NCERT 5th Class (CBSE) Mathematics: Large Numbers

Large Numbers We use only 10 digits in our place value system to write any number, no matter how large it is. The value of the digit depends on its place in the number. Can you read the number in the following sentence? India is the seventh largest country in the world with an area of 32, 87, 263 sq. km. The number given above can written in different ways. Word from: Thirty-two lakh, eighty-seven thousand, two hundred sixty-three. Long expanded from: (3×10,oo,ooo)+(2×1,oo,ooo)+(8×10,000)+(7×,1000)+(2×100)+(6×1o)+(3×1) Short expanded from: 30,00,000+2,oo,ooo+80,000+7,000+200+60+3 Standard from: 32,87,263 Question: Write the numbers in word from, long expanded from and short expanded from. (a) 2038987 (b) 198006 (c) 30706 (d) 8609645 (e) 9900051 Answer: (a) Word from: Twenty lakh, thirty-eight thousand, nine hundred eighty-seven Long expanded from: (2×10,00,000)+(3×10,ooo)+(8×1ooo)+(9×1oo)+(8×10)+(7×1) Short expanded from: 20.00,000+30,000+8000+900+80+7 (b) Word from: One lakh, ninety-eight thousand, six Long expanded from: (1×1,oo,000)+(9×10,000)+(8×1ooo)+(6×1) Short expanded from: 1,oo,ooo+90,ooo+8000+6 (c) Word from: Thirty thousand, seven hundred six. Long expanded from: (3×10,000)+(7×100)+(6×1) Short expanded from: 30,000+700+6 (d) Word from: Eighty-six lakh, nine thousand, six hundred forty-five Long expanded from: (8×1o,oo,ooo)+(9×10,000)+(6×100)+(4×10)+(5×1) Short expanded from: 8o,oo,ooo+60,000+40+5 (e) Word from: Ninety-nine lakh, fifty-one Long expanded from: (9×1o,00,000)+(9×1,00,000)+(50×1)+(1×1) Short expanded from: 99,00,000+50+1 Large Numbers Question: (a) Write the number that is 1,00,000 less than: (i) 65,78,975 (ii) 71,71,940 (iii) 10,45,701 (iv) 3,70,238 (v) 1,50,987 Answer: (i) 64,78,975 (ii) 70,71,0940 (iii) 9,45,701 (iv) 2,70,238 (v) 50,987 (b)Write the number that is 10,00,000 more than: (i) 19,78,098 (ii) 2,59,894 (iii) 36,789 (iv) 78,98,509 (v) 89,99,999 Answer: (i) 29,78,098 (ii) 12,59,894 (iii) 10,36,789 (iv) 88,98,509 (v) 99,99,999 Question: Rewrite the following numbers in ascending order and then in descending order. (a) 4,78,098 4,87,678 4,76,908 4,87,876 Answer: Ascending order: 4,78,098 → 4,76,908 → 4,87,678 → 4,87,876 Descending order: 4,87,876 → 4,87,678 → 4,76,908 → 4,78,098 (b) 21,67,654 21,76,654 21,67,456 21,67,564 Answer: Ascending order: 21,67,456 → 21,67,564 → 21,67,654 → 21,76,654 Descending order: 21,76,654 → 21,67,654 → 21,67,564 → 21,67,456 Question: Write the place value, the place and the period of the digit 4 in these numbers. (a) 34,78,987 (b) 23,84,097 (c) 42,76,901 (d) 3,45,763 (e) 7,58,462 (f) 9,13,624 Answer: (a) Place value: 4,00,000; Place: Lakhs; Period: lakhs (b) Place value: 4000; Place: Thousands; Period: Thousands (c) Place value: 40,00,000; Place: Lakhs; Period: Lakhs (d) Place value: 40,000; Place: Thousands; Period: Thousands (e) Place value: 400; Place: Hundreds; Period: Ones (f) Place value: 4; Place: Ones; Period: Ones Question: (a) Which are the greatest and smallest numbers you can from using the digits 2, 6, 7, 9, o and 5? Answer: Greatest number:976520; Smallest numbers: 205679 (b) Which is the smallest 6-digit number? Answer: 1,00,000 (c) Which is the biggest 7-digit number? Answer: 99,99,999 (d) 10 ten thousands is the same as which number? Answer: 1 lakh (e) What number has 3 in the hundreds place, 5 in the ten thousands place, 9 in the ten lakhs place, 2 in the tens place, 1 in the thousands place and 0 in all the other places? Answer: 90,51,320 CRORES Let us now learn about 8- and -9 digit numbers in the Indian place value system. 8-Digit Numbers We now move one place further to the left of the place value chart. This is the crores place. 6 in the crores place is ten times 60 lakh (10×60,00,000) 6 in the crores place in 6,00,00,000 6 in the face value and 6,00,00,000 is the place value. An 8-digit number begins with the crore place. the crore place is in the crores period. 10 ten lakhs = 1 crore We put a comma or leave space to separate the crores period from the lakhs period. Standard from: 9,27,68,435 (with commas) or 9 27 68 435 (with spaces) Word form: Nine crore, twenty-seven lakh, sixty-eight thousad, four hundred thirty five. 9-Digit Numbers A 9-digit number begins with ten crores place. The ten crores place is in the crores period. Standard form: 19,27,68,435 Word form: Nineteen crore, twenty-seven lakh, sixty-eight thousand, fore hundred thirty-five Long expanded form: (1×10,00,00,000)+(9×1,00,00,000)+(2×10,00,000)+(7×1,00,000)+(6×1o,oooo)+(8×1,000)+(4×100)+(3×10)+(5×1) Short expanded form: 10,00,00,000+9,00,00,000+20,00,000+7,00,000+60,000+8,000+ 400+30+5 Exercise Question: Give the word form and the short expanded form for these numbers. (a) 6,67,09,654 Answer: Word form: Six crore, sixty-seven lakh nine thousand six hundred fifty-four Short form: 6,00,00,000+60,00,000+7,00,000+9000+600+50+4 (b) 40, 79,00,690 Answer: Word form: Forty crore seventy-nine lakh six hundred ninety Short form: 40,00,00,000+70,00,000+9,00,000+600+90 (c) 54,00,10.809 Answer: Word form: Fifty-four crore ten thousand eight hundred nine Short form: 50,00,00,00+4,00,00,000+10,000+800+9 (d) 7,13,84,690 Answer: Word form: Seven crore thirteen lakh eighty-four thousand six hundred ninty Short form: 7,00,00,000+10,00,000+3,00,000+80,000+4,000+600+90 (e) 45,68,23,008 Answer: Word form: F0rty-five crore sixty-eight lakh twenty-three thousand eight Short form: 40,00,00,000+5,00,00,000+60,00,000+8,00,000+20,000+3,000+8 (f) 3,00,00,091 Answer: Word form: Three crore ninety-one Short form: 3,00,00,000+90+1 (g) 28,00,09,463 Answer: Word form: Twenty-eight crore nine thousand four hundred sixty-three Short form: 20,00,00,000+8,00,00,000+9,000+400+60+3 (h) 96,70,38,420 Answer: Word form: Ninety-six crore seventy lakh thirty-eight thousand four hundred twenty Short form: 90,00,00,000+6,00,00,000+70,00,000+30,000+8,000+400+20 Question: Write the numeral with commas. (a) Sixteen crore eight lakh thirty-nine thousand twenty three Answer: 16,80,39,023 (b) Forty crore twenty lakh nine hundred five Answer: 40,20,00,905 (c) Ninety-six crore thirty-five thousand eight hundred fifty-seven Answer: 96,00,35,857 (d) Eighty-three lakh nine hundred Answer: 83,00,900 (e) Four crore thirty-seven lakh nineteen thousand Answer: 40,37,90,000 (f) One hundred thirty-four crore Answer: 134,00,00,000 (g) Two hundred nineteen crore thirty-six lakh forty-eight thousand six hundred nine Answer: 219,36,48,609 (h) Two thousand seven hundred crore fifteen lakh seventy-four thousand twelve Answer: 2700,15,74,012 Question: Write the place value and period of the coloured digit. (a) 56,89,00,345 Answer: Place value: 6,00,00,000; period: Crore (b) 47,80,34,112 Answer: Place value: 7,00,00,000; period: Crore (c) 3,87,93,389 Answer: Place value: 90,000; period: Thousand (d) 5,67,23,592 Answer: Place value: 60,00,000; period: Lakh (e) 87,08,09,004 Answer: Place value: 8,00,000; period: Lakh (f) 12,98,76,239 Answer: Place value: 70,000; period: Thousand Question: Find the difference in the place values of (a) the two 8s in 50,08,18,456 Answer: 7,92,000 (b) the two 9s in 92,79,56,387 Answer: 89,91,00,000 (c) the two 7s in 17,76,58,909 Answer: 6,30,00,000 (d) the two 5s in 75,50,632 Answer: 4,50,000 (e) the two 6s in 5,34,67,562 Answer: 59,940 (f) the two 4s in 4,17,94,726 Answer: 3,99,96,000 Question: Compare using or =. (a) 35,87,90,456 45,78,23,567 Answer: (e) 453,45,11,342 45,34,11,342 Answer: > (d) 1200,67,87,432 1200,67,87,431 Answer: > Question: How much is (a) 1 lakh less than 23,65,876? Answer: 22,65,876 (b) 10 lakh more than 9,34,765? Answer: 19,34,765 (c) 1 crore more than 29,35,87,001? Answer: 30,35,87,001 (d) 10 crore less than 15,00,60,050? Answer: 5,00,60,050 (e) 1 crore less than 20,45,90,394? Answer: 19,45,90,394 (f) 10 lakh less than 19,43,864? Answer: 9,43,864 Question: Complete the pattern. (a) 36,54,23,789 37,54,23,789 38,54,23,789 _______, _______ Answer: 39,54,23,789 40,54,23,789 (b) 20,45,60,705 30,45,60,705 40,45,60,705 ________, ________ Answer: 50,45,60,705 60,45,60,705 (c) 1,14,35,009 1,15,35,009 1,16,35,009 ________, ________ Answer: 1,17,35,009 1,18,35,009 INTERNATIONAL PLACE VALUE SYSTEM The population of the United States of America is more than 300000000. That is a 9-digit number which we will read as 30 crore (30,00,00,000) in the Indian system of numeration and as 300 million (300,000,000) in the international system of numeration. The international system has three places for every period. Word from Three million, six hundred eighty-one thousand, four hundred ninety-five Twenty-three million, six hundred eighty-one thousand, four hundred ninety-five One hundred twenty-three million, six hundred eighty-one thousand, four hundred ninety-five Billion Numbers that go beyond nine digits can be expressed by extending the international place value system to the left of the hundred millions place. That goes to a new period called the billions period. A billion is a very big number and difficult to imagine. If you counted numbers one, two, three, up to one billion and used one second for every number you counted (in other words, when you reached the number sixty, one minute would have passed), then it would take you 31 years to reach the number one billion! One billion seconds takes about 31 years. One billion minutes takes about 1,903 years. One billion hours takes about 114,077 years. {1000 million= 1 billion} One billion days takes about 2,737,850 year. The billions period has similar places as the other periods in the international place value system. Comparing the Indian and the International Systems Exercise Question: Write the word from of the numbers given below. (a) 235,641,739 Answer: Two hundred thirty-five million, six hundred forty-one thousand, seven hundred thirty-nine (b) 9,302,056 Answer: Nine million, three hundred two thousand, fifty-six (c) 50,050,172 Answer: Fifty million, fifty thousand, one hundred seventy-two (d) 305,098,721 Answer: Three hundred five million, ninety-eight thousand, seven hundred twenty-one (e) 400,500,878 Answer: Four hundred million, five hundred thousand, eight hundred seventy-eight (f) 93,892,690 Answer: Ninety-three million, eight hundred ninety-two thousand, six ninty Question: Write the standard international from with commas for the numbers given below. (a) Forty-eight million three hundred thirty -nine thousand thirty-nine thousand fifty-six Answer: 48,339,056 (b) Seven million five hundred ninety-three thousand two hundred Answer: 7,593,200 (c) Six hundred forty-two thousand seven hundred nine Answer: 642,709 (d) Four hundred forty-seven million three thousand nine hundred eight Answer: 447,003,908 (e) One hundred one million six hundred thousand three hundred seventy-two Answer: 101,600,372 (f) One billion ninety-six million three hundred sixty thousand five Answer: 1,096,360,005 Question: Insert commas using the international system and then write in words. (a) 4600987 Answer: 4,600,987 (Four million six hundred thousand nine hundred eighty-seven) (b) 2307902 Answer: 2,307,902 (Two million three hundred seven thousand, nine hundred two) (c) 1188440,098 Answer: 1,188,440,098 (One billion, one hundred eighty-eight million four hundred forty thousand ninety eight) (d) 134571608 Answer: 134,571,608 (One hundred thirty-four million five hundred seventy-one thousand six hundred eight (e) 1000613563 Answer: 1,000,613,563 (One billion six hundred thirteen thousand five hundred sixty-three (f) 176500010 Answer: 176,500,010 (One hundred seventy-six million five hundred thousand ten ) Question: Rewrite the following numbers giving them in both the standard Indian form and the standard international form. Then give the word forms in both the systems. (a) 56876234 Answer: 5,68,76,234 and 56,876,234 Five crore sixty-eight lakh seventy-six thousand, two hundred thirty-four Fifty-six million, eight hundred seventy-six thousand, two hundred thirty-four (b) 987478 Answer: 9,87,478 and 987,478 Nine lakh eighty-seven thousand four hundred seventy-eight Nine hundred eighty-seven thousand four hundred seventy-eight (c) 8909876 Answer: 89,09,876 and 8,909,876 Eighty-nine lakh nine thousand eight hundred seventy-six Eight million nine hundred nine thousand eight hundred seventy-six (d) 760382869 Answer: 76,03,82,869 and 760,382,869 Seventy-six crore three lakh eighty-two thousand eighty hundred sixty-nine Seven hundred sixty million three hundred eighty-two thousand eighty hundred sixty-nine (e) 4109632791 Answer: 410,96,32,791 and 4,109,632,791 Four hundred ten crore ninety-six lakh thirty-two thousand seven hundred ninety-one Four billion, one hundred nine million, six hundred thirty-two thousand seven hundred ninety-one (f) 108409630 Answer: 10,84,09,630 and 108,409,630 Ten crore eighty-four lakh nine thousand six hundred thirty One hundred eight million four hundred nine thousand six hundred thirty Question: (a) How would you express 10 million in the Indian system? Answer: One crore (b) Give the Indian word form of two hundred fifty million. Answer: Twenty-five crore (c) One hundred thousand is the same as which number in the Indian system. Answer: One lakh (d) Give the international word form for three crore thirty lakh forty thousand. Answer: Thirty-three million forty thousand (e) Give the international word form for forty-six lakh three thousand nine. Answer: Four million six hundred three thousand nine. ROMAN NUMBERS Ancient Roman wrote numerals that did not use place value. They had seven basic symbols represented by three letters. They formed other numbers by combining these letters following certain rules. {Putting a symbol after a bigger one means you add it.} 75 = LXXV 50+10+10+5 607 = DCVII 500+100+5+1+1 1560 = MDLX 1000+500+50+10 Examples Find the Hindu-Arabic number for (a) MDCCCLXI M DCCC LX I 1000 + 800 + 60 + 1 = 1861 (b) CDLXXV CD LXX V 400 + 70 + 5 = 475 Question: Write the Hindu-Arabic numerals. (a) XL Answer: 40 (b) LX Answer: 60 (c) XC Answer: 90 (d) CLXV Answer: 165 (e) CCL Answer: 250 (f) DC Answer: 600 (g) CD Answer: 400 (h) DCL Answer: 650 (i) CDLXXV Answer: 475 (j)DCXLI Answer: 641 (k) DCXIV Answer: 614 (l) MCD Answer: 14oo (m) CDXX Answer: 420 (n) MDCL Answer: 1650 (o) MCC Answer:12oo (p) MCMLXXX Answer:1980 (q) CCCXI Answer:311 (r) MMCCXXVI Answer: 2226 (s)DCXLV Answer: 645 (t)MDCXLII Answer: 1642 2. Write the Roman numerals. (a) 45 Answer: XLV (b) 58 Answer: LVIII (c) 73 Answer: LXXIII (d) 91 Answer: XCI (e) 140 Answer: CXL (f) 282 Answer: CCLXXXII (g) 356 Answer: CCCLVI (h) 658 Answer: DCLVIII (i) 468 Answer: CDLXVIII (j)591 Answer: DXCI (k) 802 Answer: DCCCII (l) 927 Answer: CMXXVII (m) 1400 Answer: MCD (n) 1560 Answer: MDLX (o) 2500 Answer: MMD (p) 3006 Answer: MMMVI (q) 1981 Answer: MCMLXXXI (r) 2573 Answer: MMDLXXIII (s) 1090 Answer: MXC (t) 2914 Answer: MMCMXIV Question: Change the dates of these famous inventions to Hindu-Arabic numerals. (a) MC Answer: 1100 (b) MDIX Answer: 1509 (c) MDCCXLIV Answer: 2744 (d) MDCCCLXXXV Answer: 1885 Examples Round 978 to the nearest 100. Step 1. Decide between which two multiples of 100 the number lies. 978 lies between 900 and 1000. Step 2. Look at the digit to the right of the place you are rounding to. 978 You are rounding to the hundreds place. So look at the tens place. Step 3. Since 7 > 5, round up. 978 is rounded up 1000. Exercise Question: Round to the nearest 10. (a) 23 Answer: 20 (b) 85 Answer: 90 (c) 53 Answer: 50 (d) 48 Answer: 50 (e) 21 Answer: 20 (f) 51 Answer: 50 (g) 96 Answer: 100 (h) 37 Answer: 40 (i) 74 Answer: 70 (j) 19 Answer: 20 (k) 62 Answer: 60 (l) 34 Answer: 30 Question: Round these numbers to the nearest 100. (a) 139 Answer: 100 (b) 902 Answer: 900 (c) 362 Answer: 400 (d) 737 Answer: 700 (e) 591 Answer: 600 (f) 72 Answer: 100 (g) 449 Answer: 400 (h) 255 Answer: 300 (i) 846 Answer: 800 (j) 488 Answer: 500 (k) 612 Answer: 600 (l) 958 Answer: 1000 Question: Round these numbers to the nearest 1000. (a) 1178 Answer: 1000 (b) 8135 Answer: 8000 (c) 1433 Answer: 1000 (d) 5561 Answer: 6000 (e) 5770 Answer: 6000 (f) 688 Answer: 1000 (g) 9290 Answer: 9000 (h) 2060 Answer: 2000 (i) 3244 Answer: 3000 (j) 4452 Answer: 4000 (k) 9669 Answer: 10,000 (l) 7903 Answer: 8000 Question: Round to the nearest 10,000. (a) 11198 Answer: 10,000 (b) 53235 Answer: 50,000 (c) 32431 Answer: 30,000 (d) 17575 Answer: 20,000 (e) 15285 Answer: 20,000 (f) 63045 Answer: 60,000 (g) 90897 Answer: 10,000 (h) 22819 Answer: 20,000 (i) 95418 Answer: 10,000 (j) 46600 Answer: 50,000 (k) 89961 Answer: 90,000 (l) 78123 Answer: 80,000 Question: An abandoned baby elephant found by forest officials was supposed to be around 20 months old. What could be the maximum and minimum age of the elephant baby in month? Answer: Either 15 months or 24 months Question: Rewrite these sentences by rounding to the number given in brackets. (a) There are 3,15,36,000 seconds in a year (nearest crore). Answer: There are about 3,00,00,000 seconds in year. (b) There are 5,25,600 minutes in a year (nearest lakh). Answer: There are about 5,00,000 minutes in year. (c) As per the 2011 census, Nepal had a population of 2,66,20,809 (nearest crore). Answer: As per the 2011 census, Nepal had a population of 3,00,00,000. (d) The state of Uttarakhand has 69,97,433 literature people as per the 2011 census (nearest ten lakh). Answer: The state of Uttarakhand has 70,00,000 literate people as per the 2011 census.

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NCERT 5th Class (CBSE) Mathematics: Large Numbers

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