Chapter 1: Knowing Our Numbers (NCERT Solutions)
Exercise 1.1
(a) 1 lakh = ____________ ten thousand.
(b) 1 million = ____________ hundred thousand.
(c) 1 crore = ____________ ten lakh.
(d) 1 crore = ____________ million.
(e) 1 million = ____________ lakh.
Answer:
(a) 1 lakh (1,00,000) = 10 ten thousand (10 × 10,000).
(b) 1 million (1,000,000) = 10 hundred thousand (10 × 100,000).
(c) 1 crore (1,00,00,000) = 10 ten lakh (10 × 10,00,000).
(d) 1 crore (1,00,00,000) = 10 million (10 × 1,000,000).
(e) 1 million (1,000,000) = 10 lakh (10 × 1,00,000).
(a) Seventy-three lakh seventy-five thousand three hundred seven.
(b) Nine crore five lakh forty-one.
(c) Seven crore fifty-two lakh twenty-one thousand three hundred two.
(d) Fifty-eight million four hundred twenty-three thousand two hundred two.
(e) Twenty-three lakh thirty thousand ten.
Answer:
(a) Seventy-three lakh (73,), seventy-five thousand (75,), three hundred seven (307).
Numeral: 73,75,307
(b) Nine crore (9,), five lakh (05,), forty-one (041 - waiting for thousands, so 00,041).
Numeral: 9,05,00,041
(c) Seven crore (7,), fifty-two lakh (52,), twenty-one thousand (21,), three hundred two (302).
Numeral: 7,52,21,302
(d) Fifty-eight million (58,), four hundred twenty-three thousand (423,), two hundred two (202). (International System)
Numeral: 58,423,202
(e) Twenty-three lakh (23,), thirty thousand (30,), ten (010).
Numeral: 23,30,010
(a) 87595762 (b) 8546283 (c) 99900046 (d) 98432701
Answer:
(In the Indian system, commas are placed after 3 digits from the right, and then after every 2 digits.)
(a) 87595762 → 8,75,95,762
Name: Eight crore seventy-five lakh ninety-five thousand seven hundred sixty-two.
(b) 8546283 → 85,46,283
Name: Eighty-five lakh forty-six thousand two hundred eighty-three.
(c) 99900046 → 9,99,00,046
Name: Nine crore ninety-nine lakh forty-six.
(d) 98432701 → 9,84,32,701
Name: Nine crore eighty-four lakh thirty-two thousand seven hundred one.
(a) 78921092 (b) 7452283 (c) 99985102 (d) 48049831
Answer:
(In the International system, commas are placed after every 3 digits from the right.)
(a) 78921092 → 78,921,092
Name: Seventy-eight million nine hundred twenty-one thousand ninety-two.
(b) 7452283 → 7,452,283
Name: Seven million four hundred fifty-two thousand two hundred eighty-three.
(c) 99985102 → 99,985,102
Name: Ninety-nine million nine hundred eighty-five thousand one hundred two.
(d) 48049831 → 48,049,831
Name: Forty-eight million forty-nine thousand eight hundred thirty-one.
Exercise 1.2
Tickets sold on 1st day = 1,094
Tickets sold on 2nd day = 1,812
Tickets sold on 3rd day = 2,050
Tickets sold on 4th day = 2,751
Total tickets sold = 1,094 + 1,812 + 2,050 + 2,751 = 7,707
Therefore, 7,707 tickets were sold on all four days.
Runs Shekhar wishes to complete = 10,000
Runs scored so far = 6,980
Runs needed = (Runs he wishes to complete) - (Runs scored so far)
Runs needed = 10,000 - 6,980 = 3,020
Therefore, he needs 3,020 more runs.
Votes received by the successful candidate = 5,77,500
Votes received by the nearest rival = 3,48,700
Margin of victory = (Votes of successful candidate) - (Votes of nearest rival)
Margin = 5,77,500 - 3,48,700 = 2,28,800
Therefore, the successful candidate won by a margin of 2,28,800 votes.
Books sold in 1st week = ₹ 2,85,891
Books sold in 2nd week = ₹ 4,00,768
Total sale = 2,85,891 + 4,00,768 = ₹ 6,86,659
Comparing the sales: 4,00,768 > 2,85,891, so the sale was greater in the second week.
Difference = 4,00,768 - 2,85,891 = ₹ 1,14,877
Therefore, the sale in the second week was greater by ₹ 1,14,877.
Digits available: 6, 2, 7, 4, 3.
Greatest 5-digit number (arrange digits in descending order): 76432
Least 5-digit number (arrange digits in ascending order): 23467
Difference = 76432 - 23467 = 52965
Therefore, the difference is 52,965.
Number of screws manufactured in 1 day = 2,825.
Number of days in January = 31 days.
Total screws produced in January = 2825 × 31
2825 × 31 = 87,575
Therefore, the machine produced 87,575 screws in the month of January 2006.
Cost of 1 radio set = ₹ 1,200.
Cost of 40 radio sets = 1200 × 40 = ₹ 48,000.
Total money with the merchant = ₹ 78,592.
Money remaining = (Total money) - (Cost of 40 sets)
Money remaining = 78,592 - 48,000 = ₹ 30,592.
Therefore, ₹ 30,592 will remain with her.
Difference between 65 and 56 = 65 - 56 = 9.
The difference in the answers will be 7236 × 9.
Difference = 7236 × 9 = 65,124
Therefore, his answer was greater than the correct answer by 65,124.
Total length of cloth = 40 m = 40 × 100 cm = 4,000 cm.
Cloth needed to stitch 1 shirt = 2 m 15 cm = (2 × 100) cm + 15 cm = 215 cm.
Number of shirts that can be stitched = Total cloth / Cloth for 1 shirt
= 4000 ÷ 215
4000 ÷ 215 gives quotient 18 and remainder 130.
Therefore, 18 shirts can be stitched and 130 cm (or 1 m 30 cm) cloth will remain.
Weight of one box = 4 kg 500 g = (4 × 1000) g + 500 g = 4,500 g.
Maximum load the van can carry = 800 kg = 800 × 1000 g = 8,00,000 g.
Number of boxes = Total load / Weight of one box
= 8,00,000 ÷ 4,500 = 8000 ÷ 45
8000 ÷ 45 gives quotient 177 and a remainder.
Therefore, a maximum of 177 boxes can be loaded.
Distance covered each way = 1 km 875 m = 1,875 m.
Distance covered both ways in 1 day = 1875 × 2 = 3,750 m.
Distance covered in 6 days = 3750 × 6 = 22,500 m.
22,500 m can be written as 22 km 500 m.
Therefore, she covers a total distance of 22 km 500 m in six days.
Total quantity of curd in the vessel = 4 l 500 ml = (4 × 1000) ml + 500 ml = 4,500 ml.
Capacity of one glass = 25 ml.
Number of glasses = Total quantity / Capacity of one glass
= 4500 ÷ 25
4500 ÷ 25 = 180.
Therefore, it can be filled in 180 glasses.
Chapter 1: Knowing Our Numbers (Practice Questions)
RD Sharma / Extra Practice
To form the greatest number, arrange digits in descending order: 7, 5, 4, 0.
Greatest 4-digit number = 7540
To form the smallest number, arrange digits in ascending order. However, a number cannot begin
with 0. So, we place the next smallest digit (4) first, followed by 0, then 5, then 7.
Smallest 4-digit number = 4057
Expanded form:
= 5 × 1,00,00,000 + 2 × 10,00,000 + 3 × 1,00,000 + 4 × 10,000 + 0
× 1,000 + 9 × 100 + 0 × 10 + 7 × 1
= 50000000 + 2000000 + 300000 + 40000 + 900 + 7
In the number 7,65,432, the digit 6 is in the ten thousands place.
Place value of 6 = 60,000.
Face value of 6 = 6.
Difference = 60,000 - 6 = 59,994
Successor (add 1) = 99,99,999 + 1 = 1,00,00,000 (One Crore)
Predecessor (subtract 1) = 99,99,999 - 1 = 99,99,998
Comparing the numbers starting from the leftmost digit:
Crores digit: 4 = 4.
Ten Lakhs digit: 5 = 5.
Lakhs digit: 6 < 7.
Since 7 is greater than 6, the second number is larger.
Therefore, 4,57,12,000 is greater.
Largest 7-digit number = 99,99,999
Smallest 7-digit number = 10,00,000
Total number of 7-digit numbers = (Largest 7-digit number - Smallest 7-digit number) + 1
= (99,99,999 - 10,00,000) + 1
= 89,99,999 + 1 = 90,00,000
5,290 rounds off to the nearest thousand as 5,000.
17,986 rounds off to the nearest thousand as 18,000.
Estimated sum = 5,000 + 18,000 = 23,000.
5981 rounds off to the nearest hundred as 6,000.
4428 rounds off to the nearest hundred as 4,400.
Estimated product = 6,000 × 4,400 = 2,64,00,000.
Break down the number: 73 = 70 + 3 = (50 + 20) + 3.
Roman numeral for 50 is L.
Roman numeral for 20 is XX.
Roman numeral for 3 is III.
Combining them: LXXIII
Break down the number: 99 = 90 + 9.
Roman numeral for 90 is XC (100 - 10).
Roman numeral for 9 is IX (10 - 1).
Combining them: XCIX
XL = 50 - 10 = 40.
VIII = 5 + 3 = 8.
Total = 40 + 8 = 48
CV = 100 + 5 = 105.
XC = 100 - 10 = 90.
Difference = 105 - 90 = 15
Total adults (men + women) = 45,39,812 + 44,82,731 = 90,22,543.
Total population = 1,20,50,000.
Number of children = Total population - Total adults
= 1,20,50,000 - 90,22,543 = 30,27,457
Population in 2011 = 3,45,67,890.
Increase in population = 23,45,678.
Population in 2021 = 3,45,67,890 + 23,45,678 = 3,69,13,568
Total weight in mg = 2,50,000 × 20 = 50,00,000 mg.
To convert mg to grams, divide by 1000:
50,00,000 mg = 50,00,000 / 1000 = 5,000 grams.
To convert grams to kilograms, divide by 1000:
5,000 grams = 5000 / 1000 = 5 kg.
Greatest 3-digit number = 999.
Smallest 4-digit number = 1000.
Product = 999 × 1000 = 9,99,000
To form the smallest number, begin with the smallest non-zero digit: 1.
Follow with 0, then the remaining digits in ascending order: 3, 5, 8, 9.
Smallest 6-digit number = 103589.
Inserting commas according to the International System (every 3 digits from right):
103,589.
Convert both to numerals:
5 crore = 5,00,00,000.
3 million = 3,000,000.
Difference = 5,00,00,000 - 30,00,000 = 4,70,00,000.
8325 rounds off to the nearest hundred as 8,300.
491 rounds off to the nearest hundred as 500.
Estimated difference = 8,300 - 500 = 7,800.
False. The primary rule for comparing numbers is to first check the total number
of digits. A number with more digits is always greater than a number with fewer digits,
regardless of the leftmost digit.
Example: 9,999 (starts with 9) is less than 10,000 (starts with 1).
Chapter 1: Knowing Our Numbers (Concepts & Summary)
1. Comparing Numbers
- The number with the most digits is always the greatest.
- If the number of digits is the same, compare the leftmost digits. The number with the greater leftmost digit is larger.
- If the leftmost digits are the same, compare the next digits to the right, and so on.
- Ascending Order: Arranging from the smallest to the greatest.
- Descending Order: Arranging from the greatest to the smallest.
2. Place Value Systems (Reading & Writing Numbers)
Indian System of Numeration
- Values: Ones, Tens, Hundreds, Thousands, Ten Thousands, Lakhs, Ten Lakhs, Crores, Ten Crores.
- Use of Commas: The first comma comes after hundreds (3 digits from right), then after every 2
digits (thousands, lakhs, crores).
Example: 5,08,01,592 (Five crore eight lakh one thousand five hundred ninety-two).
International System of Numeration
- Values: Ones, Tens, Hundreds, Thousands, Ten Thousands, Hundred Thousands, Millions, Ten Millions, Hundred Millions.
- Use of Commas: Commas come after every 3 digits from the right.
Example: 50,801,592 (Fifty million eight hundred one thousand five hundred ninety-two).
Relationship
- 1 Lakh = 100 Thousands
- 1 Million = 10 Lakhs = 1,000 Thousands
- 1 Crore = 10 Millions = 100 Lakhs
- 1 Billion = 1,000 Millions
3. Estimation (Rounding Off)
To estimate means to find a number that is close enough to the exact answer, making mentally calculation easier and faster.
- Nearest Tens: Look at the ones place. If it is 1, 2, 3, or 4, round down (make
it 0). If it is 5, 6, 7, 8, or 9, round up (increase tens digit by 1 and make ones 0).
Example: 42 → 40; 47 → 50. - Nearest Hundreds: Look at the tens place (digits 1-49 round down, 50-99 round
up).
Example: 410 → 400; 468 → 500. - Nearest Thousands: Look at the hundreds place (digits 1-499 round down, 500-999
round up).
Example: 2,345 → 2,000; 4,790 → 5,000. - General Rule for Operations: Estimate each number to its greatest place, then perform addition, subtraction, or multiplication.
4. Expanding Numbers using Brackets
- Using brackets helps organize complex operations.
Example: 7 × 109 = 7 × (100 + 9) = 7 × 100 + 7 × 9 = 700 + 63 = 763.
5. Roman Numerals
| Roman | I | V | X | L | C | D | M |
|---|---|---|---|---|---|---|---|
| Hindu-Arabic | 1 | 5 | 10 | 50 | 100 | 500 | 1000 |
Rules:
- If a symbol is repeated, its value is added as many times as it occurs (e.g., II = 2, XX = 20). A symbol is not repeated more than three times. The symbols V, L, and D are never repeated.
- If a symbol of smaller value is written to the right of a symbol of greater value, its value gets added to the value of the greater symbol (e.g., VI = 5 + 1 = 6, LX = 50 + 10 = 60).
- If a symbol of smaller value is written to the left of a symbol of greater value, its value is subtracted from the value of the greater symbol (e.g., IV = 5 - 1 = 4, XL = 50 - 10 = 40).
- The symbols V, L, and D are never written to the left of a symbol of greater value (they are never subtracted). I can be subtracted from V and X only. X can be subtracted from L, M and C only.