TERM 2 - CLASS 5 - PRIME YEARS

When you want to shout something, you use an exclamation mark. You shout when you are excited, afraid, freaked out etc. You can spot an exclamation by the exclamation mark at the end. These types of sentences are used to ask questions. Questions end with a question mark at the end. To form questions, the basic rule is to invert the order of subject and the auxiliary verb of a sentence. 51

Example: It is raining outside. Question: Is it raining outside? Example: He can speak Spanish. Question: Can he speak Spanish? For sentences that do not have an auxiliary verb, we use words such as ‘do’ ‘does’ , , and ‘did’ . Example: I play the guitar. Do you play the guitar? You have learned about using interrogative pronouns such as ‘who’ , ‘whom’ ‘which’ , etc. in sentences. These are the most common types of sentences. These sentences give you facts. Example: I am going to school. Your house is very beautiful. Commands are sentences which order someone. Example: Get me that glass of water. Bring me your notebook. Note: Commands are impolite. When speaking to your elders or even to your colleagues, it is improper to use commands. Commands can be converted to polite sentences by use of words such as ‘please’, ‘may’ etc. Example: Please get me a glass of water. 52

1. What kind of sentences would you use for the following situations? a. You are seeing a friend after a long time. b. You have to ask directions to a place c. You are the boss in your office. You ask your employee to get you a file. d. You tell your mother you want bread and butter. e. You want to know where your mother has kept your favorite t-shirt. f. India just won the World Cup. g. You need to tell your address to your teacher. h. You ask your dog to fetch the ball. i. You are excited to go on a holiday. j. You need to tell your friend you are going to Delhi tomorrow. k. You need to ask a stranger his name. 2. Write those sentences. 53

1. Rewrite the following commands as polite sentences. a. Get me that file now. b. Deepak, play the movie. c. Neha, bring me a sandwich. d. Call the manager immediately. e. Nisha, book a taxi for me to go to the airport. f. Bring your notebook to me. • Do this as a class activity. • Divide the class into teams. • Each team shall make chits with different situations written on it. • Come up with interesting, funny situations. • The other teams shall frame sentences for the situations. • The team which gives the most number of correct responses shall win. 54

Contents 1. Roman numerals ....................... 56 2. Decimals ................................... 62 3. Fractions ................................... 79 4. Average..................................... 92 5. Percentages.............................. 95 6. Profit and loss .........................101 7. Simple interest ........................105 8. Unitary method ........................ 111 Class 5 Term 2

Roman numerals These days, Indo–Arabic numerals are universally used. However, in some places, we still see the use of Roman numerals. For example, some clocks and watches have roman numerals. Labels of classrooms are written in Roman numerals too. Roman numerals upto 10 I II III IV V VI VII VIII IX X 1 2 3 (5-1) 5 5+1 5+2 5+3 10-1 10 For higher values, we use the following Roman numerals: L C D M 50 100 500 1000 Rule of using Roman numerals: 1. If a symbol is repeated, the value is added as many times as it is repeated. e.g. I - 1 X – 10 C – 100 II - 2 XX – 20 CC – 200 III - 3 XXX – 30 CCC – 300 2. A symbol is not repeated more than thrice. 56

3. Symbols V, L and D are not repeated. 4. If a symbol of lower value is written to the right, its value is added. If it is written to the left, its value is subtracted. e.g. VI = 5+1 IX = 10-1 IV = 5-1 XI = 10+1 LX = 50 + 10 XL = 50 - 10 10 20 30 40 50 60 70 80 90 X XX XXX XL L LX LXX LXXX XC 100 150 160 170 180 190 C CL CLX CLXX CLXXX CXC 200 300 400 500 600 700 CC CCC CD D DC DCC 800 900 1000 DCCC CM M 57

1. Write the Roman numerals for the following : a. 15 f. 29 b. 18 g. 35 c. 21 h. 39 d. 25 i. 40 e. 28 j. 49 2. Write these numerals in Roman system a. 73 b. 92 c. 69 d. 98 3. Write the roman numeral for the following a. 9 b. 40 c. 60 d. 90 e. 110 f. 1200 58

Multiplication by Roman numerals If you put a bar on a numeral, it means the numeral is multiplied by 1000. e.g: = 5x 1000 = 5000 = 16 x 1000 = 16000 = 9 x 1000 = 9000 Guided Find the value of the following: a. CD e. MMC b. VIII f. DCLX c. DXXXV g. CCLV d. CDIX h. CL 59 4. Write the Indo-Arabic numerals for the following: a. LXXIII e. MXX VIII b. CCCLXVIII f. LXXX VII c. CMXLIV g. XC VII d. XCIX h. MM DCLX

1. Express the following as Roman numerals 1) 68 6) 190 11) 287 2) 93 7) 3260 12) 841 3) 578 8) 568 13) 4563 4) 1365 9) 1590 14) 6485 5) 6580 10) 9394 15) 9900 2. Express the following as Indo-Arabic numerals 1) V 6) MXX VII 2) X 7) IV DC XXV 3) VIII 8) V DCIV 4) XXX 9) VII CCLV 5) XX MCV 10) XX DCCC 3. Put > or < 1) XXIII XXXII 2) LXIV XLIX 3) CDXIII DCX 4) CL MCC 5) MCDV MDCL 6) DCX MDCCX 60

4. Find the sum: 1) VII + II 2) XIV + IX 3) XXV + VIII 4) CLV + XL 5) CDXX + IV 6) CD + MC 7) XXV + IV 8) XV DCX + MMX 5. Convert the numbers into Roman numerals and perform the indicated operation: 1. 50 + 63 2. 80 + 25 3. 253 – 50 4. 658 – 400 5. 2863 + 200 6. 9843 – 6000 7. 25000 + 316 8. 15000 + 5483 61

Decimals Decimals are fractions with multiples of 10 in the denominator. is one tenth and is written as 0.1 0.1, 0.2 …… 0.9 are tenths. In the same way, is called one hundredth and is written as 0.01 in decimal form. 0.01, 0.02 ……… 0.09 are hundredths. 1 10 1 100 Relation between tenths and hundredths Draw a square. Divide it into 10 equal strips. Colour one strip blue. The coloured part is 1 out of 10 or But = 0.1 Divide the square into 100 small squares. Now, the coloured part is 10 therefore = = 0.1 Colour one square in the grid red. The coloured square is 1 out of 100 = 0.01 1 10 1 10 10 100 1 10 1 100 62

Colour two strips yellow What fraction is yellow? = 0.2 Shade 15 squares green Green portion is = 0.15 The black portion is = 0.05 20 100 15 100 5 100 In the given grid, shade 1. 0.04 2. 0.4 3. 0.25 4. 0.15 1 100 10 100 50 100 75 100 1 2 3 4 Rupee and Paise 1 Rupee = 100 paise 1 Paisa = or 0.01 rupee 10 paisa = = 0.1 rupee 50 paise = = 0.5 rupee = rupee 75 paise = = 0.75 rupee = rupee Metre and Centimetre 100 cm = 1 m 1 cm = = 0.01m 50 cm = = 0.5 m 75 cm = = 0.75 m 1 100 50 100 75 100 Independent 63

Expanded form and place value Hundreds Tens Units . Tenth Hundredth Thousandth 0.1 0.01 0.001 1 . 5 1 . 2 5 1 . 1 2 5 2 . 5 1 2 . 0 5 2 3 2 . 0 0 5 1. One point five 2. One point two five 3. One point one two five 4. Two point five 5. Twelve point zero five 6. Two hundred and thirtytwo point zero zero five Expanded form: The above decimals can be written in the expanded form as follows: a. 1 + b. 1 + + c. 1 + + + 5 10 2 10 5 100 1 10 2 100 1000 5 5 10 5 100 5 1000 d. 2 + e. 10 + 2 + f. 200 + 30 + 2 + 64

1. Write the following in words (Read the numerals) 1) 12.5 5) 24.25 2) 20.15 7) 128.105 3) 8.134 8) 38.098 4) 12.008 2. Write the above numbers in expanded form. Independent Ordering decimals As you have learned in previous classes, when the numerator is fixed and the denominator increases, the value of a fraction decreases. Thus, > > Therefore, 0.5 0.05 0.005 > > 0.5 0.25 > > 0.125 Similarly, 21.5 21.125 21.0125 > > 1 10 1 100 1 1000 Independent Put > or < 1) 2.5 2.05 4) 10.12 10.20 2) 4.5 4.15 5) 115.5 125.05 3) 8.15 8.05 6) 85.2 79.9 65

Write the decimals in ascending order a. 2.5, 12.05, 8.92, 2.05, 2.125 b. 11.5, 10.85, 20.05, 15.001 Guided Independent Write the following in ascending order a. 1.5, 1.025, 1.125, 1.05 b. 6.3, 6.003, 6.3, 6.25, 6.025 c. 19.05, 19.5, 19.125, 19.25, 19.001 d. 2.85, 2.58, 2.08, 2.804, 0.285 Like and unlike decimals 10 .5, 11.05, 1.005 are unlike decimals because the number of decimal places in each is different. Similarly, 1.5, 1.25, 1.255 are unlike decimals. We can make them like decimals as follows 1.500, 1.250, 1.255 Zero added at the end of a decimal does not change the value of the number. 1.5 = 1.50 = 1.500 66

Conversion of decimals into fractions 0.5 = We need to reduce the fractions to lowest terms. 5 and 10 have a common factor 5. Dividing both numerator and denominator by 5, we get a fraction reduced to lowest terms. 5 10 This division can be a mental operation, cancelling and writing new numbers. 5 10 5 1 2 = 5 0. 7 = . 7 and 10 have no common factors. So, we write the fraction as it is 25.8 = 25 = 25 5 10 1 2 = 1 2 7 10 8 10 4 5 4 5 28 100 28 100 0.28 = ; = (cancelling by 4) 7 25 7 25 1. Convert the following to fractions and reduce to lowest terms a. 0.16 e. 0.15 b. 0.45 f. 8.5 c. 0.05 g. 4.006 d. 2.25 h. 0.125 Independent 67

Conversion of fractions into decimals We know that = 0.5 = 0.25 = 0.75 How do we get it? Divide the numerator by denominator. Add a zero and put a decimal point in the quotient. Put a zero again and continue. Continue the division till the remainder is zero. = 0.125 = 0.16 1 2 1 4 3 4 1 0 0.125 8 -8 2 0 - 1 6 4 0 - 4 0 0 1 8 4 25 15 10 5 10 = 1 = 1.5 Another Method: Another way to convert fractions into decimals is to make the denominator 10 or a multiple of 10 by multiplying the numerator and denominator by a suitable number. Some examples are given below. = = 0.16 = = = 0.35 = = = 0.076 4 0 0.16 25 - 2 5 1 5 0 -1 5 0 0 4 x 4 25 x 4 16 100 7 x 5 20 x 5 19 250 7 20 35 100 76 1000 19 x 4 250 x 4 68

Convert the following fractions into decimals. Follow whichever method is best suited. 1) 5) 8 9) 8 2) 6) 1 10) 2 3) 7) 3 4) 8) (Either Convert mixed fractions to improper fractions and divide the numerator by denominator; or keep the whole number as it is. Operate upon only the fractional part). Independent 15 10 128 100 999 100 9 20 1 4 1 25 3 7 49 12 1 8 4 5 Writing decimals correct to two places 1.128: The digit in the 3 decimal place is 8, which is more than 5. rd Thus, we add one to 2 and discard the 3 place. rd 1.128 = 1.13 Similarly, 8.432 = 8.43; 15.563 = 15.56; 7.726 = 7.73 Independent Round off the following correct to 2 decimal places a. 25.315 b. 8.028 c. 1.617 d 0.823 e. 22.225 f. 16.081 g. 15.009 69

1. Write in expanded form a. 125.538 2. Write the numeral a. 5 + + b. 21 + 3. Write in words a. 25.385 4. Write in ascending order 8.125, 8.008, 8.15, 8.503, 8.038 5. Convert decimals to fractions a. 2.35 b. 7.05 c. 2.125 6. Convert fractions into decimals a. 2 b. 1 c. 8 d. 7 7. Rewrite with decimals correct to 2 places a. 25.057 b. 1.008 3 10 4 1000 8 1000 1 4 4 5 3 5 1 8 70

1) 2 + + = 2.53 (true/false) 2) 3.175 > 3.017 (true/false) 3) 2 = 2.25 (true/false) 4) 6.05 = 6 (true/false) 5) 25.048 = 25.05 (true/false) 6) 75 paisa = 0.075 rupee (true/false) 7) 1 rupee 5 paisa = 1.5 rupee (true/false) 8) 1m 25 cm = 1.25 m (true/false) 5 10 3 1000 1 8 1 20 Addition and subtraction of decimals Add 7.1, 8.15, 10.225, 23.81 Make the decimals into like decimals and write in columns 8 7 . 0 0 6 - 4 3 . 5 0 0 4 3 . 5 0 6 6 10 7 . 1 0 0 8 . 1 5 0 1 0 . 2 2 5 2 3 . 8 1 0 4 9 . 2 8 5 1 Subtract 43.5 from 87.006 Again, first convert them into like decimals and write in columns. 71

1. Add 16.403 + 3.8 + 15.056 2. Add: 7.3 + 3.091 + 12.825 3. Add: 1.86 + 12.008 + 220.015 4. Subtract 23.4 from 42.016 5. Subtract 2.008 from 12.5 6. Find the value of 8.13 + 3.7 – 2.006 Independent Multiplication Multiplication by 10, 100, 1000 etc 6.234 x 10 = 62.34 6.234 x 100 = 623.4 6.234 x 1000 = 6234 While multiplying a decimal by 10, the decimal point shifts one place to the right. When multiplied by 100, the point shifts 2 places to the right, and in the case of 1000, the point shifts 3 places to the right. 0.1675 x 10 = 1.675 0.1675 x 100 = 16.75 0.1675 x 1000 = 167.5 72

Do the following multiplications a. 6.256 x 10 d. 31.251 x 100 g. 25.167 x 1000 b. 12.05 x 10 e. 3.25 x 100 h. 245.075 x 1000 c. 128.5 x 10 f. 15.2 x 100 i. 8.17 x 1000 Independent Multiplication of a decimal by a whole number 2.85 x 12 2 8 5 x 12 3 2 . 2 0 Multiply just like you multiply whole numbers, and put decimal point 2 places to the left. Ans : 32.20 Multiplication by a decimal Multiply like whole numbers. Count the total number of decimal places, and put the point at that many places to the left. e.g. 2.087 x 1.5 Total 4 decimal places So, point shifts 4 places to the left Ans : 3.1305 .061 x .02 No. of decimal places = 5 Put 2 zeros to the left to make it 5 Ans : 0.00122 2 0 8 7 x 1 5 3.1 3 0 5 . 6 1 2 . 0 0 1 2 2 73

Do the following multiplications a. 62.9 x 15 b. 6.85 x 1.3 c. 1.25 x 3.25 d. 3.81 x 4.25 e. 0.284 x 0.52 Division of decimals Division by 10, 100, 1000 etc 2.5 ÷ 10 = 0.25 2.5 ÷ 100 = 0.025 2.5 ÷ 1000 = 0.0025 When divided by 10, the decimal point shifts one place to the left. When divided by 100, the decimal point shifts 2 places to the left. When divided by 1000, the decimal point shifts 3 places to the left. In the absence of places, add zeros 2382 ÷ 10 = 238.5 2382 ÷ 100 = 23.85 2382 ÷ 1000 = 2.385 36.24 ÷ 300 = 0.1208 Independent 36.24 300 = 36.24 3 x 100 = 0.3624 3 = 0.1208 First divided by 100 Divided by 3 74

Divide decimal by a whole number 12.48 ÷ 4 Divide them as whole numbers. Put decimal point in the quotient 12.48 ÷ 4 = 3.12 Division of a decimal by a decimal The divisor should be a whole number Accordingly, we have to shift the decimal point. e.g : 25.28 ÷ 1.2 In the divisor, there is 1 decimal place. Multiply The dividend and the divisor both by 10. 25.28 x 10 ÷ 1.2 x 10 Now, do the division as above 252.8 ÷ 12 = 21.067 correct to 3 decimal places 252.8 21.0666 - 24 12 -12 08 0 12 72 8 0 72 348 58 - 30 48 48 0 6 e.g : 3.48 ÷ 0.06 Both contain 2 decimal places. Multiply both by 100 3.48 x 100 = 348 0.06 x 100 = 6 348 ÷ 6 = 58 1 2 . 4 8 3.12 - 1 2 0 4 -4 8 4 -8 0 8 0 72 8 75

e.g : 24.5 ÷ 0.05 Divisor contains 2 decimal places. Multiply both the numbers by 100 24.5 x 100 = 2450 0.05 x 100 = 5 2450 ÷ 5 = 490 245 0 49 0 20 45 45 0 0 5 Find the quotient a. 25.75 ÷ 10 b. 11.85 ÷ 100 c. 0.086 ÷ 1000 d. 24.56 ÷ 0.08 Guided e. 72.9 ÷ 0.09 f. 124.8 ÷ 0.12 g. 86.48 ÷ 0.8 h. 128.16 ÷ 16 Do the following divisions a. 12.82 ÷ 10 f. 12.82 ÷ 100 b. 285.5 ÷ 1000 g. 2.453 ÷ 10 c. 2.453 ÷ 100 h. 13.84 ÷ 0.09 d. 125.8 ÷ 0.5 i. 24.5 ÷ 0.25 e. 32.45 ÷ 0.025 j. 140.5 ÷ 15 Independent 76

1. Add: 1.82 + 3.5 + 22.008 + 4.03 2. Subtract 8.015 from 12.12 3. Find the value: 3.15 – 1.81 + 4.05 – 1.028 4. Find the value a. 0.2132 x 100 b. 8.5 x 1000 c. 0.012 x 10 d. 0.25 x 0.13 e. 31.5 x 0.08 5. Find the value a. 20.6 ÷ 0.4 b. 18.4 ÷ 4 c. 0.108 ÷ 0.12 d. 25.25 ÷ 0.015 1. Arun purchased rice for Rs.132.50, oil for Rs.68.25, tea powder for Rs.25.08, and a scrubber for Rs.5.30. What is the total amount spent by Arun? 2. From a vessel containing 25 litres of oil, 5.13 litres was removed. What is the remaining quantity of oil? 77

3. If one pen costs Rs.8.35, what is the cost of 4 pens? 4. Product of two numbers is 240.25. If one number is 2.5, what is the other number? 5. 29 kg of sugar was packed equally into 5 bags. What is the quantity of sugar in each bag? 6. Length and breadth of a room are 12.25m and 8.42 meters respectively. What is the area of the room? (Area = length X breadth) A. Find the value: 1) 8.5 x 10 2) 2.25 x 10 3) 182.5 x 100 4) 256.25 ÷ 100 5) 0.085 x 1000 B. Tick the correct answer: 1) 7.5 + 2.05 = _______ ( 10 9.55 / ) 2) 12.05 – 4.5 = _______ ( 8.00 7.55 7.50 / / ) 3) 25 ÷ 100 = _______ ( 2.5 0.25 / ) 4) 82.5 x 100 = _______ ( 0.825 8250 82500 / / ) 5) 0.005 x 100 = _______ ( 0.5 0.05 5 / / ) 6) 2.8 ÷ 10 = _______ ( 28 0.28 280 / / ) 7) 0.1 x 0.2 = _______ ( 0.02 0.2 / ) 8) 0.2 x 0.3 x 0.1 = _______ ( 0.6 0.06 0.006 / / ) 78

Fractions Part of a whole is called fraction. Activity 1: Do this activity a. The pictures given below are , , Label them. b. Now Look at the boxes given below 1. Divide the white part of 1 into 2 equal parts 2. Divide the white part of 2 into 3 equal parts 3. Divide the white part of 3 into 4 equal parts 4. Divide 4 into 5 equal parts 1 4 1 3 1 2 1 2 3 4 79

Activity 2: a. Find the given fractions from the collection. How many apples will be there in ? 1 2 , 2 4 , 4 6 , 2 6 , 3 6 , 6 12 Which among them are equivalent fractions? b. Find 4 pairs of equivalent fractions in a collection of 24 apples. 1) Find 4 equivalent fractions for 2) Find 4 equivalent fractions for 3) Check whether these are equivalent fractions (Cross multiply denominator and numerator) 2 3 48 96 4 20 a. 1 5 1 3 c. , 6 9 4 5 d. 3 4 1 3 3 9 , , Guided 2 3 b. 4 8 , e. 80

Order of fractions If there are 12 apples =6 apples = 4 apples > = 3 apples = 4 apples 1 2 1 3 1 2 1 3 1 4 1 3 1 3 1 4 1 2 2 3 1 6 1 4 or or > Which is greater: 1 4 2 5 1 x 5 = 5; 2x4 = 8 5 20 = , 8 20 To find which is the greater fraction out of two fractions which are not equivalent, cross multiply the numerator and denominator of the two fractions. The bigger product will show the bigger faction. Example: 2 5 > 1 4 Thus, second fraction is greater. 81

A) Put < or > Half of the square is divided into 8 triangles. Triangles marked are all identical. 2 3 a. 4 5 3 10 b. 2 9 3 8 c. 4 9 5 8 d. 4 7 2 3 e. 6 7 5 13 f. 8 21 1) What part of the square is each of the triangles? 2) What part of the square are triangles 1,2,4,5 together? 3) What part of the square ABCD is one of the small squares? 4) What part of the square is the rectangle in the figure? A B C D 1 2 4 3 5 7 6 8 E Guided 82

Operations on Fractions 1. Addition: 2 3 4 5 + 1. We need to have a common denominator to make them like fractions. 2. To get common denominator, we find the LCM of the denominators. 3. Make the LCM as the common denominator. 4. Multiply each fraction by LCM to get new numerator. 5. Add the numerators of the new fraction LCM of 3 and 5 =15 2 3 15 = 10 x 5 3 5 15 = 9 x 3 10 + 9 15 19 15 = 4 15 =1 4 7 3 14 + 4 7 14 = 8 ; x 2 3 14 14 = 3 x 8 + 3 14 11 14 = Multiply Multiply 1 5 = 2 5 + 1 + 2 5 3 5 = 1) When two or three fractions are added, we add only the numerator. 2) Only like fractions can be added. 3) If the given fractions are unlike, they must be converted to like fractions and then added. Example: Add Example: LCM of 7 and 14 =14 83

Addition of mixed numerals Find the sum of and Method 1: Add the whole numbers; 3 + 2 + 1 = 6 Add the fractions LCM of 6, 4, 3 = 3 x 2 x 2 x 1 x 1 = 12. Independent 3 5 a. 3 4 + 2 3 b. 4 7 - 1 8 c. 3 5 + 3 4 d. 2 5 - 1 6 3 4 3 , 2 1 3 1 2 + 9 + 4 12 15 12 = 3 12 1 = 1 4 1= Adding whole numbers and faction: Method 2: Convert the mixed numerals into improper fractions and add as you add other fractions + 16 1 4 = 7 1 4 1 6 3 4 3 + 2 1 3 1 L.C.M of 6, 4, 3 = 12 + = 1 6 3 4 + 1 3 + 19 6 11 4 + 4 3 + x 12 = 19 6 38 2 , x 12 = 11 4 33 3 , x 12 = 4 3 16 4 , 38 + 33 + 16 12 87 12 = 3 12 7 = 1 4 7= 84

Multiplication of fractions Multiplication of a fraction by a whole number Example: It means added 3 times. If half is added 3 times, we get Guided 1 2 2 3 1 + 2 1 4 3+ = 1. 1 2 2 5 3 -1 2. = Independent 4 7 3 4 3 +1 = 1. 3 5 1 3 4 -2 2. = 3 8 3 4 5 +1 = 3. 4 9 2 3 1 +2 4. = 3+ 3 8 1 4 5 +1 - 2 5. 1 6 = 1 2 3 x 1 2 3 = x 1 2 1 3 2 = 1 1 2 1 2 85

When we multiply a fraction by a whole number, we multiply only the numerator. 1 + 1 + 1 3 3 3 = 1 3 x 3= =1 1 3 x 1 3 x 1 3 = Multiplication of a fraction by a fraction Example: The statement means is again divided into 3 parts, that is, of . One third of , or half of . When is divided into 3 parts, we get . When is divided into half, we get . 1 3 x 1 2 1 2 1 3 1 2 1 2 1 3 1 2 1 6 1 3 1 6 Example: Draw a rectangle. Divided it into 5 parts and colour 3 parts of it, this is . Divided the same rectangle into 4 parts and shade 1 part in a different colour. How many parts have both the colours? 3 How many parts are there in total? 20. Thus, the product is 1 2 1 3 1 2 1 3 3 5 3 5 1 4 x 3 20 1 4 3 5 86

Example: Convert to improper fractions and multiply: 1 5 1 3 4 2 x 1) 2) 3) 4) = 6 5 11 4 x 66 20 = 3 6 20 3 10 =3 Independent 5) 6) 7) Find 8) Find Division Division in fractions is multiplication by the reciprocal. Why do we do so? Consider the division statement: 3 4 15 x 24 x 3 1 4 3 x 2 2 5 1 8 5 3 5 3 x 6 7 3 5 x 2 3 x 1 2 1 1 3 2 x 1 4 3 x 1 3 2 2 5 1 of 1 3 25 of It means if is divided into three parts, what is the value of each part? We know that if is divided into 3 parts, each part will be of half, that is, 1 2 3 1 2 1 3 1 2 1 3 is the reciprocal of 3. 1 3 1 2 1 6 x = 87

It means that if half is divided into one fourth, how many one fourths will be there? We know there are 2 one fourths in one half. We get this in this way: 1 2 2. 1 4 Do the following divisions: 1 4 x =2 1 3 7 11 5 3. 22 3 5 11 x = 3 10 3 = 1 3 2 9 11 3 22 4. 9 11 22 3 x = 6 3 2 Independent 1 4 3 1. 1 4 2. 2 1 3 1 3. 24 4 5 3 4 4. 1 8 5. 11 5 22 7 6. 45 5 3 5 6 7. 9 1 6 5 3 5 8. 7 19 3 8 9. 10 3 4 2 3 8 10. 10 1 4 x ( 2 4 ) 88 2

a. In proper faction, numerator is ___________ than the denominator b. Like factions have the same ——————— c. Reciprocal of is —————— d. Division is multiplication by ————————— e. = = f. The simplest form of is ————— g. = ———— h. i. If 4 is divided into 2 equal parts, each part is equal to ——— j. 3 5 1 2 2 1 60 84 2 5 3 4 1 2 x x 3 4 5 16 12 15 = ( 12 5 15 12 ) 1 2 17 72 1 2 = + 1. A shirt needs 2½ m of cloth and a pair of trousers requires 1¾ m of cloth. What is the total length required for a safari suit? 2. A cake was cut into 24 pieces and distributed among 8 children. What fraction of the cake did each child get? If there were 12 children, what fraction of the cake would each have got? 3. Arun studies 2 hours in school, and he has the rest of the day free for other activities. What fraction of the day does he have for other activities? 4. If 1 kg tomato costs Rs 20, what is the price of 2½ kg tomatoes? If the price of tomatoes increases by 10 rupees per kg, how much tomatoes one can buy for Rs 20? 89

1. What is the fraction shaded in the figure? (fig. 1) 2. Divide the triangle into 3 equal parts (fig. 2) 3. Guess what fraction is shaded in each figure. Check whether your guess is correct? fig.1 fig. 2 4. What fraction of the grid is coloured? fig.1 fig. 2 fig. 3 90

1. A rich man had 17 elephants when he died, in his will he allotted of the elephants to his eldest son, to his second son and remaining to his youngest son. How could they divide 17 elephants among them? How many did each get? What was the fraction of the elephants given to the youngest son? ( Hint: 17 is a prime number. It cannot be divided. But if it is made 18, it can be divided according to the will.) 2. Fill in the remaining rows. 1 2 1 2 1 3 1 3 1 4 1 5 1 6 5 6 7 12 9 20 11 30 91

Averages Average is a means to assess the general trend of an event. For example, the average income of an employee, average monthly income of a household in a given city or village, average age of a team in a game, average students in a class, average performance in a class etc. Average helps us to assess the general performance, without taking into account individual attributes. Properties of average 1. Average lies between the highest and the lowest values. 2. Average is obtained by adding up all the individual values and dividing the sum by the number of entries. Example: 1. In a theatre, the collection for 4 days was as follows: Rs. 4580, Rs. 5000, Rs. 2530, Rs. 1680 What is the average collection of the theatre? Average collection = Rs. 4580+5000+2530+1680 4 = Rs. 3447.50 2. Average height of 5 students in a class is 154 cm. One student of height 160m left the class and one student of height 150 joined the class. What is the new average height of the class? 92

Follow these steps: Average height = 154cm Total height of 5 students = Average x total number of students = 154 x 5 = 770 Height of the student that left = 160 cm Total height of 4 students that remain = 770 cm – 160 cm = 610 cm Height of the new student that joined the class = 150 cm Total new height of the 5 students = 610 cm + 150 cm = 760 cm New average = 760 5 = 152 cm 1. Marks of 5 students in an entrance test were 83, 65, 72, 58, 90. Find the average score 2. Average weight of 8 men is 120kg. One more man with a weight of 150kg joins the group. What is the new average? 3. Rahul scored 71, 83, 57, and 68 runs in 4 innings. Sachin scored 67, 81, 72 and 59 in another 4 innings. Whose is a better performer? Find out from their average. 4. Marks of 5 subjects for 4 students are given below. Name Eng Maths Sc S.st Hindi Vimal 80 58 90 96 85 Anuj 65 80 82 54 63 Vidya 75 90 72 60 58 Sameer 60 82 85 70 64 93

i) Calculate the average marks of each student? ii) Calculate the average marks in each subject? iii) In which subject is the average performance better? iv) Who has scored the highest average marks? 5) Rainfall in the month of June, July, August, September, and October was 90cm, 110cm, 80cm, 75cm and 60cm respectively. What was the average rainfall for the period of June to October? 1. The marks in mathematics for 20 students of a class are given below. 80, 70, 62, 54, 62, 70, 62, 48, 92, 35, 48, 54, 62, 80, 70, 54, 62, 35, 35, 35, a. Sort out the number of same scores b. Find the average performance of the class (Hint: Some scores are repeating. Take them only once) 2. In a school cultural meet, the average scores for various competitions are given below. a. 3 schools from New Delhi had an average score of 150 out of 200 b. 4 schools from Kolkata had an average score of 130 out of 200 c. 5 schools from Chennai had an average score of 140 out of 200 What was the average score of all the schools taken together? • Collect data from 5 car sales centers to find the number of cars of a popular brand sold in that year. • Calculate the average number of cars sold in the city in that year. 94

Percentage Percent means per hundred, centum is the Greek word for hundred. Percent is a fraction with 100 in the denominator. It is expressed as %. e.g: 50 100 50%= 5 100 5%= 15 100 15% etc. = Fractions and decimals can be expressed as percentage Percentages are equivalent fractions of any given fractions, with 100 as the denominator. To convert a fraction or decimal into percentage, just multiply the same by 100. 50 100 50% = 1 4 = 25 100 25% = 1 x 2 x 50 50 1 x 4 x 25 25 = 3 x 5 x 20 20 3 5 x 60 100 60% = = Acitivity 1: Take a paper cut out. Fold into 2 equal parts. Each is ½. Now divide the cutout into 100 small squares. There will be 50 squares in 1 half. 1 2 1 2 95

Acitivity 2: Have 10 X 10 grid papers. 1. Represent the following fractions as percent on the grid. a. b. c. 2. Represent the following decimals as percentage on the grid paper. a. 0.1 b. 0.5 c. 0.25 d. 0.2 e. 0.01 Stick the paper in your notebook. Acitivity 3: 1. Express the number of vowels in your name as a percentage. 2. Express the number of vowels in English alphabets as a percentage. Acitivity 4: You spend time in school as follows. Academics 4 hrs. Sports 1 hr Lunch hr Art class 1 hr If the school functions from 8:30 a.m to 3:30 p.m, express time spent for each as a percentage. 1 10 1 10 1 2 1 10 Time spent for one item total school time % = x 100 96

Activity 5: Draw a Pie Chart with the given data after changing it into percentage. A family’s income and expenditure are as follows. Monthly Income - Rs 50,000/- House rent - Rs 15,000/- School fees - Rs 6,000/- Domestic expenses - Rs 10,000/- Outings and recreation - Rs 3,000/- Savings - Remaining. 1. Convert each expenditure into percent. 2. Express it as a pie chart. Draw a circle. Divide it into 10 equal parts, each of these 10 parts again divided into 10 equal parts. house rent 30% 100 10 20 90 80 70 30 50 40 60 House rent = In the same way mark the others. Use different colours for various items. 15000 x 100 50000 =30% Activity 6 : Take your marks in the class test which are out of 25. Convert them into percentage. 97

Conversion involving percentage 5% of 120 = 2. What is 30% of Rs 1200? Ans : 5 100 x 120 = 6 6 2 30 100 x 1200 =Rs. 360 3. Out of Rs 150, Rs 40 was spent. What % was spent? Ans: 40 150 x 100 = 400 15 = 26.6% 4. 5% of a number is 15. What is the number? 5 100 = 15 The whole = = 300 (multiply by reciprocal) 15 5 x 100 5. 8% of an amount is Rs 64. What is the total amount? Ans: 8 100 = 64 64 x = Rs. 800 = 100 8 1. What is 5% of 120? Guided Amount 98

1. What is 25% of 4850? 2. Out of 500m of cloth, 125m was cut off. What % was cut off? What percent remains? 3. In a theatre out of 950 seats, only 480 seats were filled. What % seat was vacant? 4. What percentage of 350 is 7? 5. A company gives 10% commission for every Rs 4800 worth of sales to an agent. Find the commission the agent gets. 6. Sale tax on a sale of Rs 3500 is Rs 256, find the percentage of sale tax. 7. The population of a city has increased by 20% over the last two years. If the original population was 10,58,270. Find the present population. 8. A money lender collects Rs 150 for every 1000 rupee he gives as loan for a month. What is the percentage of interest he charges? 99

Choose the correct answers 1. 1 paisa = ___ % of rupee. a. 10% b. 1% c. 0.1% d. %. 2. 40 % = ____ . a. b. c. d. 3. 42 % = ____ . a. 0.42 b. 4.2 c. 42 d. 0.042 4. 20% of 50 = ____ . a. 40 b. 10 c. 5 d. 25 5. 0.008 = ____ . a. 8% b. 0.8% c. 0.08% d. 80 %. 6. 7cm = ___ % of a metre. a. 7% b. 70% c. 0.7% 7. If 50% of an amount is Rs. 400, what is the total amount? a. 200 b. 800 c. 450 d. 4000 8. If 20% of an amount is Rs. 70, what is the total amount? a. 200 b. 35 c. 350 d. 200. 9. If 10% of a sum is Rs 3,000, total amount is? a. 30,000 b. 20,000 c. 35,000 d. 15,000 10. A medical shop gives 8% discount on medicines. If a customer get a discount of Rs 40, what is the cost of medicines he purchased. a. 800 b. 500 c. 400 d. 80. 1 100 1 2 1 5 2 5 1 4 Time: 10 minutes 100