Knowing Our Numbers. Introduction. Natural Numbers. Whole Numbers. Digits. Numeral. Numeration. Number Patterns

Transcription

1 1 Knowing Our Numbers Introduction By now we have learnt how to deal with small as well as large numbers and Indian and International place value system of numbers. We have also done addition, subtraction, multiplication and division of numbers in our previous classes. We have also seen patterns in numbers and some sequences, etc. In this lesson, we will move further and learn how to handle large numbers. We will also learn about the conversion of length from one unit to another and relation between different units, comparison of numbers, estimation of numbers and Roman numerals. Natural Numbers The counting numbers 1, 2, 3, 4, are called natural numbers. Whole Numbers The natural numbers along with 0 are called whole numbers. Digits In Hindu-Arabic system of writing numbers, the ten symbols namely 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 are called digits or figures. Numeral A single digit or group of digits denoting a number is called a numeral. For example, 8; 3451 and are numerals. The process of writing a number in digits or figures is called notation. Numeration Writing a number in words is called numeration. Number Patterns Numbers can have some interesting patterns. Complete the table given alongside to find out for yourself. Indian Place Value Chart = = =... 9, =... 99, =... CRORES LAKHS THOUSANDS ONES Ten Crore Crore Ten Lakh Lakh Ten Thousand Thousand Hundred Ten One 10,00,00,000 1,00,00,000 10,00,000 1,00,000 10,000 1,

2 Example 1: Some numbers are written in the Indian place value chart. Read the table carefully and answer the following questions. CRORES LAKHS THOUSANDS ONES NUMBER Ten Crore Crore Ten Lakh Lakh Ten Thousand Thousand Hundred Ten One EXPANDED FORM 4,75, ,47, ,76,45, ,00, , , ,00, ,00, , , ,00,00, ,00, ,00, , , (a) Which is the smallest number? (b) Which is the greatest number? (c) Arrange these numbers in ascending and descending orders. (d) Write the number names of the above numbers. Solution: (a) 4,75,320 (b) 9,76,45,310 (c) Ascending order: 4,75,320 < 98,47,215 < 9,76,45,310 Descending order: 9,76,45,310 > 98,47,215 > 4,75,320 (d) Number Number name Remember 1 hundred = 10 tens 1 thousand = 10 hundreds 1 lakh = 100 thousands 1 crore = 100 lakhs 4,75,320 Four lakh seventy-five thousand three hundred twenty 98,47,215 Ninety-eight lakh forty-seven thousand two hundred fifteen 9,76,45,310 Nine crore seventy-six lakh forty-five thousand three hundred ten Face Value The face value of a digit is the number or a symbol it represents irrespective of its position in the numeral. For example, the face value of 5 in 3,76,451 as well as in 57,869 is 5. Place Value The place value of a digit in a number is its face value multiplied by its position value in the place value chart. The place value of a digit depends on its position in the number. For example, the place value of 5 in 6,285 is 5 1 = 5 and 5,41,008 is 5 1,00,000 = 5,00,000. 2

3 The face value as well as place value of the digit 0 is always 0, irrespective of its position in the number. International Place Value Chart BILLIONS MILLIONS THOUSANDS ONES Let s learn to write numbers according to international place value chart and their corresponding number names. Writing Numbers in International Place Value Chart BILLIONS MILLIONS THOUSANDS ONES Number Ten Billion Billion Hundred Million Ten Million 100,000,000,000 Hundred Billion 10,000,000,000 Ten Billion 1,000,000,000 Billion 100,000,000 Hundred Million 10,000,000 Ten Million 1,000,000 Million 100,000 Hundred Thousand 10,000 Ten Thousand 1,000 Thousand 100 Hundred 10 Ten 1 One Million Hundred Thousand Ten Thousand Thousand Hundred Ten One 34,561,675, ,738,190, ,494,383, Number Names of the Given Numbers Number Number name 34,561,675,891 Thirty-four billion five hundred sixty-one million six hundred seventyfive thousand eight hundred ninety-one 46,738,190,325 Forty-six billion seven hundred thirty-eight million one hundred ninety thousand three hundred twenty-five 6,494,383,501 Six billion four hundred ninety-four million three hundred eighty-three thousand five hundred one 3

4 Use of Commas in Writing Large Numbers In Indian system of numeration, commas are used to mark thousands, lakhs and crores. The first comma is used after hundreds place, i.e., three places from the right, which marks thousands. The second comma is used two digits later, i.e., five digits from the right, which marks lakhs and the third comma is used further two digits later, i.e., seven digits from the right, which marks crores. In International system of numeration, commas are used to mark thousands, millions and billions. The first comma marks thousands, second marks millions and third marks billions, etc. It means the first comma is used three places from the right, the second comma is used three digits later, i.e., six digits from the right and finally the third comma is used further three digits later, i.e., nine digits from the right. While writing the number names, we do not use commas. Example 2: Fill in the blanks: (a) 1 lakh = thousand (b) 1 lakh = ten thousand (c) 1 million = lakh (d) 1 million = hundred thousand (e) 1 crore = lakh (f) 1 crore = million Solution: (a) 100 (b) 10 (c) 10 (d) 10 (e) 100 (f) 10 Example 3: Write the numerals for the following with commas at correct places. (a) Seventy-two lakh thirty-five thousand four hundred eight (b) Eight crore ninety-four lakh five hundred two (c) Seventeen million three thousand four (d) Twenty-nine billion one hundred seventy-one million four hundred two thousand three hundred seventy-two Solution: (a) 72,35,408 (b) 8,94,00,502 (c) 17,003,004 (d) 29,171,402,372 Example 4: Insert commas suitably and write the number names according to Indian system of numeration: (a) (b) (c) Solution: Number 4 Number with commas Number name (a) ,75,95,762 Eight crore seventy-five lakh ninety-five thousand seven hundred sixty-two (b) ,13,121 Seventy lakh thirteen thousand one hundred twenty-one (c) ,34,32,701 Six crore thirty-four lakh thirty-two thousand seven hundred one Example 5: Insert commas suitably and write the number names according to International system of numeration: (a) (b) (c)

5 Solution: Number Number with commas Number name (a) ,921,301 Seventy-eight million nine hundred twenty-one thousand three hundred one (b) ,452,283 Seven million four hundred fifty-two thousand two hundred eighty-three (c) ,049,831 One hundred forty-eight million forty-nine thousand eight hundred thirty-one Example 6: Write the following numbers in expanded form: (a) 3,45,892 (b) 6,90,39,283 Solution: (a) 3,45,892 = (3 1,00,000) + (4 10,000) + (5 1,000) + (8 100) + (9 10) + (2 1) (b) 6,90,39,283 = (6 1,00,00,000) + (9 10,00,000) + (0 1,00,000) + (3 10,000) + (9 1,000) + (2 100) + (8 10) + (3 1) Example 7: Find the difference between the place values of two 5s in 4,65,73,89,507. Solution: The place value of 5 at crores place = 5,00,00,000 The place value of 5 at hundreds place = 500 The difference between the two place values = 5,00,00, = 4,99,99,500. Hence, the required difference in place values of two 5s is 4,99,99,500. Example 8: How many six-digit numbers are there in all? Solution: To find the number of six-digit numbers, we need to know the largest and smallest six-digit numbers. Largest six-digit number = 9,99,999 Smallest six-digit number = 1,00,000 Total number of six-digit numbers = 9,99,999 1,00, = 9,00,000 Hence, there are 9,00,000 six-digit numbers in all. Comparison of Numbers We have done comparison of small numbers in our previous classes also. We were not only finding greater or smaller of the two given numbers, but also arranging the numbers in ascending and descending orders and finding the greatest or smallest amongst them. A step-wise method to compare two numbers is given below: Procedure: Step 1: First check the number of digits in the given numbers. The number having less number of digits is always smaller than the number having more number of digits. Step 2: If the number of digits are same in the given numbers, then start comparing the digits from the extreme left of the given numbers. Step 3: If the corresponding digits at the same position have unequal value, then the number having the bigger value is greater than the number having the smaller value. 5

6 Step 4: If the value of digits at the same position in two numbers is equal, then move to the next digit from the left and continue repeating steps 3 and 4 till you arrive at a set of unequal digits and hence are able to ascertain a greater or a smaller number. Example 9: Which is greater? 67,83,45,810 or 67,79,89,783 Solution: First, arrange the given numbers in a place value chart as shown below. Number Ten Crore Crore Ten Lakh Lakh Ten Thousand Thousand Hundred Ten One 67,83,45, ,79,89, Step 1: The number of digits in both the numbers are same, i.e., nine. Step 2: At ten crore and crore place, the corresponding digits are of the same value. Step 3: At the ten lakh place, the digits have different values, i.e., the first number has digit 8 while the second has digit 7. Since 8 > 7, therefore, 67,83,45,810 > 67,79,89,783 Example 10: Arrange the following numbers in ascending order: 13,45,678, 43,56,120, 2,35,67,890, 34,52,678 and 1,45,62,187. Solution: First, arrange the given numbers in a place value chart as shown below. Number Ten Crore Crore Ten Lakh Lakh Ten Thousand Thousand Hundred Ten One 13,45, ,56, ,35,67, ,52, ,45,62, Step 1: Step 2: The numbers having minimum number of digits, i.e., seven are 13,45,678; 43,56,120 and 34,52,678. Now their extreme left digits can be arranged as 1 < 3 < 4. Therefore, the three numbers in ascending order are 13,45,678 < 34,52,678 < 43,56,120. Now we have two numbers having 8 digits each, i.e., 2,35,67,890 and 1,45,62,187. Their extreme left digits are unequal and can be arranged as 1 < 2. Therefore, the numbers in ascending order are 1,45,62,187 < 2,35,67,890. 6

7 Step 3: Now every number with less digits is smaller than every number with more digits. Therefore, the given numbers in ascending order are as follows: 13,45,678 < 34,52,678 < 43,56,120 < 1,45,62,187 < 2,35,67,890 Exercise Write the numerals for each of the following with commas at correct places: (a) Ninety-five lakh three hundred twenty-two (b) Forty crore one thousand thirty (c) Two billion three hundred sixty-five thousand seven hundred twenty-eight (d) Two hundred twenty-five million seven hundred sixteen thousand four hundred two 2. Insert commas suitably and write the number names according to Indian system of numeration. (a) (b) (c) (d) Insert commas suitably and write the number names according to International system of numeration. (a) (b) (c) (d) Fill in the blanks. (a) lakh = 100 million (b) lakh = 10 crore (c) 1 billion = million 5. Write the following numbers in expanded form: (a) 2,43,45,892 (b) 6,78,92,831 (c) 45,73,910 (d) 2,34,56, Find the difference between the place values of two 7s in the number 2,85,73,89, How many seven-digit numbers are there in all? 8. Find the difference between the place value and face value of 8 in 67,85,90, Which is smaller 5,67,34,523 or 5,67,35,587? 10. Arrange the following numbers in ascending order: 34,56,678; 78,46,120; 3,49,67,880; 73,62,678 and 8,97,62, Arrange the following numbers in descending order: 34,56,658; 78,98,120; 49,67,890; 79,02,678 and 7,97,62,187. Standard Units of Measurement A. Length The standard unit of length is metre. There exist smaller as well as bigger units of length. Their relation are as follows: 1 centimetre (cm) = 10 millimetres (mm) 1 decimetre (dm) = 10 centimetres (cm) 1 metre (m) = 10 decimetres (dm) 1 decametre (dam) = 10 metres (m) 1 hectometre (hm) = 10 decametres (dam) 1 kilometre (km) = 10 hectometres (hm) Extension Some commonly used relations: 1 m = 100 cm 1 m = 1,000 mm 1 km = 1,000 m 7

8 B. Mass The standard unit of mass is kilogram (kg). There exist smaller as well as bigger units of mass. Their relation are as follows: 1 centigram (cg) = 10 milligrams (mg) Extension 1 decigram (dg) = 10 centigrams (cg) tonne kilogram gram milligram 1 gram (g) = 10 decigrams (dg) 1 decagram (dag) = 10 grams (g) 1 hectogram (hg) = 10 decagrams (dag) 1,000 1,000 1,000 1 kilogram (kg) = 10 hectograms (hg) C. Capacity Remember To convert a larger unit to smaller, we multiply by 10 p. To convert a smaller unit to larger, we divide by 10 p, where p is the number of steps from one unit to the other. The standard unit of capacity is litre (L). There exist smaller as well as bigger units of capacity. Their relation is as follows: 1 centilitre (cl) = 10 millilitres (ml) 1 decilitre (dl) = 10 centilitres (cl) 1 litre (L) = 10 decilitres (dl) 1 decalitre (dal) = 10 litres (L) 1 hectolitre (hl) = 10 decalitres (dal) 1 kilolitre (kl) = 10 hectolitres (hl) Example 11: A book exhibition was held for four days in a school. The number of tickets sold at the counter on the first, second, third and last day was 1,094; 1,812; 2,050 and 2,751 respectively. Find the total number of tickets sold on all the four days. (NCERT) Solution: Number of tickets sold on the first day = 1,094 Number of tickets sold on the second day = 1,812 1,094 1,812 Number of tickets sold on the third day = 2,050 2,050 Number of tickets sold on the final day = 2, ,751 Total number of tickets sold on all the four days 7,707 = 1, , , ,751 = 7,707 The total number of tickets sold on all the four days = 7,707 Example 12: In an election, the successful candidate registered 6,83,400 votes and his nearest rival secured 3,48,700 votes. By what margin did the successful candidate win the election? Solution: Number of votes registered by the successful candidate = 6,83,400 8 For example, to convert kilo into hecto, deca, and so on, we multiply by 10, 10 2 and so on. Similarly, to convert milli into centi, deci and so on, we divide by 10, 10 2 and so on. 10 hm mm 10 ml km cm L Number of votes registered by his nearest rival = 3,48,700 Margin of votes by which the successful candidate won = 6,83,400 3,48,700 = 3,34,700 Therefore, the margin of votes by which the successful candidate won = 3,34,700 Example 13: Royal bookstore sold books worth ` 3,85,891 in the first week of August and books worth ` 4,00,768 in the second week of the month. How much was the sale for the two weeks together? In which week was the sale greater and by how much? ,83,400 3,48,700 3,34,700

9 Solution: Sale during the first week of August = ` 3,85,891 Sale during the second week of August = ` 4,00,768 Total sale for the two weeks together = ` 3,85,891 + ` 4,00,768 = ` 7,86,659 Therefore, the sale for two weeks together = ` 7,86,659 The extreme left digit of the number representing sales of the second week is greater than that of the first week. Therefore, the sale in the second week is greater. In the second week, the sale is greater by = ` 4,00,768 ` 3,85,891 = ` 14,877 Example 14: Find the difference between the greatest and the least number that can be formed using the digits 8, 2, 7, 4, 3 with each digit coming only once. Solution: The greatest number which can be formed using the digits 8, 2, 7, 4 and 3 with each occurring only once is 87,432. The smallest number which can be formed using the digits 8, 2, 7, 4 and 3 with each occurring only once is 23,478. The difference between the greatest and smallest numbers so formed = 87,432 23,478 = 63,954 Therefore, the difference between the greatest and smallest numbers using the digits 8, 2, 7, 4, 3 = 63,954 Example 15: A machine, on an average, manufactures 3,215 screws a day. How many screws did it produce in the month of January 2009? 3,215 Solution: Average number of screws manufactured in a day = 3, Number of days in the month of January = 31 3,215 Number of screws produced in the month of January ,645X = 3, ,665 = 99,665 Therefore, the number of screws produced in the month of January 2009 = 99,665 Example 16: A merchant had ` 88,592 with her. She placed an order for purchasing 40 video games at ` 1,200 each. How much money will remain with her after the purchase? Solution: Cost of each video game = ` 1,200 Cost of 40 video games = ` 1, = ` 48,000 The merchant had ` 88,592 and she placed an order for ` 48,000. Money which will be left with her after the purchase of 40 video games = ` 88,592 ` 48,000 = ` 40,592 Therefore, the money left with her after the purchase of 40 video games = ` 40,592 ` 3,85,891 + ` 4,00,768 ` 7,86,659 ` 4,00,768 ` 3,85,891 ` 14,877 87,432 23,478 63,954 1, ,800X 48,000 88,592 48,000 40,592 9

10 Example 17: To stitch a shirt, 2 m 15 cm cloth is needed. Out of 40 m cloth, how many shirts can be stitched and how much cloth will remain? Solution: Cloth required to stitch a shirt = 2 m 15 cm = 215 cm (Q m 4,000 = 10018cm) 215 Total cloth available for stitching shirts = 40 m = 4,000 cm 1,850 Number of shirts that can be stitched 1,720 = 4,000 cm 215 cm 130 Since the quotient is 18 and remainder is 130 therefore, 18 shirts can be stitched and 130 cm or 1 m 30 cm cloth will remain. Example 18: A medicine is packed in boxes, each weighing 5 kg 300 g. How many such boxes can be loaded in a van, which cannot carry beyond 800 kg? Solution: Weight of each box of medicine = 5 kg 300 g = 5,300 g (Q 1 kg = 1,000 g) Maximum load which the van can carry = 800 kg = 8,00,000 g Number of boxes which can be loaded in the van = 8,00,000 5,300 = 8, Since the quotient is 150 and remainder is 50 therefore, not more than 150 medicine boxes can be loaded in the van. Example 19: The distance between the school and the house of a student is 1 km 875 m. Everyday she walks both ways. Find the total distance covered by her in six days. Solution: Distance between the school and the house = 1 km 875 m = 1,875 m (Q 1 km = 1,000 m) Distance covered by the student everyday = 1,875 2 = 3,750 m Total distance covered in 6 days = 3,750 6 = 22,500 m Therefore, the total distance covered in 6 days = 22 km 500 m Example 20: A large tanker of capacity 15 kl 500 L is full of petrol. How many petrol pumps can be supplied each with a capacity of 720 L of petrol? Solution: Total petrol in the tanker = 15 kl 500 L = 15,500 L (Q 1 kl = 1,000 L) Capacity of each petrol pump = 720 L Number of petrol pumps that can be supplied with petrol = 15, = 1, Since the quotient is 21 and remainder is 38 therefore, 21 petrol pumps can be supplied with petrol. 53 8, , , ,

11 Exercise Add 3,45,634; 2,37,854 and 1,23,00, A businessman earned a profit of ` 2,34,561 in the month of January, ` 1,95,467 in the month of February and ` 3,44,560 in the month of March. What is the total profit earned in these months? 3. A multinational company imported 78,000 cars in the year ; 81,347 cars in the year and 90,189 cars in the year How many cars in all were imported during the period of three years? 4. A man bought a piece of land for ` 18,70,900. He paid ` 13,45,670 for the construction material and ` 2,25,000 to the labour who constructed the building. How much money did he spend in all? 5. Fill in the blanks with suitable digits at each place. (a) 100 million = crore (b) 1 billion = crore 6. The population of a city is 30,54,678. If the number of males in the city is 17,11,987, find the number of females. 7. There was a stock of 2,75,67,890 sacks of wheat in a godown of Food Corporation of India (FCI). During drought and flood situations in Orissa and Assam respectively FCI sent 87,89,045 and 96,73,500 sacks of wheat to these states. What is the remaining stock with FCI? 8. The difference between two numbers is 8,67,593. If the smaller number is 6,34,289, find the greater number. 9. A container contains 15 L of oil. 345 ml of oil drained out of the container due to leakage. How much oil is left in the container? 10. The cost of a steel almirah is ` 5,495. What is the cost of 345 such almirahs? 11. The cost of a flat is ` 21,35,690 in a colony. A company decided to purchase 18 such flats for its executives in the colony. What will be the total expenditure of the company? 12. A Maruti showroom sold 456 Zen cars in the month of March at ` 3,13,495 each. What is the total sale of the showroom with respect to Zen cars? 13. A student multiplied 8,355 by 89 instead of multiplying by 98. By how much was his answer less than the actual answer? 14. A steel wire of 100 m is divided into 8 equal pieces. What is the length of each piece? 15. A vessel has 8 L 280 ml of curd. How many bowls each of 1 L 35 ml capacity can be filled? 16. A tank holds 1,000 L of water. If the capacity of a bucket is 12 L 600 ml, find the maximum number of buckets that can be filled by the water tank. 17. To stitch a trouser, 1 m 30 cm cloth is needed. Out of 15 m cloth, how many trousers can be stitched and how much cloth will remain? Estimation You might have heard statements like: Nine hundred people enjoyed Shahrukh Khan s movie in a theatre. Thirty thousand people witnessed Sachin Tendulkar s century in a cricket match between India and Pakistan. 11

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13 Example 22: Using the above mentioned procedure, estimate the following numbers to their nearest hundreds. (a) 374 (b) 143 (c) 3,421 (d) 46,785 Solution: (a) In the number 374, the digit at tens place is 7 which is greater than 4. As per the procedure, we need to replace digits at tens and ones place by 0 and add 1 to hundreds place. Therefore, estimate of the given number to its nearest hundreds is 400. (b) In the number 143, the digit at tens place is 4 which is less than 5. As per the procedure, we need to replace digits at tens and ones place by 0. Therefore, estimate of the given number to its nearest hundreds is 100. (c) In the number 3,421, the digit at tens place is 2 which is less than 5. As per the procedure, we need to replace digits at tens and ones place by 0. Therefore, estimate of the given number to its nearest hundreds is 3,400. (d) In the number 46,785, the digit at tens place is 8 which is greater than 4. As per the procedure, we need to replace digits at tens and ones place by 0 and add 1 to the digit at hundreds place. Therefore, estimate of the given number to its nearest hundreds is 46,800. Estimation or Rounding Off the Numbers to the Nearest Thousands Procedure: Step 1: In the given number, examine the digit at hundreds place. Step 2: If the digit at hundreds place is 0, 1, 2, 3, or 4, replace the digits at hundreds, tens and ones place by 0 each and rest of the digits remain unchanged. Step 3: If the digit at hundreds place is 5, 6, 7, 8 or 9, replace the digits at hundreds, tens and ones place by 0 and add 1 to the digit at thousands place. Example 23: Using the above mentioned procedure, estimate the following numbers to their nearest thousands. (a) 1,374 (b) 3,621 (c) 46,785 Solution: (a) In the number 1,374, the digit at hundreds place is 3 which is less than 5. As per the procedure, we need to replace digits at hundreds, tens and ones place by 0. Therefore, estimate of the given number to its nearest thousands is 1,000. (b) In the number 3,621, the digit at hundreds place is 6 which is greater than 4. As per the procedure, we need to replace digits at hundreds, tens and ones place by 0 and add 1 to the digit at thousands place. Therefore, estimate of the given number to its nearest thousands is 4,000. (c) In the number 46,785, the digit at hundreds place is 7 which is greater than 4. As per the procedure, we need to replace the digits at hundreds, tens and ones place by 0 and add 1 to the digit at thousands place. Therefore, estimate of the given number to its nearest thousands is 47,000. Representation of Whole Numbers on a Number Line To represent whole numbers on a number line, follow these steps. Draw a straight line. 13

14 Mark a point O on it. Label the point O as 0. Now mark points on the right hand side of 0 at equal distances as 1, 2, 3, 4, 5,... and so on. O The line shown above is a number line representing whole numbers. We observe that the number on the left hand side is smaller than the number on the right hand side. For example, since 2 is on the left of 5 on the number line therefore 2 < 5. Rounding Off the Numbers Using Number Line Let s learn the concept of rounding off the numbers with the help of number line. (i) Estimate 43 to its nearest tens. Step 1: Let s draw a number line including numbers from 40 to Step 2: Now, observe whether 43 is nearer to 40 or 50. Step 3: Clearly, 43 is nearer to 40. Therefore, estimated value of 43 to its nearest tens is 40. (ii) Estimate 478 to its nearest hundreds. Step 1: Let s draw a number line including numbers from 400 to Step 2: Now, observe whether 478 is nearer to 400 or 500. Step 3: Clearly, 478 is nearer to 500. Therefore, estimated value of 478 to its nearest hundreds is 500. (iii) Estimate 2,500 to its nearest thousands. Step 1: Let s draw a number line including numbers from 2,000 to 3, ,000 2,000 2,500 3,000 4,000 5,000 6,000 Step 2: Now, observe whether 2,500 is nearer to 2,000 or 3,000. Step 3: Clearly, 2,500 is neither nearer to 2,000 nor to 3,000. Actually, it is in the middle of both. In such a situation, as a rule we do not take the smaller estimated number. Therefore, estimated value of 2,500 to its nearest thousands is 3,000. Estimation in Sum or Difference There is no rigid rule to estimate a number. We can round off a number to any place (tens, hundreds or thousands) depending upon the degree of accuracy required. The most important aspect of estimation or rounding off is that the estimated value should make sense.

15 Example 24: Estimate each of the following using the general rule: (a) (NCERT) (b) (c) 12, ,888 (d) 28,292 21,496 (NCERT) Solution: (a) Let s round off to the nearest hundreds. 730 is rounded off to is rounded off to 1,000. Therefore, estimated sum of 730 and 998 = ,000 = 1,700 (b) Let s round off to the nearest hundreds. 891 is rounded off to is rounded off to 200. Therefore, estimated difference of 891 and 244 = = 700 (c) 12, ,888 Let s round off to the nearest thousands. 12,780 is rounded off to 13,000. 2,888 is rounded off to 3,000. Therefore, estimated sum of 12,780 and 2,888 = 13, ,000 = 16,000 (d) 28,292 21,496 Let s round off to the nearest thousands. 28,292 is rounded off to 28, ,496 is rounded off to 21,000. Therefore, estimated difference of 28,292 and 21,496 = 28,000 21,000 = 7,000 Example 25: Give a rough estimate by rounding off to the nearest hundreds and also a closer estimate by rounding off to the nearest tens: (a) ,317 (b) 3,25,215 25,370 (c) 4,89,348 48,365 (NCERT) Solution: (a) ,317 On rounding off to the nearest hundreds: 439 is rounded off to is rounded off to ,317 is rounded off to 4,300. Therefore, estimated sum of ,317 = ,300 = 5, ,000 1, , ,000 16,000 28,000 21,000 7, ,300 5,000 15

16 On rounding off to the nearest tens: 439 is rounded off to is rounded off to ,317 is rounded off to 4,320. Therefore, estimated sum of ,317 = ,320 = 5,090 (b) 3,25,215 25,370 On rounding off to the nearest hundreds: 3,25,215 is rounded off to 3,25, ,370 on rounded off to 25,400. Therefore, estimated difference of 3,25,215 25,370 = 3,25,200 25,400 = 2,99,800 On rounding off to the nearest tens: 3,25,215 is rounded off to 3,25, ,370 on rounding off remains the same, i.e., 25,370. Therefore, estimated difference of 3,25,215 and 25,370 = 3,25,220 25,370 = 2,99,850 (c) 4,89,348 48,365 On rounding off to the nearest hundreds: 4,89,348 is rounded off to 4,89, ,365 is rounded off to 48,400. Therefore, estimated difference of 4,89,348 and 48,365 = 4,89,300 48,400 = 4,40,900 On rounding off to the nearest tens: 4,89,348 is rounded off to 4,89, ,365 is rounded off to 48,370. Therefore, estimated difference of 4,89,348 and 48,365 = 4,89,350 48,370 = 4,40, ,320 5,090 3,25,200 25,400 2,99,800 3,25,220 25,370 2,99,850 4,89,300 48,400 4,40,900 4,89,350 48,370 4,40,980 Estimation in Product Let s estimate the product If we approximate both to their nearest tens, we get = 13,300. This is a reasonable estimate, but is not quick enough. If we approximate both to the nearest hundreds, we get = 20,000. This is quick but not a good estimate. To get a better estimate, we try rounding off 74 to the nearest tens, i.e., 70, and also 189 to the nearest hundreds, i.e., 200. We get = 14,000 which is both quick and a good estimate. The 16

17 general rule that we follow is, therefore, round off each factor to its greatest place and then multiply the rounded off factors. Example 26: Estimate the following products using general rule: (a) 9, (b) 5,281 3,849 (c) 1, (NCERT) Solution: (a) 9, Rounding off each factor to its greatest place: 9,250 rounded off to the nearest thousands is 9, rounded off to the nearest tens is 30. Therefore, estimated product of 9,250 and 29 = 9, = 2,70,000 (b) 5,281 3,849 Rounding off each factor to the nearest thousands: 5,281 is rounded off to 5,000. 3,849 is rounded off to 4,000. Therefore, estimated product of 5,281 and 3,849 = 5,000 4,000 = 2,00,00,000 (c) 1, Rounding off each factor to its greatest place: 1,291 rounded off to the nearest thousands is 1, rounded off to the nearest hundreds is 600. Therefore, estimated product of 1,291 and 592 = 1, = 6,00,000 Example 27: Estimate and compare with the actual sum: (a) (b) 10, ,368 (c) 2, ,679 Solution: (a) Let s round off to the nearest hundreds. 870 is rounded off to is rounded off to 1,000. Therefore, estimated sum = ,000 = 1,900 However, actual sum = = 1,856 On rounding off 1,856 to its nearest hundreds, it becomes 1,900. Therefore, estimation is reasonable. (b) 10, ,368 Let s round off to the nearest thousands. 10,866 is rounded off to 11,000. 9, ,70,000 5,000 4,000 2,00,00,000 1, ,00, ,000 1, ,856 17

18 2,368 is rounded off to 2,000. Therefore, estimated sum = 11, ,000 = 13,000 However, actual sum = 10, ,368 = 13,234 On rounding off 13,234 to its nearest thousands, it becomes 13,000. Therefore, estimation is reasonable. (c) 2, ,679 Let s round off to the nearest thousands. 2,345 is rounded off to 2,000. 5,679 is rounded off to 6,000. Therefore, estimated sum = 2, ,000 = 8,000 However, actual sum = 2, ,679 = 8,024 On rounding off 8,024 to its nearest thousands, it becomes 8,000. Therefore, estimation is reasonable. Example 28: Estimate and compare with the actual difference: (a) (b) 1, (c) 67,983 23,865 Solution: (a) Let s round off to the nearest hundreds. 840 is rounded off to is rounded off to 200. Therefore, estimated difference = = 600 However, actual difference = = 624 On rounding off 624 to its nearest hundreds, it becomes 600. Therefore, estimation is quite reasonable. (b) 1, Let s round off to the nearest hundreds. 1,355 is rounded off to 1, is rounded off to 600. Therefore, estimated difference = 1, = 800 However, actual difference = 1, = 789 On rounding off 789 to its nearest hundreds, it becomes 800. Therefore, estimation is quite reasonable. 11, ,000 13,000 10, ,368 13,234 2, ,000 8,000 2, ,679 8, , ,

19 (c) 67,983 23,865 Let s round off to the nearest thousands. 67,983 is rounded off to 68, ,865 is rounded off to 24,000. Therefore, estimated difference = 68,000 24,000 = 44,000 However, actual difference = 67,983 23,865 = 44,118 On rounding off 44,118 to its nearest thousands, it becomes 44,000. Therefore, estimation is quite reasonable. Example 29: Estimate each of the following products by rounding off each number to its nearest hundreds: (a) (b) Solution: (a) Rounding off each number to the nearest hundreds: 377 its rounded off to is rounded off to 400. Therefore, estimated product = = 1,60,000 (b) Rounding off each number to the nearest hundreds: 245 is rounded off to is rounded off to 600. Therefore, estimated product = = 1,20,000 Example 30: Find the estimated quotient for (a) (b) Solution: (a) Rounding off each number to the greatest place: 477 rounded off to the nearest hundreds is rounded off to the nearst tens is 20. Therefore, estimated quotient = = 25 (b) Rounding off each number to the greatest place: 862 rounded off to the nearest hundreds is rounded off to the nearst tens is 30. Therefore, estimated quotient = = 30 Exercise Estimate each of the following sum to its nearest tens: (a) (b) (c) (d) Estimate each of the following sum to its nearest hundreds: (a) (b) (c) (d) 5, ,653 Do you know? 68,000 24,000 44,000 67,983 23,865 44, inches = 1 foot (ft) 3 feet = 1 yard (yd) 1 decade = 10 years 10 decades = 1 century 19

20 3. Estimate each of the following sum to its nearest thousands: (a) 56, ,834 (b) 43, ,784 (c) 24, ,118 (d) 21, , Estimate each of the following difference to its nearest tens: (a) (b) (c) (d) Estimate each of the following difference to its nearest hundreds: (a) (b) (c) (d) Estimate each of the following difference to its nearest thousands: (a) 7,674 3,432 (b) 3,689 2,532 (c) 6,764 5,321 (d) 8,956 7, Estimate the following products by rounding off each number to its nearest tens: (a) (b) (c) (d) Estimate the following products by rounding off each number to its nearest hundreds: (a) (b) (c) (d) Estimate the following products using general rule: (a) (b) 3, (c) 2, (d) 9, Find the estimated quotient for each of the following: (a) (b) (c) (d) Brackets Vani bought 5 sketch pens from a stationery shop for ` 3 each. Vani s brother Nikhil bought 7 sketch pens for ` 3 each. Vani calculated that the total money spent by them is = = 36, i.e., ` 36. Nikhil calculated it by a different method but got the same amount, i.e., (5 + 7) 3 = 12 3 = 36, i.e., ` 36. Nikhil combined the total number of pens, i.e., and treated it as a single number by putting brackets and then multiplied it by the cost of 1 pen, i.e., ` 3. The rule is to first simplify everything in the bracket to a single number and then perform the other operations outside. Expanding Brackets We will now see how the use of brackets allows us to make calculations simple and systematic. (i) 13 1,013 = 13 (1, ) = 13 1, = 13, = 13,169 (ii) = 102 ( ) = = ( ) ( ) 5 = = 10, = 10,710 Roman Numerals We are well aware of the numerals 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. These numerals are used in Hindu- Arabic numeral system. However, one of the old systems of numeration developed by Romans is also in common use. This system is called Roman system of numeration. There are seven distinct 20

21 numeral symbols in this system. These symbols and their corresponding Hindu-Arabic numerals are given below: Roman Numerals I V X L C D M Hindu-Arabic Numerals ,000 Rules followed in the Roman System of Numeration Rule 1: If a symbol is repeated, its value is added as many times as it occurs. Examples: Ø III = = 3 Ø XX = = 20 Ø CC = = 200 Ø MMM = 1, , ,000 = 3,000 Remarks: (i) A symbol can be repeated at the most three times. (ii) Symbols V, L and D are never repeated. (iii) The only symbols which can repeat are I, X, C and M. Do you know? In Roman system of numeration, there is no symbol for zero. Rule 2: A symbol of smaller value written to the right of a symbol of greater value always gets added to the symbol of greater value. Examples: Ø VI = = 6 Ø VII = = 7 Ø XI = = 11 Ø XIII = = 13 Ø LX = = 60 Ø CXI = = 111 Rule 3: A symbol of smaller value written to the left of a symbol of greater value always gets subtracted from the symbol of greater value. Examples: Ø IV = 5 1 = 4 Ø IX = 10 1 = 9 Ø XL = = 40 Ø XC = = 90 Ø CD = = 400 Ø CM = 1, = 900 Remarks: V, L and D are never subtracted from a symbol of larger value. I can be subtracted from V and X only. X can be subtracted from L and C only. C can be subtracted from D and M only. Rule 4: When a symbol of smaller value is written between two symbols of larger values, it is always subtracted from the symbol of larger value, which comes immediately after the symbol of smaller value. Examples: Ø XIX = 10 + (10 1) = 19 Ø CXIV= (5 1) = 114 Ø CLIV = (5 1) = 154 Ø CCIX = (10 1) = 209 Rule 5: If a bar is placed over a symbol, its value gets multiplied by 1,000. Examples: Ø V = 5 1,000 = 5,000 Ø L = 50 1,000 = 50,000 21

22 22 Exercise Simplify: (a) 7 (11+ 9) (b) 42 + (73 23) (c) 248 ( ) (d) ( ) (68 58) (e) (f) Write the equivalent Roman numerals for the following Hindu-Arabic numerals: (a) 22 (b) 28 (c) 32 (d) 45 (e) 47 (f) 55 (g) 64 (h) 81 (i) 93 (j) 97 (k) 326 (l) 548 (m) 991 (n) 1,207 (o) 3, Express each of the following in Hindu-Arabic numerals: (a) XVII (b) XXIV (c) XLV (d) LI (e) XCII (f) XCV (g) CLXXXVII (h) CCCLXXXIII (i) CDXLIX (j) MCDLV (k) MDCCXXV (l) MMMDXXI SUMMARY 1. The counting numbers 1, 2, 3, 4,... are called natural numbers. 2. The natural numbers along with 0 are called whole numbers. 3. The ten symbols, i.e., 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 are called digits and a single digit or a group of digits denoting a number is called a numeral. 4. The process of writing a number in digits is called notation and the process of writing a number in words is called numeration. We can express numbers in two ways: (i) Indian system of numeration or (ii) International system of numeration. 5. The face value of a digit is the number or a symbol it represents irrespective of its position in the number. 6. The place value of a digit in a number is its face value multiplied by its position value. The place value of a digit is the value of the digit because of its position in the number. 7. In the Indian system of numeration commas are placed after 3 digits starting from the right. The other periods to the left of the ones period have two places each. 8. In the International system of numeration commas are placed after every 3 digits from the right. 9. Comparison of numbers: Given two numbers, the one having less number of digits is smaller. If the number of digits is the same then we start comparing the digits from the extreme left till we get a pair of digits which are unequal and the one having first greater leftmost digit is greater. 10. Every whole number can be marked on a number line. 11. In certain situations we do not need the exact number but only a reasonable guess or an estimate. Estimation involves approximating a quantity according to the situation and accuracy required. 12. In some situations, we have to obtain a quick estimate of the outcome of number operations. We do it by rounding off the numbers involved and then applying the operations. 13. The Roman numeral system uses seven basic symbols to represent numbers. I = 1, V = 5, X = 10, L = 50, C = 100, D = 500, M = 1,000 We also follow certain rules specific to the Roman numeral system.

23 MULTIPLE CHOICE QUESTIONS REVIEW EXERCISES For Formative and Summative Assessment 1. The sum of the smallest whole number and the smallest natural number is: (a) 0 (b) 1 (c) 2 (d) 3 2. The face value of 7 in the numeral 6,782 is: (a) 7 (b) 700 (c) 782 (d) none of these 3. The difference between the place value and face value of 3 in the numeral 6,530 is: (a) 3 (b) 27 (c) 33 (d) 0 4. The equivalent of 1 million in the Indian system of numeration is: (a) 10 thousand (b) 1 lakh (c) 10 lakh (d) 10 crore 5. The greatest whole number is: (a) 1 billion (b) 10 crore (c) 10 lakh (d) none of these 6. The total number of five-digit numbers is: (a) 1,00,000 (b) 99,999 (c) 89,999 (d) 90, The largest four-digit number having distinct digits is: (a) 9,000 (b) 9,867 (c) 9,768 (d) 9, The number 14,349 when rounded off to the nearest hundreds is: (a) 14,000 (b) 14,300 (c) 15,000 (d) 14, The equivalent of 44 in Roman numeral is: (a) XXXXIV (b) XLIIII (c) XLIV (d) IVIV 10. The smallest natural number which on rounding off to the nearest hundreds gives 400 is: (a) 399 (b) 401 (c) 449 (d) The sum of place value of all the digits of the number 6,001 is: (a) 7 (b) 61 (c) 6,001 (d) 6, While adding two four-digit numbers the closest estimate will be obtained by rounding each number to the nearest: (a) ones (b) tens (c) hundreds (d) thousands MENTAL MATHS True or False 1. The place value of a digit is independent of whether the number is written in Indian system or International system of numeration. 2. In Roman numerals, the digits do not have any place value. 3. In the Roman numeral system, the symbol VC represents the number In International system of numeration, a number having more number of digits is always greater than the number having less number of digits. 5. For the Hindu-Arabic numeral 15, the corresponding Roman number can also be written as VVV. 6. Rounding off each number to the nearest tens before performing the required operation gives more accurate estimate than rounding off each number to the nearest hundreds or thousands. 23

24 Fill in the Blanks 1. A single digit or group of digits denoting a number is called a. 2. The face value of digit 3 in the number 3,284 is. 3. The place value of digit 7 in the number 6,721 is. 4. In Roman numerals, X can be subtracted from and only. 5. The estimate of the number 764 when rounded off to nearest hundreds is. 6. A number greater than or equal to 500 and less than 1500 on rounding off to nearest thousands gives. Answer Orally 1. Which digit in the number 1,234 has the highest face value? 2. Which digit in the number 1,234 has the highest place value? 3. How many kilograms are there in 1 quintal? 4. How many centimetres are there in 1 km? 5. What is Hindu-Arabic numeral system? 6. How many distinct symbols (numerals) are there in the Roman numeral system? 7. Which symbol is used to represent 10,000 in the Roman numeral? 8. What number is obtained on rounding off 6,292 to the nearest hundreds? 9. What is a palindromic number? LET S EVALUATE 1. Write the numerals for each of the following: (a) Sixteen crore forty lakh ten thousand two hundred forty-nine (b) Seven crore two lakh eighty-seven 2. Write number names for (a) 7,23,56,708 (b) 27,57, Write each in expanded form: (a) 5,35,23,981 (b) 34,49,28, Find the difference between the place values of two 7s in 78,65,49, How many five-digit numbers are there in all? 6. Arrange the following numbers in ascending as well as descending order: 4,75,63,892; 56,45,389; 3,27,896; 5,64,585 and 45,87, The population of a city in the year 2002 was 2,45,67,890. In the year 2007, population rose to 2,45,69,923. What was the increase in population? 8. The construction cost of 18 duplex houses constructed by a builder is ` 4,56,24,564. What is the cost of one such duplex house? 9. A student multiplied 7,236 by 75 instead of multiplying by 57. By how much was his answer greater than the correct answer? 10. A vessel has 5 L, 500 ml of ice cream. How many ice cream cups, each of 50 ml capacity, can be filled? 11. Express each of the following as a Hindu-Arabic numeral: (a) XXXII (b) XCV (c) DCCLXIV (d) CCXX (e) MVI (f) LXXXIV 12. Estimate and compare with the actual sum: (a) (b) 9, , Estimate the product of by rounding off each number to its nearest hundreds. 24

25 14. Find the estimated quotient for Express the following numbers as Roman numerals: (a) 446 (b) 341 (c) 66 (d) 227 (e) 49 (f) 999 HIGHER ORDER THINKING SKILLS 1. Write the greatest four-digit number using 2 different digits. 2. The place value of which digit is always equal to its face value irrespective of its position? 3. Find the difference between the greatest and the smallest numbers each of which on rounding off gives 5, After some decades, India s population according to a survey will be 1,329,854,134. Write this number in words according to: (a) International system of numeration (b) Indian system 5. Find the difference between the largest and the smallest four-digit numbers formed by the digits 0, 2, 5 and The distance between two towns is 36 km 500 m. A bus makes 8 rounds between the two towns in a day. How much distance will it cover in a week? Group Activity Make the Greatest Number Objective: To learn about place value of digits Material required: Flash cards numbered 0 to 9 Directions: Try to build the greatest number The students of the class are divided into groups of 10 each. Each student of the group is assigned a digit from 0 to 9, whose flash card they can hold. Ten slips of paper with digits 0 to 9 written on them are folded and put in a box. The teacher draws a slip and reads the digit. The digit called once is not repeated. Students from each group holding the flash card bearing the digit called take positions of their choice marked U (unit), T (ten), H (hundred), Th (thousand), T.Th (ten thousand) on the floor designated for their group. T.Th Th H T U T.Th Th H T U T.Th Th H T U Group I Group II Group III The procedure is repeated till all the five slots are occupied by the students. The positions once taken cannot be changed or interchanged with others. Result The group with the greatest number formed wins the game and the one with the second highest number comes second. The group with next number in descending order is third and so on. The group with the smallest number comes last. Note: The game can be played for few more rounds. 25