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h dSoeclmcn coPY onlYrncY bl th6 MIIP montlrhud on trvi(d 0a tho lurlhaqlhe FdCt rcAs ldentifyyourstrongestskills,trackyearonyear performance See clearlywhere you need to improve Ivleasure your true capabilities on a global scale n 2017 the most comprehensive, 6nnua, global benchrnark of lnreLap\"rern.inen.g.e-;p0Fr\"or:egr,r.er5isrs'!.wr.le.llb5,e.tratou'_noc,Dchgredr.'rodi'n_eeIrno.1d,11iao<'CJo.gA.eeS.5asss1(eCVslsjdes'1t0h\"e\"\"r ... Find out more at www.macmillaneducetion.inlicas $ uNsw\"ct\"uat A macmillan eoucallon cAs € a oroduct of UNSW G oba , a who ly owned enrerpnse ol uNswAlstrali6flheunilersitvof NewsolthwaesJ lisdistributed erc us vey in lndi6 by Macmi an Educanon

Mot Ypq\"gp-p LIERARY I sPEcIrEtt cow DAIE.. ltlt;, Usho Vorodorajon Former Vice Principal & HOD Mathematics PS. Senior Secondary School, Mylapore, chennai W\"r..,flfi @t ert^eft l0I r Ut.fIe educat on

A 1 iii MACMlLLAN a Momilldn Publishe$ La(tia Private Lt'I 2013 20t7 A-;;L.u:;riel;;1\"\"i;;ir;;*ra\".\"ii*eUi.\"h,e^'ri*s\"r0o.,e,\"c\"\"srnt\"0;'e'm.,\"\"*nr\"io.en,.a.d-a*i-\".vt.\"l,npF.*'dtoavu*e'nneoi,rciec*rccua,orrhrhrauuiiabeoerrnhe\"ncoidrcoohraaeipr'snalv,edp\"nnodnmgcsmoh'meavrrpigrilptunaieicetnec\"nlda'trtri'iicetmssd\\'lsarsosoooruipronlebatpnidrcsoaraedrifonlrl'm'hroiheacantih8'creearpeohsmnutrPvrbsii,cel'irrpcavoiasular'rhbmlimolsibo(avoanmnrrmueioeabmanivrv' Reptinted 2a14 (hn e) 20|5 (lwi'e),20t6 RePri\\ted 2017 (Thtice) 2418 MACMILLAN PUBLISHERS INDIA PRJVATE LIMITED Delhi B.nialuru chennai Kolkata Mumbai Ahmedabad Bhopal chandigarh coimbalore cuttaok Cuwaha;i HydeFbad JaDur Lucklow Maduai Naepur ?atna PuDe Thiruvdanthapuram visa\\hapatnam ISBN: 978-9352-52166_l Published by Macmillan Publishen India ?rivate Ltd' 21, Patullos Road, Chemai 600002, lndia Printed al: Repro India Ltd , Suat \"* -y *\".*,hD;,bTa€ahtvisserehrb.enorcoorolkirfnhyitseetnmhe€*veeaopnnnfttbtrhflhioeserhbaeeourdotuhikncoao.MttoOioit\"ni.nrarugl ra\"lt,fn\"odLrulaer\"naovm.cin*ogr\"ucopc*uptoirvypieorJiseiea.isci.otrio.aTnh\"lett\"t\"au.nuylih'\"oi(rs\")\"*oufuthrnedpib-fpooo\"n.kv,yh\"aorepigtvh8rtvissehottfahkmaesnvbapelelen*ieominnoaindnavebmrletvencmtelavennin1eofrtienwnghsealrdte-'

Pref ace Maths Xpress for Classes 1 to 8 has been developed by practising teachers and maths experts. Based on the latest pedagogical approaches to teaching mathematics, the series provides a strong mathematical focus and plenty of opportunities for Continuous and Comprehensive Evaluation. The series is based on the premise that all students can become powerful mathematical learners. Special Features a Let's Recall recapitulates previous knowledge of concepts a Maths Around Us helps students apply mathematical concepts in real-life situations . HOTS/Target Olympiad/Problem Solving empowers students to think out-of-the-box a Speed Maths/Mental Maths prepares students to carry out quick calculations a Take Care helDs students avoid common errors a Maths Lab Activity and Project connect maths with real-life situations a Fun With Maths and Maths Game reinforce the concepts taught in a fun way a Enrich Yourself provides deeper insight into the concept under study a Revision Station helps students in assessing their understanding ofthe concepts learnt a Worksheet helps in reinforcing the concepts learnt a Glossary serves as a ready reference a Practice Sheets aid in comprehensive evaluation a Maths Engine includes tasks that foster creative thinking and problem-solving skills among students

a\\le]sson .SJ ilructure l\"si:'.: liecali Recapitulates previous knowledge of concept5. \"1\":,i':*\"'=\"' :-'.t41i li';, \".:.. .iL:lt.\"i \":11;:,' rr' \"--- ;- , Itr'laths Around Us Helps students apply mathematical concepts in real-life situations. i /)'l'S/Tijr et Olyrrpiad/Prolclern ;r.1\" l,:rln g* Empowers students to think out-of-the box. r !qdia, :$dh,@,rsoi!\"., tI \" .5pelcl irn alih s/r:.lr: i.i ::,:t i I,r,;ritl- :i Ju.'/AAJ\\,,;tii) Prepares students to carry out quick calcuiations. l. i r I rh i.i'';.r.r.'i.'/)!.''':.n-'.r! ilrii,l '4'; ,,, .l)l Connect maths with real-life srtuationS.

Reinforce the concePts taught in a fun way. . Helps students in assessing their understanding of the concepts Iearnt. aWoersHEEri 'A Helps in reinforcing the concepts 'l learnt. Serves as a ready reference. Lr 3 'a\"*,'.@\"--\"'5] Aid in comprehensive evaluation.

Dlgltul Resoutloes : 0 glopdld r. More on Large Numbers 2. Operations on Large Numbers Mult 5. Decimals 6. Basic Geometry --: 8. Metric Measures 9. Perimetcr nnd Area r0. Vrlunc and Nets r1. Timc and 12. Life Mathemntics 13. Mapping Skills 1.1. Data Handling ts. Problem Solving Skills

for fuo- erbonclog |'qlbs to n ake for l€ornlrg tblokhg leornlDg s|lnple I'qtlF skllls r€ol llfe Oge-oppropriote digitol resources io eDrich leorning experience * Two lDteroctive 'srrokes oDd Lodders'gopes coverigg flve rDotbemoticol tttelr'es eoclt. ** Two interoctive gomes bosed on the tbeme 'crlcket', coverirrg importont mott emotlcol concepts. *** 'quick drlll' octivities cover dtfferent toplcs deqlt within tt € book.

Contents 'L L. More on Large Numbers T7 2. Operations on Large Numbers 3, Factors and Multiples 35 4. Fractions 55 5. Decimals 78 6. Basic Geometry 7. Patterns and Symmetry to2 8. Metric Measures 9. Perimeter and Area t1,4 10. Volume and Nets 1.27 1L. Time and Temperature 12. Life Mathematics 138 13. Mapping Skills 14. Data Handling r52 1-5. Problem Solving Skills 164 179 Pra ctice Sheets 190 Glossary 199 2IO Answers to Selected Questions 217 221. 223 ti*

[aryeNffire Let us see the numbe rs oetow. r E t m 5 A 5-dlgit number begins with the ten thousands place and a 6-digit number begins with the lakhs place. A comma separates the thousands period from the ones period and the lakhs period from the thousands period. In the examples above, the first number is wntten as 43,257 and the second number is written as 6,78,354. Write the number names for 7,23,5L2;6,04,818 and 2O,OO2. Circle the larger n umber. a. 54,25L;23,574 b. 9,06,524;35,264 o3. Put the sign >,<or = in the boxes. ob.75,2L5 a.63,421 63,472 55,21s c. C54,337 .l q q, (1q 5,52,51,5 >+,5 t 5 7,34,287 to].f 7,00,s49 e. 10,010 {] 11,000 4. Circle the number that can be rounded off to the nu mber in the box. a. 4ss,46a,442,+sr Gzo-i @b. zzz, zze, zoz,zza 5. Arrange the nu mbers in ascend ing and descending order. 78,449; 78,45L;77,45t; 77,449

Moths Around Us Rahul and Raghav are cousins who live in different cities. Once, Rahul visits Raghav at his home during vacations. They decided to go to a museum and were waiting at the bus station for a public vehicle. Raghav Rahul! You know something, the number of public service vehicles in my city is 1,00,000. Rahul So what Raghav, the number of PUblic service vehicles in my city is 94,355. Raghav But Rahul, I am sure that my city has more number of public vehicles. Rahul That is not true Raghav! 9 in the first place is bigger than 1. So, we have more number of public vehicles! Raghav But Rahul my number has more number of digits than yours. So, my number is greater Rahul Ha ! Ha ! Most of the digits in your number are zeros! My number is greater. The discussion continues... What do you think? Whlch city has more number of public service vehicles-Rahul's or Raghav's? Which number is greater? Large Numbers Lakhs and Crores 9 99 Let us learn more about large numbers. 1 Recall that the largest 6-digit number is 9,99,999. 10000 00 What happens if you add 1 to it? We get the smallest 7-digit number, fhere ony number that is 10,00,000. bigger thon It is read as ten lakh. 9,99 ,99,999? What is the largest 7-digit number? lt is 99,99,999. lf you add L to it, you get 1,00,00,000, which is the smallest 8-digit number. lt is read as one crore. ln the previous class, we learnt about different periods and place values in the number system. Now we shall extend this concept beyond six digits. The place to the left of lakhs is read as ten lakh and one place to the left of ten lakhs is read as one crore.

Indian System of Numeration Place Value Chart <-Periods <- Places 4 0 5 3 21 75 4 0 82 62 1 75 . The first number is read as three crore fortv-two lakh five thousand three hundred twenty-one. . The second number is read as seventy-five lakh forty-two thousand eighty-two. How do you read the third number? . This number is read as eight crore sixty-two lakh fifty-three thousand one hundred seventy-five. lvll vIro-z--;;-----tr J-;J-I----) How do we write 'fifty lakh five'? II Lakhs | [Thousands I Note: The empty places are filled with zeros. Periods @ffi',fi i#, 00 So, the number is 50,00,005. Inserting Commas The digits in a number can be separated by putting commas according to the period they lie in. The digits under the same period are read together and the name of the period is read alongside. See an example below for the number 65483179. 6,54,83,I79 8v3 Thousands <-Periods It is read as six crore fifty-four lakh eighty-three thousand one hundred seventy-nine.

Read these numbers after separating the periods using commas' a. 48!25739 b. 82OOO82 c. 4572rLO5 d' 6273192 Remember The place value of a particular digit in a number is given by its position in the number. The place value of 5 75,42,Oa2 is 5,00,000. The place value of 8 a,62,53,17 5 is 8,00,00,000. 1. Read the following numbers and illustrate using a place value chart' a. 8,72,73,52r b. 3,72,4I,175 c 4,00,00'150 d. 66,85,633 e. 3,03,03,003 t 7,85,00'001 g. 35,42,19L h. 9,78,00,578 2. In the following numbers, insert commas to separate periods and write the numbers in words. 792834t c. 78242L4I 42000420 f. t7542807 a. 5572L47 r d. z>zt t2t L to025420 h. 623842L 3. Write the numerals for the following. a. Eighty lakh forty thousand twenty b. Two crore fortv-two lakh seventy-five thousand two hundred two c. Two lakh fifty-seven thousand three hundred fifty-seven d. Two crore thirty lakh twelve thousand three hundred twenty-five e. Fifty-three lakh forty-seven thousand four hundred thirteen f. Eight crore eighty-two lakh five thousand three hundred eight

4. Find the place values ofthe coloured digits in the following numbers. a. 4,83,2L,725 b. 8,4I,32,L27 c. 42,37,228 d. t8,48,255 e. L,34,52,L32 .f 4,82,47,a2I g. 7,23,75,482 h. 4,87 ,29,382 i. 7,63,98,500 Expanded Form Recall the expanded form of 54,t27. 54,L27 = 50,000 + 4000 + 100 + 20 + 7 simila(ly,48,25,L27 can be written as 40,00,000 + 8,00,000 + 20,000 + 5000 + l-00 + 20 + 7. 7,39,27,52L= 7,00,00,000 + 30,00,000 + 9,00,000 + 20,000 + 7000 + 500 + 20 + L. Standard Form tr Let us see how we can reduce a number from its expanded form to standard form. 5UUU 0 00 0 To reduce the expanded form 0 00 0 5,00,00,000 + 40,000 + 5000 + 600 + 2 J 00 0 in standard form, arrange the numbers in appropriate columns and then add. 0 -The standard form of the number is 5 ,00,45,602. 2 500 4 5o 02 1. Write the expanded form of the following numbers. d.2,51,79,256 a. 97,28,321 b. 3,52,60,713 c. L2,59,000 2. Write the following numbers in standard form. a. 5,00,000 + 1000 + 700 + 8 b. 10,00,000 + 10,000 + 1000 + L c. 20,00,000 + 7,00,000 + 50,000 + 8000 + 100 + 50 + 3 d. 80,00,000 + 7,00,000 + 20,000 + 1000 + 300 + 1.0 + 7 Comparison of Numbers Let us consider two numbers 22,41,780 and 5,35,27,72L. lf the number of digits in the numbers to be compared are different, the number having more number of digits is larger. .'. 5,35,27,721. > 22,4I,780 lsince 22,41,780 has 7 digits and 5,35,27,72L has 8 digits].

Example 1: Compare 3,52,47,8L2 and 6,21.,53,42L Both the numbers have the same number of digits. Hence, compare the digits starting from the highest place value till the digits are different. So, compare the digits in the crores column. .We know that 3 < 6 .'. 3,52,47 ,aI2 < 6,21-,53,42t Example 2: Compare 49,73,157 and 48,32,178. In these two numbers, the number of digits is equal and the digits with the highest place value are also equal, that is, 4. So, we move to the next place value digit. In the first number, the digit in lakhs column is 9. In the second number, the digit in lakhs column is 8. We know that 9 > 8. '. 49,73,1,57 > 48,32,178 Write the numbers that come just before and just after L0,00,001; 28,57,002 and 5,78,7L,IIL. Ascending Order When numbers are arranged in the order of the smallest to the largest, they are in ascending order. Example: Arrange 77 ,64,645, 27 ,36,389, 64,53,927 and I,08,74,739 in ascending order. Here, three numbers are 7-digit numbers and one number is an 8-digit number. So, the 8-digit number will be the largest numDer, NoW compare the digits with the highest place value among the other three numbers. Cleatly, 2 < 6 < 7. Thus, the ascending order is 27,36,389 < 64,53,927 < 77,64,645 < 1,,08,7 4,7 39. Descending Order When numbers are arranged in the order of the largest to the smallest, they are in descending order. Example: Arrange 78,68,97 4,73,43,638,73,42,903 and 79,55,822 tn descending order. Allthe numbers are 7-digit numbers. For all the numbers, the digit at the highest place value is also same, that is,7. So, compare the digits at the next place value. .Fs

One number has 8, two numbers have 3 and one number has 9 as the digit. Clearly,9>8>3. Hence,79,55,822 is the largest number and 78,68,974 is the second largest number. NoW let us compare 73,43,638 and 73,42,903. In this case, the three left most digits are same. Compare the fourth digit from the left . Clearly, 3 > 2. Hence, 73,43,638 > 73,42,903 Thus, the descending order is 79,55,822 > 78,68,974 > 73,43,638 > 73,42,9O3. Forming Numbers To form the largest number from the given digits, arrange the digits in descending order. Example: Form the largest 7-digit number using the digits 2, 8, 6, 1, 3,7 and 5. Arrange the digits in descending order. 8>7>6>5>3>2>1\" Thus, the largest number that can be formed using the given digits is 87,65,321. . To form the smallest number from the given digits, arrange the digits in ascending order. Example: Form the smallest 8-digit number using the digits O, 5, L, 7 , 2, 4,3 and 8. Arrange the digits in ascending order. 0<1<2<3<4<5<7<8 The left most digit of a number cannot be zero. So, we put zero in the second Dosition. Thus, the smallest 8-digit number that can be formed is I,02,34,578. 1. Compare the following numbers by using >, < or = . a.23,73,452 oo ot]c. 72,52,255 o oe. 25,67,624 23,47 ,2s3 b.8,26,72,273 38,64,743 65,23,765 d.44,63,776 44,63,776 27 ,02,534 9,24,23,725 2. Write the following numbers in ascending order. a. 56,34,7 54t 54,7 2,34L; 7 2,6t,567; 24,LL,1 L2 b. 7 3,I3,537 ; 9 4,24,27 2; 57,32,37 3 ; 3,46,25,7 22

3. Write the following numbers in descending order. a. L,55,38,244; 5 4,27,425 ; L2,7 3,253; 8,36,58,27 7 b. 2,7 2,31,445; 46,56,53L; 7 5,62,436 ; 36,3L,468 4. Form the largest 7- or 8-digit number using the fo lowing digits. a. 3,5,1.,7,2,0,9 b. 2,O,4,I,8,3,7,5 c. 9,4,1,5,2,8,O 5. Form the smallest 7-digit number using the folJowing digits. a. 5,L,3,8,2,O,4 b. 7,3,1,5,9,2,6 c. 7,2,6,L,8,3,9 Rounding off Numbers It is not the exact number. lt has What are been rounded off to the the rules of rounding off? nearest lakh. ldon't remember the rules I \\lvr^ \\vh. Do vou remember the rules of rounding off? Let us revise 1. Find the rounding digit. The rounding digit is the digit at tens place of a number, if we have to rou nd off that number to the nearest 10. Similarlv if we have to round off a number to the nearest 100, the rounding digit is the digit at hundreds place, and so on. 2. lf the digit just to the right of the rounding digit is 0, I,2,3 ot 4, keepthe rounding digit as it is and change all the digits to the right of the rounding digit to zero.

Examples: L33 rounded offto the nearest ten is 130. 3128 rounded off to the nearest hundred is 3100. 3. lf the digit just to the right of the rounding digit is 5, 6,7,8 or 9, add L to the rounding digit and change all the digits to the right of the rounding digit to zero. Examples: 3759 rounded off to the nearest hundred is 3800. 1,03,692 rounded off to the nearest thousand is 1,04,000. Round off 28,72,179 and 35,42,821 to the nearest 10, 100 and 1000. L. Mr Sinha bought a house for { 78,45,321. Round off the cost of the house to the nearest 100. 2. Suppose 12,57,821 scientists work in the research iabs of India. Round off this number to the nearest a. 10: b. 100: c. L000: 3. Complete the table by rounding off the numbers to the nearest 10, 100 and 1000. Number Rounded to the nearest 10 100 1000 88,84,630 D. 67 ,72,639 c. 23,65,76t d. 9,99,99,748 5,33,00,671

L5 The 2O1O Haiti Earthquake measured 7.0 on the Richter scale. As per reports, it caused 31,65,894 deaths. Find the estimated number of casualty to the nearest thousand. Find out the reasons why earthquakes occur. Do you think earthquake incidences can be reduced? ryP 7. Use the digits 9, 8, 6, 4,3,1, 0 to form a 7-digit number which can be rounded off as 95,43,100 to the nearest 100. 2. Write a 7-digit number which can be rounded off as 83,14,000 to the nearest 1000. 3. Form an 8-digit number using the digits 7,0,6, 5,4, 2, 3, L which is as close as possible to the largest 7-digit number. Indian System and lnternational System of Numeration So far, we have learnt Indian or Hindu-Arabic system of numeration Let us learn another system of numeration called the international system of numeration. The international system has three places for every period. The place value chart for the international svstem of numeration is as follows: Periods Places 25 7 z 1 18 2 3 In the international system, 2572L31 is written as 2,572,131. It is read as: Two million five hundred seventy-two thousand one hundred thirty-one. Now consider the number 51823457. lt is written as 51 ,823,457. It is read as: Fifty-one million eight hundred twenty-three thousand four hundred fiftv-seven. A comparison of the place value charts between the Indian and international systems is given below. c (crores) TL (Ten L (Lakhs) TTh lndian System Lakhs) TTh HTh (Hundred lnternational TM (Ten M (Millions) Thousands) System Millions) Fl Different Same _Hlt

One lakh in the Indian svstem is read as one hundred thousand in the international system. Ten lakh in the Indian system is read as one million in the international system. One lakh of Indian system belongs to the thousands period in the international system. Remember that numbers in the periods have to be separated by commas. 1.. Insert commas and write the following numbers in words in the international svstem. a. 3547t92 b. 4Lr5302 c. 25730002 d. 8200091 e. 10004731 2. Write the following numbers in the international svstem of numeration. a. Twenty-two lakh forty-four thousand twenty b. Three lakh two thousand three hundred thirtv-four c. Eleven lakh twenty-four thousa nd three hundred twenty-five d. Three lakh fiftv-one thousand three e. Thirty-three lakh five hundred fifty-four 3. Rewrite the following numbers in the international system of numeration. a. The total collection from a musical show is { 33,47,L27. b. Nearly 2,74,83,420 foreigners visited the Taj Mahal last year. c. The sales during the festive season added up to ? 17,48,345. Roman Numerals Roman numerals originated in ancient Rome. Romans did not have zero in their number system and were not aware of the place value system either. There are seven basic symbols in Roman numerals. 1 5 10 50 1oO r5oO 1OOO lM Rules for writing numbers in Roman numerals are as follows: 1. Numerals V L and D are never repeated and they are never subtracted. 2. I can be subtracted from V and X only. 3. X can be subtracted from L and C only. 4. C can be subtracted from D and M only, 5. lf a smaller numeral follows a larger numeral, the numerals are added.

Example 1: How will you write 522 in Roman numeral? 522=500+20+2=DXXll D XX II Example 2: Which number is represented by CCLXXV? ccLXXV=C +C +L +X +X + V = 100 + 100 + 50 + 10 + 10 + 5 = 275 100 100 50 10 10 5 6. lf a smaller numeral appears before a larger numeral, subtract the value of the smaller numeral from the larger numeral. Example L: Write XL in Hindu-Arabic numerals. XL = 50 - 10 = 40 [Subtracting the value of X from L] LX Example 2: Write CM in Hindu-Arabic numerals. CM = 1000 - L00 = 900 [Subtracting the value of C from M] 7. Numerals l, X, C and M can be repeated to represent a number but cannot be repeated more than three times. Example: CCC is 300 but 400 cannot be represented by CCCC. It has to be written as cD (500 - 100). 8. When a smaller numeral is written in between two larger numerals, it is always subtracted from the larger numeraljust after it. Example: XIX is 10 + (L0 - 1) = 19 but not L1, + lO = 27. XX|Visl0+ 10+(5-1)-24butnot 10+11+5 =26. Adding Roman Numerals Example: Add CLXVlll and XXV and write the answer in Roman numerals. CLXVlll = L00+50+ 10+8= L68 XXV=L0+L0+5=25 Hence, CLXV|ll+ XXV = 168 + 25 = 193 = CXC T (Since, ,109 +-90 + 3_) c xc I Subtracting Roman Numerals Example: Subtract CCV from DCXC and write the answer in Roman numerats. CCV=100+100+5=205 DCXC=500+ 100+ (100- 10)=596* 100+90=690 -baFt Hence, DCXC - CCV = 690 - 205 = 485 = CDLXXXV l12 lf

1. Write the Roman numerals for the following Hindu-Arabic numerals. a. 37 b. 93 c, 56 d. 49 e. 68 c. 1035 h. 2072 t. 276 2. Write the Hindu-Arabic numerals for the followine Roman numera ls. A. XLVII b. CXLVI c. MXLIV d. MXCVIII E. CDLV n. DCXLIII f. cccv CCXXVI 3. Solve and write the answer in Roman numerals. a. LXVII- XLIX b. XCV - XtVl c. XXXVI _ XXII d. CXCII- XLV e. MCD-CCCXXII t CDU _ CLXXX 4. Add the following Roman numerals and write the answer in Roman numerals. a. XXVlll + XIX b. LX + CL c. LV + lll d. XXXV+ MXVI E. DC+CD I MCCLX + LXIV Target Olympiod Pinky, while copying a number from her textbook, wrote 98,75,281 in place of the original number by mistake. The original number, when rounded off to L00, became 98,75,800. Also, if the number was rounded off to 10, it became 98,75,820. Find the original number. MY Proiect oufrilfnund\"tttorm, we see the llghtning first and then hear the thunder. Why does it happen like this? This happens because light travels faster than sound. Find the speed of light and the speed of sound from Your science teacher'

D[gtH'S}EAD A-g4n7 Obiective: To compare large numbers using an abacus' Materials required: Notebook, pencil, abacus, beads Method: Students use an abacus to represent large numbers Startfromthe leftand insert the same number of beads as the digits in the given number' Represent 14,22,757 on the abacus. Represent 36,35,242 on the abacus. TL ThH Represent 25,31,406 on the abacus. ) c TL TO The number of beads in the highest place value tells you which number is bigger. L. Write the largest 8-digit number. 2. Write the smallest 7-digit number. 3. Write the number that comes iust altet 7 ,21'75,291' 4. Write the number you get when you add 1 to the largest 7-digit number' 5. Write the number you get when you subtract 1 from the smallest S-digit number' 6. Write the number that comes just before 91,27,840'

1. Circle the smallest number and tick the largest number. a. 53,27,789, 53,2L,8a9, 54,2L,789, 55,3L,987 b. 37,27,089, 37,27,088, 34,25,789, 37,O7,O89 2. Write the smallest and the largest number that you can form using the digits 6,2,3,7,0,9 and 5. 3. Answer the following questions. a. I2,2O,TS0peoplevisitedamuseuminoneyear.Rounditofftothenearestl00.; b.3,T2,365peoplevisitedtheTajMahalduringlastyear.RounditofftothenearestlO.i c. CDLX in Roman numerals is in the Hindu-Arabic numerals. 4. Solve the following and write the answer in Roman numerals. a. LXXIV + CX : b. XXV - VIII : 5. International airports in different cities are some ofthe busiest places in India. Here is a table showing the approximate number of passengers at the airports in different cities. City I Name ofthe airport Number of passengers Delh i Indira Gandhi lnternational Forty-six million eight hundred Airport twenty-three thousand KOtKata Netaii Subhash Chandra Bose Eight million five hundred International Airport seventy-two thousand .. ChhatrapatiShivaiilnternational Twenty-nine million one hundred rvtumDal Atrpoft thousand Chennai AnnalnternationalAirport Eight million seven hundred thousand a. Write the number of passengers, in words, using each airport in the Indian number svstem, b. Arrange the names of the cities in ascending order based on the number of passengers. c. Which two airports are not catering to more than L crore passengers? d. The airport at Hyderabad is used by eight million four hundred thousano passengers. Which place would you give Hyderabad in the ascending order of passengers? oo ............iFt - -d8_

4h\" A WORK5HEET Once, little Johnny dreamt of a magic carpet. On the carpet, he could cover the whole universe. In his dream, he saw himself flying on the carpet. He flew 2,57,500 km towards south, then 31 ,42,700 km towards north and finally 71,28,32J. km towards east. Then he landed in an unknown place. ii. Arrange the distances which he flew in ascending order and write the place vaiue of 7 in each number. The place value of 7 in 2,S7,SOO km rs The place va lue of 7 in 3j,42,IOO km is The place value of 7 in j!,28,327 Km B Johnny was greeted by an alien in that unknown place. He offered to take Johnny around. The unknown place was 34,50,000 square metres in area. b. Writc the number 34,50,000 in the international number system. The a lien offered Johnny 27,32,51,000 (gold coins) to buy the magjc carpet. But Johnny refused to sell his carpet. C. Write the number name of 21,32,5j,,0OO in the international number system. On his return trip, Johnny took a straight route and travelled 5OOO km less than 7I,28,32L km towards the east direction. d. Rou nd off the distance covered by Johnny in the east direction to the nearest 1000. ! ! ! ! ! I ; ! r - i t i I i - i I t I!!EEFFt FFEFFtFr E H H s rH HE E IH H gi t !1 f,! F EE F Ft E E rt Ft E ri Er s rria ii ii ii - E r HH i.F-

[aryeNmhcre ir Addition and Subtraction ii ii ii ii Addition and subtraction of Iarge numbers are performed in the same way as addition and li li subtraction of smaller numbers. We start from the ones place and regroup if necessary. il \"EooEtr \" E,..lltr7289 <-- Addend @94@53@@ <- Minuend +1.523 <-- Addend -2 9 7 <-8 Subtrahend 88I2 +Sum 1,475 <- Difference Thus,7289+1523=8812 Thus, 4453 - 2978 = L475 Multiplication and Division Let us revise the terms of multiolication and division with exampres. ET Multiplicand D. 2 <- Quotient I2 <- - M ultiplier otvtdeno fZs.-Divisor ------+ 20 -40 Remainder <- 5 5 1 6 <-- Product Thus,43x12=516 Thus,45+ 20= 2 (Q) and 5 (R) Ft d#.

1. Match the following: 6480 ii. Ll2S + 3271+ 4258 l. L6392 7278 + 9 8654 c, 3Zt x I72 802 48 x 135 52t90 - 35798 Addition We are already familiar with addition of 5- and 6-digit numbers with regrouping' Addition of 7- and 8-digit numbers is also done in the same way' start adding from the ones place. lf the sum of any column is more than ten, then regroup with the next column. For example, if the sum in the ones column is 13, regroup it as 1 ten and 3 ones. Write 3 in the ones column and carry over one to the tens column' Look at the following examPles. 1:Example Add 4,32,576 and 2,5L,762 3265@il;o,, 7 I 6 62 6845 38 Let's solvei 4,74,52,758 + fhus, 4,32,576 + 2,5I,762 = 6,a4338 6,24,34,255 + 4,55,?7,700 Exa mple 2: Add 5, 09z and 2,43,65,L88. 53624 ,,o0+9otE 2 +24365 8 77 892 80 .FHt rhus, 5,36,24,092 + 2,43,65,7a8 = 7'79'89'28O

Properties of Addition Property 1: lf P and Q are two numbers, then P + Q= e+ p Example: Consider the numbers 24,62,381 and 35,67 ,890. 24,62,387 + 35,67,890 = 60,3Q,27t and 35,67 ,89O + 24,62,38t = 6O,3O,27L rhus, 24,62,387 + 35,67,a90 = 35,G7,890 + 24,6238r Property 2: lf P, Q and R are three numbers, then (P + Q) + R = p + (e + R) Example: Consider the numbers 36,52,798,56,87 ,456 and 10,29,385. (36,52,798 + 56,87,456) + 10,29,385 = 93,40,254 + 10,29,385 = 1,03,69,639 49ain,36,52,798 + (56,87,455 + 10,29,385) = 36,52,798 + 67 ,L6,841 = 1,03,69,639 Thus, (36,52,798 + 56,87,4561 + 10,29,385 = 36,52,798 + 156,a7,456 + 10,29,395) Property 3: When zero is added to any number or a number is added to zero, then the sum is the number itself. Example: 79,48,567 + O = 79,48,567 Similarly, 0 + 79,48,567 = 7 9,48,567 Property 4: When L is added to a number, you get the next number which is called the successor of the number. Example: 3,56,84,598 + 1 = 3,56,84,599, which is the successor of 3,56,94,598. Su btraction We are familiar with the subtraction of 5- and 6-digit numbers. Subtraction of 7- and 8-digit numbers is also done in the same way. !rh.'anht,.hoo|r- Example 1: Subtact 44,24,897 lrom 47,52,7!9. LIERARY m'- : trEtr l:sPEcllrElt coPY 78@ @z @72@9 :loArE.lLI'6.1 42 897 3278 2 -Thus, 47,52,719 44,24,897 = 3,27 ,822

Example 2: Subtract 7,81,601 from 9,87,50,21'0' I H I s8@7@8s@@ @z@v @ -781 60 s 1 Thus, 9,87,50,210 - 7 ,8L,60:. = 9,79 '68'609 Properties of Subtraction iPfoDcrLv When zero is subtracted from any number' then the difference is the number itselt Exam ple: 59,57,643 - O = 59,57'643 Piollcfi\\.r.l: When a number is subtracted from itself' then the difference is zero -Exam ple: 36,53,7 84 36,53,7 a4 = O P f op..i iV 1: Whenlissubtractedfromanynumber,yougetthepreviousnumberwhich is called the predecessor of the number' ple:Exam 7 ,64,82,457 - 7 = 7,64,82'456' which is the predecessor of 7 '64'82'457 ' Word Problem Example 1: Mr Smith bought a bike for { 2'46'745 and a car for { l'5'80'480 What amount of money did he sPend in all? So lutlo n: tr tr .-D2 .D @, Cost of bike 4 5 Cost of car + I] 80 0 Total amount sPent Total amount spent bY Mr Smith is< Ia,27 '225 900000 + 1305684 = 1305684 - 900000 =

Example 2: Sabina had { L5,00,000 with her. She bought a new car for { 8,42,800 and an Solution: trulltrused car for { 2,82,L70. How much monev is left with her? Cost of the new car 08 2800 Cost of the used car 2 8 2L70 Money spent by Sabina r 7249 70 Amount of money left with Sabina = Total money Sabina had - Total money spent by her trEliilltrtrEtr had tTotal money sabina @S @6 /6\\ @s 0 a Amountspent - 1 I 90 Amount left 7 0 J0 Thus, Sabina is left with { 3,75,030. Moths Around Us Lima goes to a shopping mall with her father She finds out that a few thousand people visit the shopping mall every weekend, and the sale transaction runs into some crores. She was curious to know how a shop owner calculates his/her profit over the weeKeno. Seeing this curiosity, her father explained that a shop owner adds all the sales done in a week. Then he/she subtracts the rent of the shop, electricity charges, wages to be paid to the employees, etc., from the total sales. The remaining amount is his/her profit. Llma understands the importance of operations on large numbers. Add the following: b. 35,78,928 + 4,75,935 d. 18,59,231+ 48,92,L93 + 53,52,476 a. 78,91,253 + 4,95,865 c. L8,25,723 + 7,21,,93A + 5,42,t53 .f 65,41,854 + 9,58,42r + 4,52,L44 e. 67,23,579 + 52,38,165 + 7,24,9L,725

2. Subtract the following; b. 73,52,L92 - 8,39,L87 a. 89,2L,352- 4,85,789 35,2t,792 -t8,39,Lga c. 52,L5,372 - 25,06,789 e. 5,39,t8,729 - 43,79,857 f. 52,1,4,720 - 23 ,L9 ,857 3. Solve the b. 78,L5,929 + 25,28,59t - 59,72,89L a. 2,54,27,792 + 32,51,678 + 58,41,,902 4. Ms Shrishti invested { 1\"5,50,500 in her business last year. The total sales was { 8,78,450. What is the difference between her sales and investment? 5. Rahult father saved { 12,85,925 in the last L5 years. How much more should he save to make it { 15.00.000? 6. Solve the following: a. What should be added to 5,37,93,210 to get 8,89,06,972? b. What should be subtracted from 9,65,05,398 to get 4,53,98,932? c. Find the difference between the largest 8-digit number and the smallest 6-digit number. 7. 2L,32,481people live in State A.2,42,745 people moved from State B to State A and 18,452 people moved from State A to State C. How many people now live in State A? 8. Fill in the missing digits. l tr'.iEtr trWEtr T8 5 8 2 tr4 2 T7T l nL 7 9598 0 555 4 l Erol'mE eto. trffiHtr T 8L] u7 TL Il 2 -TT t] 8 tr T.J 2 85J 1 2L 8 07

1. Write the largest 7-digit number that ends with 1 and the smallesr 7-digit number that ends with 2, using the digits 7 ,9,9,2,0,1, 4 without repeating any digit. 2. What number should be added to the sum ofthe above two numoers ro make it t,25,42,t79? Multiplication We have already learnt how to multiply two 3-digit numbers and how to multiply a 4_digit number by a 2-digit number Let us discuss the steps involved in the murtiplication of anv number by a 3-digit or a 4-digit number. Multiplication of a Number by a 3-digit Number Multiplication of a number by a 3-digit number involves 3 steps. First, multiply the multiplicand by the place value of ones digit of the multiplier; then by the place value of tens digit of the multiplier, and then by the place value of hundreds digit of the multiplier. Then add the three answers. Example 1: Multiply 1275 by 428. Here,1275 is the multiplicand and 428 is the multiplier. FtET 2 7 5 <-- Multiplicand 4 2 8 <-Multiplier Step 1: 1275 x 8 = L0 00 Step 2: 1,275 x 20 = Step 3: 1275 x 400 = + 0 <-5 4 5 -, 0 0 Product Therefore, 1275 x 428 = 5,45,7OO Multiplication of a Number by a 4-digit Number Multiplication of a number by a 4-digit number involves 4 steps. First, multiply the multiplicand by the place value of ones digit, then by the place value of tens digit, then by the place value of hundreds digit and then by the place value of thousands digit. Then add the four answers. Example: Muhiply 5297 by 1025. Here,5297 is the multiplicand and 1025 is the multiplier.

I i1*l :''.' Er,',,,EEl 5 2 9 7 <-Multiplicand 1 O 2 5 <-Multiplier Step 1: 5297 x 5 26 85 Slep 2t 5297 x 20 Step 3: 5297 x 0 1.05940 2 5 <- Product Step 4; 5297 x 1\"000 000000 +5297000 A 29 Therefore, 5297 x lO25 = 54,29,425 Properties of MultiPlication Property 1: The product of any number and f. is the number itseli Example: 852'J- x L = 8521; 24,6a3 x L = 24,683 obout the product o Property 2: The product of any number with 0 is 0. on even ond on odd Example: 38,567 xO = 0 ; 4,29,732x O =O Property 3; lf P and Q are two numbers, then P x Q= Qx P. number? Will it be odd or even? Example: 24,673 x I24 = 30,52,OL2 Also,I24 x 24,613 = 3O,52,O1'2 So, 24,613 x L24 = L24 x 24,673 Property 4: The product oftwo even numbers is an even number' Example: 2I4x4=856 Property 5: The product of any two odd numbers is an odd number' Example: 7L7 x7 =8L9 Word Problem EllExample: A basket of apples costs ( 1346. What is the cost of 96 such baskets? 134 cost of 1 basket of apples Cost of 96 baskets of apples x 6 1292 16 Hence,96 baskets of apples will cost { 1,29,216' ...Ft

1. Multiply the following: 'b. 472I x 52I c. 123I x 482 e. 4837 x L03 f. 342Lx LzO a. L028 x 321 h. 2527 x 3985 i. 7L08 x 8015 d. 1845 x 240 g. 5005 x 1210 2. A silk saree costs ? 2428. What will be the cost of 134 such sarees? 3. There are 1520 children in a school. The school collects { 125 from each student for charitv Find the total amount collected. 4. Find the product ofthe largest 3-digit number and the smallest 4-digit number. 5. A factory produces 363 dolls in a day. How many dolls will be produced in 1268 days? 6. A factory manufactures 455 soaps in a day. How many soaps will the company manufacture in 15 weeks? Short Cut Multiplication Consider 48 x 102. 102 is 100 + 2. 48 x toz 48 x 100 4800 o. 234 x 2O5= b.754 x 97= 48x2 + 96 Try 121 x 39. t2I x 40 4896 t2LxL 39is40-1. 4840 'J,21, x 39 - 1,21 47 79

Solve: 2. 725 x99 3. 48x49 4. 58 x 1O2 1. 158 x 101 -,Wt t- o---JfiqrtrIltr{ Patterns in M ultiplication Observe the pattern, Observe the pattern. 1x9=9 121 x 11= 133L 12x9=108 1,2!x11,t=7343I I23x9=1197 L27x11II=73443t 1234x9=17106 L27xL171t=134443r 12345x9=111L05 Guess, what will be 12L x 11i.111) Guess, what will be .123456 x 9 ? Find out: _2. 101 x 111 = 3. 1O1 x 1111 = 1. 101 x 11= Division choose the answers from the box and fil in the branks with the retter beside each answer in the correct order. What name do you get? 726 (Ul 427 lAl es (M) 4s1 (J) L2622 (N) 31713 (A) 7722 (Nl 28 (R) 808 (A) L. 448 divided by 15 2. 5052 divided by 12 3. 9500 divided by 100 A 95139 divided by 3 5. 69498 divided by 9 6. 7986 divided by 1L 7. 8118 divided by 1\"8 8. 80800 divided by 100 9. 75732 divided by 6 l\"T- -+a_

Dividing a Number by 2- and 3-digit Numbers Example 1: Divide 4932 by 38. Step 1: 38 is a 2-digit number. So, consider the number formed by the thousands and hundreds place in the dividend. 1 49 > 38, so divide 49 by 38. 38 Step 2: Bring down the next digit and divide. Step 3: Bring down the last digit and divide. -7 6 Step 4: No more digits are left to bring down. Therefore, quotient is 129 and remainder is 30. Checking Division 30 You can check your division by using the following formula. ++++ llQuotient x Divisor + Remainder = Dividend I29x38+30=4932 Example 2: 320 metres of cloth is needed to make a circus tent. How many tents can be made with 64,320 metres of cloth? Number of tents made from 320 metres of cloth = 1 Number of tents made from 64,320 metres of cloth 320 = 64,320 + 32O (Here,32 < 320, so we put a zero in the quotient and bring down -320 the next digit.) Therefore, 201 tents can be made from 64,320 metres of cloth. Properties of Division Property 1: lf a number is divided by itsell then the quotient is always 1. Example: 95,462+95,462=7 Property 2: lf a number is divided by 1, then the quotient is the number itself. Example: 95,462+t=95,462 Property 3: lf zero is divided by a number other than zero, then the quotient is always zero. Example: 0+721=0;0+4t24=0

1. Divide the following and verify your answer. a. 10,427 + 28 h q1 7qq: ,< c. L2,579 + 33 f. 68,572+ t24 d. 78,4L2 + 6a e. 78,415 + 35 2. The cost of 1L identical mobile phones is { 9L,3OO. What is the cost of 1 mobile phone? 3. 100 kg of rice is required to feed soldiers in a military camp in one day. How many days will 3200 kg of rice last in the camp? 4. 68 pearls are needed to make a necklace. lf there are 1.4,620 pearls, then how manv necklaces can be made? LS A factory produces 18 bicycles per day. How many days will it take to produce 10,962 bicycles? Give two advantages of riding a bicycle over the other modes of transport. lyEl Ourine an epidemic of malaria, the village panchayat decided to distribute free '-'- medicines to a|| the affected people. tf each person has to take 3 tablets a dav for 15 days, how many tablets will be ordered for 15,242 people? tf the village panchayat has 22,485 tablets in reserve, how many more patients can be treated? What quality of the village panchayat is evident here? A. Social awareness B. Social responsibility C. Moral responsibiliw Prepare a chart on the symptoms, causes and prevention of malaria. 1,. A family eats 52 kg of rice in a month. How many complete months witl 768 kg of rice last? How much more rice is needed to make it last for one more month? lf the family reduces its consumption of rice by 4 kg per month, how many months will the given quantity last? 2. Fill in the circles with >, < or = sign without actually performing the calculations. a. Q Qb.482542 + 32 +Z+ +azs+z 8r7 - 4Ls +rs, ros

A Average What is an average? Why do we need averages? An average is a single value which represents a group of values. Amit and Arjun are two friends who go to play volleyball on the beach every day and on their way back, they collect seashells. The number of shells that they collected during a week is given below. Monday :lJ,-, _, ... Tuesday Wednesday :,' Th ursday -t ,t -' Friday ',-.-. Saturday Su nday At the end of the week, thev find that the total number of shells collected by both is the same. But Amit would not accept this and said only on the last day, Arjun collected more shells than him because, he was very tired that day and could not collect more. on all the other days, he had collected more than Arjun. Does it mean that Amit has collected more seashells? fNo, on an average, Amit has collecteO = 4 shells per day. Arjun's average is also the same (f = 4 shells) as Amit's. This means that both have the same number of shells. How to calculate the average? step 1: Find the sum of all the given numbers. \\tcp 2: Divide the sum by the number of addends given. \\tcp 3: The quotient is the average. I xamDle 1: The runs scored bv a cricketer in 1.1 matches are as follows: Ltt, 98, 42, 79, 1-01-, 89, 97 , 82, 89, 67 , 58 Find his average score. tro rtiof: Total runs scored by the cricketer = LL1+98+ 42 + 79 + 101 +89+97+82+89+67+58=913 -----.o-H- -

. -Averase 9L3 score = Igtal funs. played TL = 83 No. of matches .'. The average score of the cricketer is 83. Example 2: Mr Khan has decided to take his family to his village on a weekend by car. During the first hour, he covered 55 km. During the second hour, he covered 52 km. During the third hour, he covered 48 km and during the fourth hour, he covered 57 km. What is his average speed? Total distance covered in 4 hours = 55km+52km+48km+57km = 2I2 km -Averase sDeed Tota] distance travelled -_27A2 _-\"-ra kilometres per hou r Total time taken 1. Find the average of the first 10 even numbers. 2. Find the average ofthe first five multiples of4. 3. The marks scored by Ravi in 5 maths tests are given as 8L,78,93,85 and 88. His marks in 5 tests of English are given as 68, 72,90, 88 and 82. Find his average marks in both maths and English. In which subject did he score better? L+ I I I .i '6 No rain 5un Tues Wed tnurs Days of the wee The above bar graph shows the amount of rainfall in centimetres during a week. a. What is the average rainfall? b. on which day the rainfall was less than the average?

c. Which are the days when the rainfall was more than the average? d. Was the rainfall equal to the average rainfall on any day? The runs scored by two teams of cricket players in 7 matches are as follows: Match 123456 7 247 Tea m 247 328 521 r28 272 391 72 Team B 198 228 361 408 52t 452 a. Which team's performance is better? b. ln how many matches has team A scored better than their average? c. ln how many matches has team B scored less than their average? 1. The average marks scored by a class of 25 students in English exam is 82. What is the total marks scored by them? Hint: -Averaae = N-.umTrtoetraloml saturkdsen. t.sTotal marks = Averaee x Number ot students 2. The average daily wage paid to 10 male workers of a factory is { 180 and the average daily wage paid to 8 female workers is < 140. What is the average wage of the whole group? Torget Olympiod A.313CD E2GI lf the sum of any three consecutive numbers is 18, find l. B. 6 6 Given above is a grid to fill 8-digit number. Fill in the empty boxes with digits so that the sum of any three consecutive digits is 19 and that 3-digit number is an even number. Can there be more than one answer? Story Time Long ago there lived a poet. He was so poor that he could not even feed his family. On hearing about the generosity of the king, he decided to meet him and recite a l)oem. He reached the royal palace and begged for an appointment with the king. The king agreed to meet the poet. The poet sang a song in praise of the king. The king was vcry happy and told the poet to ask for anything as his reward.

The poet spread a chessboard in front of the king and said, \"Your Majesty! | want your men to keep one food grain on the Lst square, 2 on the 2nd,4 on the 3rd and so on ! Go on doubling the food grains till they reach the last square.\" The king was astonished. He asked the poet, 'Are you sure? Only food grains, not gold coins?\" The poet said, \"Yes, l'm very sure!\" The king's soldiers started keeping the grains as per the poet's wish. Can you guess how many food grains had to be kept on the 2oth square and 25th square? lt is 5,24,288 grains on the 20th square and I,67,77,2!6 grains on the 25th souare. By the time they reached the 64th square, the king had to empty his granary. The king understood the wisdom of the poet and bowed before him. Since you are familiar with large numbers, can you find out how many grains were kept on 30th, 35th and 40th squares? MORAL: Think before you speak. Do you think the king wanted to exhibit his wealth when he suggested to place gold coins on the chessboard? Did he realise his mistake when the puzzle is solved? r-Ay:.E_u-$qIEiaBv, Napier's method is an unconventional method of multiplication. t/0o'2//ll1t2,/,//4//llt,3/ 6/ 2 It works even for large numbers. 4 4),/8V2 Consider L78 x 242. 3 ';l%l% 2 Draw a 3 x 3 grid, since both the numbers have 3 digits. Then, write the numbers to be multiplied as shown. Draw the diagonals for all the squares. 7 Multiply the num bers 1 x 2 = 2, and write it as 02 in the first square as shown. Then, 7 x 2 = J-4 and write it in the second square.

Next comes 8 x 2 = 16, write it in the third square. Similarly, fill up allthe squares. Add the numbers that lie between the two diagonals. Write the answer by writing the digits from top left. The product is 43076. Check your answer with actual multiplication. NoW try for 2471 x 1252. Check your answer with actual multiplication. lt really works! lsn,t it? 2 30245 5 LI27 7392 5 2478 11 4835 + 4 + 245 6 LO971 48 In the multiplication problem, find AA suitable values for A, B and C to make the multiplication meaningful, A ABCBA

a. 4,85,9t6 + 25,432 78,9I,728 - 4,59,278 1 c. 35,72,897 - 4,59,1,89 37 ,59,821, + 5,97 ,257 + 87,2I,407 : e. /6,4L,532 + 65,419 - 28,3L,572 f. 4I,352 + 3,7 5,928 + 31,2 - 2,92.572 :i z. )otve Tne tollowtng: :i d. sLZx+>/L b. 8943 +132 d. 5921 x 290 i c. 79135 + 10 : f. 7005 + 204 . T. IUL'Z ^ +UJ 3.i Atul had { 28,49,450 with him. He bought two machines. One for ( 4,78,500 and : anorthner Tfoorr ({ 68,4d8,9955U0. How much money is left with him? lf he wants to buv a third ;: machlne tor < 20,00,000, how much more money will he need? i4. The price of a second hand car is ? 4,80,250. lf thepriceof a new car is { 2,18,750 more than the price of the second hand car, what is the price of the new car? i5. One bundle of notebooks costs { 2i.6. What is the cost of i.825 : such bundles of notebooks? :6. lf the train fare from Kolkata to Bijapur for one person is ? 33, i then how many people can travel in an amount of I g0,223? :7. The cost of a silk saree is { 3445 and the cost of a shirt is : { 375. What is the total cost of 121 sarees and 245 shirts? : 8. 1OO apples can be packed in one box. Can L,OO,gOO apples be i packed in 1008 such boxes? i 9. Fill in the missing digits. .3 oE \"i.' Emrur4 6 en2D 96 + 3 En t o 3 -fl z 3 oo sfl8 2 2 sEs 8 7 z oo

\" Woarr\"enf On the day before the mathematics exam, Reshma and her friend were discussing about a 'Maths Machine'. The 'Maths Machine' could convert a given number into another number in one step. For example, if the number 275895 was given to the machine and it was asked to convert the number into 275340, it would immediatelv subtract 555 from 275895. 1. The following conversions of numbers were done from the machine. Check what the machine has done to the numbers given on the left to make them the numbers given on the right. Fill in the blanks accordingly. One has been done for vou. a. 375842 ---------------- 380842 Add 5000 D. 4228589---------------- 422aL98 c. 7849t70 ---------------- 784917 o. 424700---------------- 2123500 Reshma was so happy with the machine that she decided to allow the machine to take two steps to get the required answer. 2. Choose the correct two steps from the given options which has to be taken by the machine to do the given conversions. a. 373242 ---------------- La'OOI i. Subtract 198241 and then add 1000. ii. Divide by 2 and then subtract 1620. iii. Subtract 1620 and then divide by 2. b. 7782428 ---------------- 77a40 i. Add 1572 and then divide by 100. ii. Add 1572 and then divide by 1000. iii. Subtract 428 and then divide by 1.00. .E,EE!LEEEE!!EII gIIIIIITHT IBI r tt ll

:l'\"' lt Factors of a given number are numbers which divide the given number comoletelv wrthout ii i; leaving a remainder. ii 6 = 1 x 2x 3 lI, 2,3 and 6 are factors of 6l ,i :: 72 = t x 2 x 2 x 3 I!, 2, 3, 4, 6 and 12 are factors of 121 ,i i 1, 2 and 3 are common factors of 6 and 12. 3 is the highest common factor {HCF) of 6 and 12. i lvtultiples of a numbercan beobtained by multiplying the given numberby1,2,3,...,and i so on. i Multiples of 2 are 2,4,6, ..., ...,... Multiplesof 3 are 3,6,9, ..., ..., ... i 6, 1,2, L8,... are common multipies of 2 and 3. 6 is the least com mon m ultiple {LCM) of i z ano 5. 123 5 6 7 8 9 10 In the given grid, circle all the multiples of 3 in red colour, all the multiples of 4 in blue colour and all LT L2 13 '1,4 17 18 19 20 the multiples of 6 in yellow colour and answer the 2t 22 23 24 25 26 27 28 29 30 following questions. 31 33 34 35 36 38 39 40 1. List the numbers circled in all the three colours. 4L 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 2, How many such numbers are there? 62 63 64 65 66 67 68 69 70 3. What is the smallest multiple of 3, 4 and 6? 7't 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 4. Cross out the prime numbers between 50 and 60. 91 92 93 94 95 96 97 98 99 t00 5. Circle the factors of 100 in the given grid in black.

Moths Around Us the 1-digit numbers What was special about the year 2016? 2016 is a number which is divisible bv 1,2,3, 4, 6, 7,8 and 9. This is so because 2016 is a common multiple of all these numbers. Can you find out which year before 2016 had this special property? Also find out which year will be divisible except 0. Also tell how often does this happen. Divisibility Rules Let us learn the following divisibility rules. A number is divisible by 2, if the digit in its ones place is 0, 2, 4, 6 or 8. A number is divisible by 3, if the sum of its digits is a multiple of 3. A number is divisible by 4, if the number formed by its last two digits is divisible by 4. A number is divisible by 5, if the digit in its ones place is 0 or 5. A number is divisible by 9, if the sum of its digits is a multiple of 9. A number is divisible by 10, if the digit in its ones place is 0. A number is divisible by 11, if the difference between the sums of alternate digits of the number is 0 or 11. See an examole below. Number Divisible bv 2 3 45 r232 Divisible, since the Not divisible, since Divisible, Not divisible, number has 2 in its 1+2+3+2=8and since 32 is slnce the digit in ones place. This is 8 is not a multiple divisible by its ones place is an even number, oT 5. 4. neither 5 nor 0. 9 10 11 1232 Not divisible, since, Not divisible, since Divisible, since the difference t+2+3+2=8 the digit in its ones between the sums of alternate and 8 is not a place is not 0. multiple of 9. digits is (1 + 3)- (2 +2) =0 Now what about the divisibility test for 6? A number is divisible by 6, if it is divisible by both 2 and 5. Check if 1232 is divisible by 6.

From the above rules, it is clear that 1232 is divisible by 2 but not by 3' So, 1232 is not divisible by 6. Example: Replace # in 273# by the smallest possible digit such that the number formed is divisible bv 3. We know that a number is divisible by 3, if the sum of its digits is a multiple of 3. .. 2 + 7 + 3 +#should be a multiple of 3. 2 + 7 + 3 = 12, which is a multiple of 3, so we can write # as 0. Hence, the number is 2730. 1.. Complete the following table by checking whether the given number is divisible by 2, 3, 4, 5, 6,9 and 10. Write 'Yes' if it is divisible and 'No' if it is not divisible. Number 2 10 529L 7380 32L5 5332 8534 76L5 2. Check whether 12925 is divisible by 5 or not using the divisibility test. Give reasons to support your answer. 3. Check if 1848 is divisible by 4 and lL using the divisibility test. 4. State true or false. a. A number divisible bv 12 must be divisible by 3 and 4. b. A number divisible by 9 is also divisible by 18. c. A number divisible by 4 is always even. d. A number divisible bv 30 is also divisible by 2, 3 and 5' 5. Replace # in the number L87# by the smallest possible digit so that the number formed is divisible bv 3. 6. With which digit you \\[,ill replace # in the number 348# so that it is divisible by 5 but not bv 10?

7. Which of the following numbers are divisible by i) 2 ii) 10 and iii) 6? a. 9504 b. 5770 c. 2456 d. 6990 e. 59540 f. 8S2Z 8. Test each ofthe following numbers for divisibility by i) 3 ii) 5 and iii) 9. a. 5049 b. 7705 c. 4265 d. 5990 e. 45350 f. 239I 8. t475 h. 46299 i. 77s5 dlygl Vrs tvtattrotra has raised a charity fund from the Orphanage winter carnival organised by her on the occasion of Christmas. She wants to donate ? 197358 equally among six orphanages and { 2346748 equally among four old age homes. Help her check if she can do equal division of monev. State some ways in which you can help the needy. L. I am a 4-digit number exactly divisible by 25. My two middle digits are the same and 1 is one of my digits. When 11\" is added to me, I am exactly divisible by 12. When 1.0 is subtracted from me, lam exactly divisible by 15. Who am l? 2. Are all the numbers divisible by 4 also divisible by 8? Explain with exampres. 3. Write the smallest 4-digit number that is divisible by 2,3, 4, 5 and 6. Factors and Multiples Factors Multiples . 1 is a factor of every number. Every number is a multiple of 1. . Every number is a factor of itself. Every multiple of a number is greater than or equal to that number. . Every factor of a number is an exact The number of multiples of a given divisor of that number. number is countless. Example: 1, 2, 3 and 6 are the only Exa m ple: 6, L2,1.8,24,30,36...arc factors of 6. multiples of 6. . Every number except L has at least 2 . Every number is a multiple of itself. factors, L and the number itself.

Prime Numbers neither prime nor A number which has only two factors, that is, composite, while 1 and the number itself, is called a prime number. 2 is the only even Examples: 5, 7, L1., 13... are all prime numbers. prime number Composite Numbers Numbers which have more than two factors are called composite numbers. Examples: 10,12,24,35... are all composite numbers. lmagine that this grid represents rows of houses in a villase. AAAAfiAAAAAAAA AAAAAAAAAAAAA AAAAAAAAAAAAA AAAAAAAAAAAAA AAAAAAAAAAAAA