from 11 with the coloured fractions from 5 . 12 12 4 1 12 4 We see that we get a whole and a fraction of 12 . So, the sum of 5 + 11 is 1 4 . 12 12 12 Example 8: Solve: 13 – 1 1 11 11 Solution: We can subtract the fractions with the following representations. 13 can be converted to mixed fraction as 1121. 11 Now, we can represent the numbers as: 12 11 11 11 Subtracting the whole parts in both the figures, we get 1 figure with 2 parts coloured and the other figure with 1 part coloured. Subtracting 1 from 2 we have only 1 part coloured. Hence, the subtraction can be represented as So, the difference of 13 – 1 1 is 1 . 11 11 11 Fractions 19

8.2 Comparison of Fractions I Think Pooja eats two pieces of a cake that was cut into four equal pieces. Farhan eats three pieces of a cake of same size that was cut into six equal pieces. Do they eat the same amount of cake? I Recall In class 4, we have learnt about equivalent fractions. Let us revise them here. Suppose a pizza is cut as shown. Rohan eats 2 of the pizza. Then, the piece of pizza he gets 8 is = . Suraj eats 1 of the pizza. Then, the piece of pizza he gets is 4 . We see that the pieces of pizza eaten by both are of the same size. So, we say that the fractions 2 and 1 are equivalent. 8 4 We write them as 2 = 1 . 84 Recall the following: • The multiples of a number are obtained by multiplying it by 1, 2, 3 and so on. For example, the first six multiples of 6 are 6, 12, 18, 24, 30 and 36. • The numbers that divide a given number exactly are called its factors. For example, the factors of 6 are 1, 2, 3 and 6. • Equivalent fractions are obtained by multiplying or dividing the numerator and the denominator of the given fraction by the same number. For example, 1 , 3 , 5 , 2 and so on are equivalent fractions. 7 21 35 14 20

I Remember and Understand We have learnt how to find equivalent fractions using pictures. Let us see some more examples to find equivalent fractions. Example 9: Find two fractions equivalent to the given fractions. a) 24 b) 33 Solution: 46 66 To find fractions equivalent to the given fractions, we either multiply or divide both the numerator and denominator by the same number. a) 24 46 Let us multiply both the numerator and denominator by the same numbers, say 2 and 3. 24×2 = 48 46×2 92 24×3 72 46×3 = 138 Thus, 48 and 72 are the fractions equivalent to 24 . 92 138 46 b) 33 66 We see that 33 and 66 have common factors 3, 11 and 33. So, dividing both the numerator and the denominator by 3, 11 or 33, we get fractions equivalent to the given fraction 33 . 66 33 ÷ 3 = 11 , 33 ÷11= 3 or 33 ÷ 33 = 1 66 ÷ 3 22 66 ÷11 6 66 ÷ 33 2 Therefore, 11 , 3 and 1 are three fractions equivalent to 33 . 22 6 2 66 4 Example 10: Find the equivalent fraction of 10 with denominator: a) 30 b) 60 Solution: 4 a) T o find the equivalent fraction of 10 with denominator 30, we multiply the denominator and numerator by 3. 4×3 12 10×3 = 30 Fractions 21

4 b) To find the equivalent fraction of 10 with denominator 60, we multiply the denominator and numerator by 6. 4×6 24 10×6 = 60 We have learnt to compare like fractions. Let us now learn to compare three or more like and unlike fractions. Example 11: Compare these like fractions. 5 13 a) 7 , 2 , 4 b) 8 , 8 , 8 9 9 9 Solution: To compare like fractions, we compare their numerators. a) Comparing the numerators of the given fractions, we get 2 < 4 < 7. Therefore, 2 < 4 < 7 . 9 9 9 b) Comparing the numerators of the fractions, we get 1 < 3 < 5. Therefore, 1 < 3 < 5 . 8 8 8 Note: We can compare them by writing the numerators in descending order too. Example12: Compare these unlike fractions. To compare unlike 135 b) 2 , 1 1 fractions, we first a) 2 , 4 , 8 3 3, 9 convert the fractions to Solution: a) 1 , 3 , 5 like fractions. 248 T o compare unlike fractions, we first have to convert them into like fractions. To do so, we find the L.C.M. of their denominators. The L.C.M. of 2, 4 and 8 is 8. Equivalent fraction of 1 = 1×4 = 4 2 2×4 8 Equivalent fraction of 3 = 3×2 = 6 4 4×2 8 Thus, the required like fractions are 84 , 6 and 5 . 8 8 Comparing the numerators, we get 4 < 5 < 6. 6 1 5 As 4 < 5 < 8, 2 < 8 < 3 8 8 4. b) 2 , 1 1 3 3, 9 The L.C.M. of 3 and 9 is 9. 22

Equivalent fraction of 2 = 2×3 = 6 3 3×3 9 Equivalent fraction of 1 = 1× 3 = 3 3 3×3 9 63 1 Thus, the required like fractions are 9 , 9 and 9 . Comparing the numerators, we get 1 < 3 < 6. As 1 < 3 < 6 , 1 < 1 < 2 . 9 9 9 9 3 3 Note: Comparing fractions helps us to find the least or greatest of a set of given fractions. ? Train My Brain Compare these fractions. a) 1 , 2 , 4 b) 4, 3 , 9 123 235 16 16 16 c) 8 , 6 , 4 I Apply Let us see some real-life situations where we compare unlike fractions. 12 Example 13: Esha ate 4 of an apple in the morning and 3 of the apple in the Solution: evening. When did she eat a larger part of TthreaainppMle?y Brain 1 Fraction of the apple Esha ate in the morning = 4 2 Fraction of the apple she ate in the evening = 3 To find when she ate a larger part, we must compare the two fractions. Step 1: 12 with the L.C.M. of 4 and 3 as Write like fractions equivalent to and their denominators. 4 3 The L.C.M. of 4 and 3 is 12. So, the required like fractions are: Step 2: 1 = 1×3 = 3 and 2 = 2×4 = 8 4 4×3 12 3 3×4 12 Compare the numerators of the equivalent fractions. Fractions 23

Since 8 > 3, 8 > 3 . 12 12 Hence, 2 > 1 . 34 Therefore, Esha ate a larger part of the apple in the evening. 12 Example 14: Kumar saves 4 of his salary and Pavan saves 6 of his salary. If they earn the same amount every month, who saves a lesser amount? Solution: To find who saves lesser, we must find the lesser of the given fractions. The L.C.M. of 4 and 6 is 12. 123 4 Equivalent fractions of 4 and 6 are 12 and 12 . 3 4 Since 3 < 4, 12 < 12 . So, 1 < 2 . 4 6 Therefore, Kumar saves a lesser amount than Pavan. I Explore (H.O.T.S.) Let us see some more examples using comparison of unlike fractions. Example 15: Colour each figure to represent the fraction given below it and then compare them. 22 9 7 Solution: 2 9 2 7 Clearly, the part of the figure represented by 2 is greater than that 7 22 2 represented by 9 . Therefore, 7 is greater than 9 . 24

Using comparison of fractions, we can arrange some unlike fractions in the ascending and descending orders. Consider these examples. Example 16: Arrange 2 , 1 , 2 , 3 and 1 in the ascending order. 3 2 5 4 6 Solution: Write the equivalent fractions of the given unlike fractions. The L.C.M. of the denominators 2, 3, 4, 5 and 6 is 60. So, the fractions equivalent to 2 , 1 , 2 , 3 and 1 with the L.C.M. as their denominator are: 3 2 5 4 6 2 = 2 × 20 = 40 1 = 1 × 30 = 30 2 = 2 ×12 = 24 , 3 = 3 ×15 = 45 3 3 × 20 60 , 2 2 × 30 60 , 5 5 ×12 60 4 4 ×15 60 and 1 = 1×10 = 10 . 6 6 ×10 60 Comparing the numerators, 10 < 24 < 30 < 40 < 45. So, 10 < 24 < 30 < 40 < 45 . 60 60 60 60 60 12 12 3 Therefore, the required ascending order is 6 , 5 , 2 , 3 , 4 . 2 1 1 5 3 Example 17: Arrange 7 , 4 , 8 , 14 and 16 in the descending order. Solution: Write the equivalent fractions of the given unlike fractions. The L.C.M. of the denominators 7, 4, 8, 14 and 16 is 112. So, the fractions equivalent to 2 , 1 , 1 , 5 and 3 with the L.C.M. as their denominator are: 7 4 8 14 16 2= 2×16 32 , 1 = 1×28 = 28 , 1 = 1×14 = 14 , 5 = 5×8 = 40 7 713×6 1=61=361××1727 4 4×28 112 8 8×14 112 14 14×8 112 and 21 = 112 . Comparing the numerators, 40 > 32 > 28 > 21 > 14. So, 40 > 32 28 21 14 . 112 112 > 112 > 112 > 5 21 3 1 112 Therefore, the required descending order is 14 , 7 , 4 , 16 , 8 . Fractions 25

Maths Munchies Comparing unit fractions While comparing unit fractions, the fraction with the largest denominator is the smallest and the fraction with the smallest denominator is the greatest. Of the unit fractions 1 1 and 1 , 1 is the greatest while 1 is the smallest. 12 , 7 10 7 12 Connect the Dots English Fun Write your name and write the number of letters in it. Write fractions to show the number of letters in half of your name and one-fourth of your name. Social Studies Fun About 3 of the Earth is covered with water. Of this water, 97 is 4 100 salt water and is not suitable for drinking. Drill Time 8.1 Classification of Fractions 1) Convert the following improper fractions to mixed fractions: a) 35 b) 11221 c) 93 d) 100 e) 115 4 12 26 20 2) Convert the following mixed fractions to improper fractions: a) 15 6 b) 23 2 c) 2 7 d) 125190 e) 40 3 8 3 13 5 26

3) Word Problems a) At Sudhir’s birthday party, there are 19 sandwiches to be shared equally among 13 children. What part of the sandwiches will each child get? Give your answer as a mixed fraction. b) I bought 2 14 litres of paint but used only 3 litres. How much paint is left with 26 2 me? 8.2 Comparison of Fractions 4) Compare these fractions. 5 91 4 23 143 a) 16 , 4 , 8 b) 21, 3 , 7 c) 5 , 5 , 5 d) 5 , 2 , 1 e) 7 , 4 , 2 999 12 15 3 5) Word problems a) Ali played the keyboard for 7 of an hour and did his homework for 5 of 30 12 an hour. Did he spend the same amount of time for both the activities? b) Sheeba ate 3 of a chocolate bar and Francis ate 1 of a similar chocolate 16 4 bar. Did they eat the equal amount of the chocolate? If not, who ate less? A Note to Parent Give different currency notes to your child. Ask him or her to find if some of them are half, one-fourth or three-fourths of some others of the given currency notes. For example, ` 50 is one-fifth of ` 250. Fractions 27

CHAPTER 9 FOrpaecrtaitoinons9Chapter I Will Learn About • adding and subtracting unlike and mixed fractions. • multiplying fractions by whole numbers and fractions. • dividing whole numbers and fractions by fractions. 9.1 Add and Subtract Fractions I Think Pooja has a round cardboard with some of its portions coloured. She knows that the fractions that represent the coloured portions are unlike fractions. She wanted to find the coloured and uncoloured parts of the cardboard. How do you think Pooja can find that? I Recall We have learnt about the types of fractions. Let us recall them here. 1) A fraction whose numerator is greater than its denominator is called an improper fraction. 2) A fraction whose numerator is less than its denominator is called a proper fraction. 3) A fraction with a combination of a whole number and a proper fraction is called a mixed fraction. 28

I Remember and Understand Addition and Subtraction of Unlike Fractions Let us understand the addition and subtraction of Unlike fractions can be unlike fractions through some examples. added or subtracted by first writing their equivalent like 31 72 fractions and then adding or Example 1: Solve: a) 15 + 10 b) 13 + 39 subtracting the numerators. c) 22 + 7 100 10 31 Solution: a) 15 + 10 [L.C.M. of 15 and 10 is 30.] 3×2 1×3 = 15×2 + 10×3 = 6 + 3 30 30 6+3 9 3 = 30 = 30 = 10 [H.C.F. of 9 and 30 is 3.] 7 2 21 2 21+ 2 23 b) 13 + 39 = 39 + 39 = 39 = 39 [L.C.M. of 13 and 39 is 39.] 22 7 22 70 22 + 70 92 23 c) 100 + 10 = 100 + 100 = 100 =100 = 25 [The L.C.M. of 100 and 10 is 100, and the H.C.F. of 92 and 100 is 4.] Example 2: Solve: a) 8 - 4 b) 17 - 5 c) 14 -– 17 9 11 30 24 25 50 Solution: a) 8 -– 4 = 88 -– 36 [L.C.M. of 9 and 11 is 99.] 9 11 99 99 88- 36 = 52 =99 99 b) 17 - 5 = 68 - 25 [L.C.M. of 24 and 30 is 120.] 30 24 120 120 68 -25 43    = 120 = 120 Fraction Operations 29

c) 14 - 17 = 28 - 17 [L. C. M. of 25 and 50 is 50.] 25 50 50 50 28 - 17 11   = 50 = 50 Addition and Subtraction of Mixed, Improper and Proper Fractions The addition and subtraction of mixed fractions are similar to that of unlike fractions. Let us understand the same through some examples. Example 3: Add: 2 3 + 3 2 5 7 Solved Solve this Steps 23 + 32 12 1 + 15 1 5 7 43 Step 1: Convert all the mixed 2 3 = 2´5+ 3 = 13 ; fractions into improper fractions. 5 5 5 Step 2: Find the L.C.M. and add 32 = 3´7 + 2 = 23 the improper fractions. 7 7 7 23 + 32 = 13 + 23 5 7 5 7 [L.C.M. of 5 and 7 is 35.] Step 3: Find the H.C.F. of the 7´13 + 5´ 23 numerator and the denominator = 35 of the sum. Then reduce the improper fraction to its simplest 91+115 206 form. = 35 = 35 Step 4: Convert the improper The H.C.F. of 206 and 35 is fraction into a mixed fraction. 1. So, we cannot reduce the fraction any further. 206 = 5 31 35 35 Therefore, 23 + 32 5 7 = 5 31 . 35 30

Example 4: Subtract 2 3 from 3 2 57 Steps Solve Solve this 2 3 from 3 2 12 1 from 15 1 Step 1: Convert all the mixed fractions into improper 57 43 fractions. 32 = 3´7 + 2 = 23 ; 7 7 7 2 3 = 2´5 + 3 = 13 55 5 Step 2: Find the L.C.M. 32 – 23 = 23 – 13 and subtract the improper 7 5 7 5 fractions. [L.C.M. of 5 and 7 is 35] Step 3: Find the H.C.F. of the numerator and = 5´23 - 7´13 115 - 91 24 the denominator of the 35 = 35 = 35 difference. Then reduce the proper fraction to its simplest The H.C.F. of 24 and 35 is 1. So, form. we cannot reduce the fraction any further. Step 4: If the difference is an 24 is a proper fraction. So, we improper fraction, convert it 35 into a mixed fraction. cannot convert it into a mixed fraction. Therefore, 32 – 23 = 24 7 5 35 ? Train My Brain Solve the following: a) 13 + 1 b) 41 – 21 c) 25 – 2 1 4 6 4 8 9 3 Fraction Operations 31

I Apply In some real-life situations, we use the addition or subtraction of unlike fractions. Let us solve a few such examples. Example 5: The figure shows the coloured portion of two strips of paper. Find the total part that is coloured in both the strips. What part of the strips is not coloured? Solution: Total number of parts of the first strip = 9 2 Part of the pink coloured strip = 9 Total number of parts of the second strip = 7 4 Part of the orange coloured strip = 7 24 Total coloured part of the strips = 9 + 7 14 36 =63 + 63 [L.C.M. of 9 and 7 is 63.] =146+336 = 50 63 Part of the strip that is not coloured is 2 – 50 [Since 9+7 = 1+1= 2 .] 63 97 = 2 - 50 = 126 - 50 = 76 1 63 63 63 In some real-life situations, we use the addition or subtraction of mixed fractions. Example 6: Ajit ate 5 3 biscuits and Arun ate 8 1 biscuits. How many biscuits did they 54 eat in all? How many biscuits were remaining if the box had 20 biscuits in it? Solution: Total number of biscuits in the box = 20 Number of biscuits eaten by Ajit = 5 3 5 32

Number of biscuits eaten by Arun = 8 1 4 Total number of biscuits eaten by both Ajit and Arun =5 3 + 8 1 = 28 + 33 = 112 +165 = 277 = 13 17 5 4 5 4 20 20 20 - Number of biscuits remaining = 20 – 13 17 = 20 – 277 = 400 277 20 1 20 20 [L.C.M. of 1 and 20 is 20.] =123 = 6 3 20 20 Therefore, Ajit and Arun ate 13 17 biscuits. 6 3 biscuits are remaining. 20 20 Example 7: Veena covered 34 2 km in 2 hours and 16 1 km in the next hour. 3 4 If she has to travel a total of 65 3 km, how much more distance must she 5 cover? Solution: Total distance to be covered by Veena = 65 3 km 5 Distance covered by her in the first 2 hours = 34 2 km 3 Distance covered by her in the next hour = 16 1 km 4 Total distance she travelled = 34 2 km + 16 1 km 34 104 65 416 +195 611 11 3 km + 4 km = 12 km = 12 km = 5012 km 3 11 Distance yet to be covered = 65 5 km – 50 12 328 611 = 5 km – 12 km 3936 - 3055 = 60 km [L.C.M. of 5 and 12 is 60.] 881 41 = 60 km = 14 60 km 41 Therefore, the distance Veena has to cover is 14 60 km. Fraction Operations 33

I Explore (H.O.T.S.) Let us see some more examples of addition and subtraction of mixed fractions. Example 8: By how much is 4161 greater than 39 2 ? 5 Solution: The required number = 4161 – 39 2 = 247 – 197 = 1235 -1182 5 6 5 30 53 123 =30 = 30 Therefore, 411 is greater than 39 2 by 123 . 6 5 30 Example 9: By how much is 22 3 less than 50 1 ? 4 7 -The required number = Solution: 50 1 – 22 3 = 351 91 1404 637 7 4 7– 4= 28 767 27 11 =28 = 28 Therefore, 22 3 is less than 50 1 by 27 11 . 4 7 28 9.2 Multiply and Divide Fractions I Think Pooja and each of her 15 friends had a bar of chocolate. Each of them ate 5 of the 12 chocolate. Pooja wants to know how much of the chocolate bar did they eat in all. How do you think Pooja can find this? I Recall Recall that when we find the fraction of a number, we multiply the number by the fraction. After multiplication, we simplify the product to its lowest terms. Similarly, we can multiply a fraction by another fraction too. 34

• Fraction in the simplest terms: A fraction is said to be in its simplest form if its numerator and denominator do not have a common factor other than 1. • Reducing or simplifying fractions: Writing a fraction in such a way that its numerator and denominator have no common factor other than 1 is called reducing or simplifying the fraction to its lowest terms. • Methods used to reduce a fraction: A fraction can be reduced to its lowest terms using: 1) division 2) H.C. F. I Remember and Understand Multiply fractions by whole numbers Let us now learn to find the fraction of a number. Suppose there are 20 shells in a bowl. Vani wants to use 1 of them for an art project. So, she divides the shells into 5 5 (the number in the denominator) equal groups and takes 1 group (the number in the numerator). This gives 5 groups with 4 shells in each group. Therefore, we can say that 1 of 20 is 4. 5 3 Vani’s sister Rani wants to use 10 of the 20 shells. So, she divides the shells into 10 (the number in the denominator) equal groups, and takes 3 groups (the number in the numerator). This gives 2 shells in each group. Hence, Rani takes 6 shells. Therefore, 3 10 of 20 is 6. We write 1 of 20 as 1 × 20 = 20 = 4. 5 5 5 Similarly, 3 of 20 = 3 × 20 = 6. 10 10 Example 10: Find the following: 21 a) 5 of a metre (in cm) b) 10 of a kilogram (in g) Solution: 2 of a metre = 2 ×1m= 2 × 100 cm = 2 × 100 cm = 200 cm a) 5 5 5 5 5 = 40 cm b) 110 of a kilogram = 1 × 1 kg 10 Fraction Operations 35

=110 × 1000 g = 1000 g = 100 g 10 Example 11: Find the following: b) 1 of a day (in hours) 2 4 a) 3 of an hour (in minutes) Solution: a) 2 of an hour = 2 ×1h= 2 × 60 min = 2 × 60 = 120 = 40 min 3 3 3 3 3 b) 1 of a day = 1 × 1 day = 1 × 24 h = 24 h=6h 4 4 4 4 Multiplying a fraction by 2-digit or 3-digit numbers is the same as finding the fraction of a number. Example 12: Find the following: a) 23 of 90 b) 15 of 128 45 32 Solution: a) 23 of 90 = 23 × 90 = 23´90 45 45 45 2070 = 45 = 46 Multiplying the numbers in the numerator and then dividing is tedious. It is especially so when the numbers are large. Therefore, we shall find if any of the numbers in the numerator and the denominator have a common factor. If yes, we take the H.C.F. of the numbers. We then divide the numbers to reduce the fraction to its lowest terms. 23 23 Hence, 45 of 90 = 45 × 90. Here, 45 and 90 have common factors, 3, 5, 9, 15 and 45. The H.C.F. of 45 and 90 is 45. So, divide both 45 and 90 by their H.C.F. 23 × 90 = 23 ´ 90 2 [Divide using the H.C.F. of the numbers.] Therefore, 45 451 = 23 × 2 = 46 b) 15 of 128 = 15 × 128 [H.C.F of 32 and 128 is 32.] 32 32 Divide 32 and 128 by 32, and simplify the multiplication. 15 4 32 × 128 = 15 × 4 = 60 1 36

Multiply fractions by fractions Multiplication of two fractions is simple. If a and c are two fractions where b and d  are not equal to zero, b d then  a × c = a × c . b d b × d Therefore, product of the fractions = Product of numerators . Product of denominators To multiply mixed numbers, we change them into improper fractions and then proceed. As we know, multiplying the numbers in the numerator and then dividing is tedious. It is especially so when the numbers are large. So, we shall check if any of the numbers in the numerator and the denominator have a common factor. We then reduce the fractions into their lowest terms and then multiply them. Consider an example to understand the concept. 23 15 Example 13: Solve: 45 × 46 Solution: To multiply the given fractions, follow these steps: Step 1: Check if the numerator and denominator have any common factors. Observing the given fractions, we see that, a) (23, 45) and (15, 46) do not have any common factors to be reduced. b) (23, 46) and (15, 45) have common factors. Step 2: Find the H.C.F. of the numerator and the denominator that have common factors. The H.C.F. of 23 and 46 is 23. The H.C.F. of 15 and 45 is 15. Step 3: Reduce the numerator and the denominator that have common factors using their H.C.F. 1 23 ´ 15 1 = 1´1 = 1 45 46 2 3´2 6 3 Therefore, 23 × 15 = 1 . 45 46 6 Fraction Operations 37

Example 14: Solve: a) 2 × 5 7 70 84 45 5 6 b) 35 × 63 c) 54 × 60 11 1´1 1´3 Solution: a) 2 × 5 = 1 × 1 = = 1 5 6 1 3 3 13 b) 1 2 1 2 1´2 =2 = 1 × 9 = 1´9 9 7 70 35 × 63 19 75 84 × 45 = 7 × 5 = 7´5 = 35 = 7 1 c) 54 60 6 5 6´5 30 6 = 16 65 Reciprocal of a fraction To find the reciprocal of a fraction, we A number or a fraction which when multiplied by interchange its numerator and denominator. a given number gives 1 as the product is called • The reciprocal of a number is a fraction. the reciprocal or the multiplicative inverse of 1 the given number. For example, the reciprocal of 20 is 20 . • The reciprocal of a unit fraction is a number. 1 For example, the reciprocal of 7 is 7. • The reciprocal of a proper fraction is an improper fraction. It can be left as it is or converted into a mixed fraction. For example, the reciprocal of 3 is 7 or 2 1 . 7 3 3 • The reciprocal of an improper fraction is a proper fraction. 95 For example, the reciprocal of 5 is 9 . • The reciprocal of a mixed fraction is a proper fraction. For example, the reciprocal of 2 3 is 8. 8 19 Note: 1) The reciprocal of 1 is 1. 2) The reciprocal of 0 does not exist as division by zero is not defined. 38

3) Numbers such as 4, 6, 9 and so on are converted into improper fractions by writing them as 4 , 6 , 9 before finding their reciprocals. 111 4) Fractions are reduced to their lowest terms (if possible) before finding their reciprocals. Let us find the reciprocals of some fractions. Example 15: Find the reciprocals of these fractions. a) 8 b) 4 c) 131 d) 4 17 19 5 Solution: To find the reciprocal of a fraction, we interchange its numerator and denominator. Therefore, the reciprocals of the given fractions are: 17 19 11 5 a) 8 b) 4 c) 3 d) 4 Example 16: Find the multiplicative inverses of these fractions. a) 5 b) 7 5 c) 0 d) 1 e) 33 1 9 3 Solution: To find the multiplicative inverse of a fraction, we interchange its numerator and denominator. The multiplicative inverses of the given fractions are: 19 3 a) 5 b) 68 c) no multiplicative inverse d) 1 e) 100 Note: 0 has no reciprocal or multiplicative inverse because we cannot multiply any number by it to get 1. Zero multiplied by any number is zero. So, 0 is the only number that does not have a multiplicative inverse. ? Train My Brain Solve the following: 14 94 2 14 c) 4 × 3 a) 54 7 b) 7 × 21 15 12 Fraction Operations 39

I Apply Divide a number by a fraction The division of a number by another means to find how many of the divisors are present in the dividend. 1 For example, 8 ÷ 4 means to find the number of fours in 8. Similarly, 10 ÷ 5 means to find the number of one-fifths in 10. Let us understand the division by fractions through some examples. Example 17: Divide: a) 15 ÷ 1 b) 75 ÷ 3 3 5 Solution: To divide the given, follow these steps: 1 a) 15 ÷ 3 Solved Solve this Steps 15 ÷ 1 75 ÷ 3 3 5 Step 1: Write the number as a 15 fraction. 15 = 1 Step 2: Find the reciprocal of the 13 divisor. The reciprocal of 3 is 1 . Step 3: Multiply the dividend with 1 15 3 45 the reciprocal of the divisor. 15 ÷ 3 = 1 × 1 = 1 Step 4: Reduce the product to its 45 lowest terms. 1 = 45 1 Therefore, 15 ÷ 3 = 45. Note: T o divide a number by a fraction is to multiply it by the reciprocal of the divisor. Divide a fraction by a number The division of a fraction by a number is similar to the division of a number by a fraction. Let us understand the division of fractions by numbers through some examples. 1 Example 18: Solve: 3 ÷ 67 40

Solution: To divide the given, follow these steps: Steps Solved Solve this 1 ÷ 67 3 ÷ 54 3 5 Step 1: Write the number 67 as a fraction. 67 = 1 Step 2: Find the reciprocal 67 1 of the divisor. The reciprocal of 1 is 67 . Step 3: Multiply the 1 11 1 dividend by the reciprocal 3 ÷ 67 = 3 × 67 = 3´67 of the divisor. Step 4: Reduce the 1 = 1 product to its lowest terms. 3´67 201 11 Therefore, 3 ÷ 67 = 201 . Divide a fraction by another fraction Division of a fraction by another fraction is similar to the division of a number by a fraction. Let us understand this through some examples. Example 19: Solve: 1 ÷ 1 3 21 Solution: To solve the given sums, follow these steps: Solved Solve this 3 210 Steps 1 ÷ 1 25 ÷ 75 3 21 Step 1: Find the reciprocal of the divisor. 1 21 The reciprocal of 21 is 1 . Step 2: Multiply the dividend by the reciprocal of the 1 1 1 21 divisor. 3 ÷ 21 = 3 × 1 Step 3: Reduce the product into its lowest terms. 1 ´ 217 =7 31 11 Therefore, 3 ÷ 21 = 7. Fraction Operations 41

I Explore (H.O.T.S.) Let us see some real-life examples using division of fractions. Example 20: Sakshi had 7 apples. She cut them into quarters. How many pieces did she get? Solution: To find the number of pieces that Sakshi got, we must find the number of quarters in 7. That is, we must divide the total number of apples by the size of each piece of apple. 11 Number of quarter pieces = 7 ÷ 4 = 7 × reciprocal of 4 = 7 × 4 = 28 Therefore, Sakshi got 28 pieces of apple. 3 Example 21: Nani had 5 of a kilogram of sugar. He poured it equally into 4 bowls. How many grams of sugar is in each bowl? Solution: 3 Total quantity of sugar = 5 kg Number of bowls = 4 33 Quantity of sugar in each bowl = 5 kg ÷ 4 = 5 kg × reciprocal of 4 3 13 kg = 3 ´ 1000 50g = 3´50 g = 150 g =5 kg × 4 = 20 20 1 Therefore, each bowl has 150 g of sugar. 16 8 Example 22: There is 25 litres of orange juice in a bottle. 25 litres of it is poured in each glass. How many glasses can be filled? Solution: Total quantity of orange juice = 16 litres 25 8 Quantity of juice poured in each glass = 25 litres 16 8 Number of glasses filled with juice = 25 litres ÷ 25 litres 16 8 = 16 2 ´ 25 1 = 2 =25 × reciprocal of 25 25 1 8 1 Therefore, 2 glasses can be filled. 42

Maths Munchies Another method to find the sum or difference of mixed fractions. We can add mixed fraction by adding the integral parts and then the proper fractions a) 23 + 32 5 7 Step 1: Add the integral parts of the mixed fractions. 2 + 3 = 5 Step 2: Add the proper fraction parts by using the L.C.M. 3 2 7´3 + 5´2 21+10 31 5 + 7 = 35 = 35 = 35 [L.C.M. of 5 and 7 is 35.] Step 3: If we get an improper fraction here, convert it to a mixed fraction. Step 4: Add the sums obtained in steps 1 and 2 to get the required sum. Therefore, 23 + 32 31 5 7 = 5 35 . Connect the Dots n Social Studies Fun 1 Did you know that the Moon is 4 the size of the Earth? Interestingly, your weight on the Moon is 16 of that on the Earth. Try calculating what your weight on the Moon will be. English Fun The English alphabet has 26 letters. Out of these, 5 are vowels. 26 Express the number of consonants in the form of a fraction. Fraction Operations 43

Drill Time 9.1 Add and Subtract Fractions 1) Add: a) 3 + 5 b) 4 + 3 c) 12 1 + 13 2 4 13 14 12 75 d) 10 1 +12 4 e) 2 + 6 33 16 30 2) Subtract: a) 4 − 3 b) 14 − 3 c) 13 − 14 9 11 30 24 30 60 d) 7 2 – 4 1 e) 12 3 – 112 84 89 3) Word problems a) Sudheer saves ` 360 per month from his salary of ` 3600. Hari saves ` 200 per month from his salary of ` 2400. What fraction of their salary do each of them save? b) Pavani used 450 cm of satin ribbon from a bundle of length 3000 cm. What fraction of the satin ribbon did she use? 9.2 Multiply and Divide Fractions 4) Multiply fractions by a whole number 12 3 4 a) 32 × 64 b) 8 × 80 c) 20 × 100 3 e) 4 × 27 d) 7 × 49 9 5) Multiply fractions by fractions a) 22 × 26 b) 4 ×16 c) 3 × 51 13 44 12 24 17 21 d) 7 × 45 e) 5 × 4 15 49 20 25 44

6) Find the reciprocal of the following: a) 27 b) 2 c) 5 1 d) 50 e) 38 53 2 23 7) Divide: a) 16 by 1 2 c) 1 by 3 4 b) 14 by 7 42 d) 1 by 1 e) 1 by 5 7 49 15 A Note to Parent While having meals, engage your child with the concept of fractions as you serve food items such as chapatti and so on. Remember to emphasise the fact that fractions deal with equal parts. Fraction Operations 45

Money10Chapter I Will Learn About • using four operations of numbers for calculating money. 10.1 Unitary Method in Money I Think Pooja went to a grocery store to buy a few packets of chocolates. She found that 1 packet has 40 chocolates. She tried calculating the cost of 20 chocolates. Can you find the cost? I Recall We have already learnt about mathematical operations such as addition, subtraction, multiplication and division. We have also learnt about decimals. Let us answer the following to recall the different operations involving money. a) ` 436.25 + ` 703.75 b) ` 565 − ` 209.50 c) ` 36.80 × 6 d) ` 91.25 ÷ 7 e) ` 495 ÷ 5 46