--- title: "NCERT Solutions Class 7 Maths Chapter 1 Large Numbers Around Us" url: https://www.swavid.com/maths/class/7/chapter/large-numbers-around-us/ncert-solutions dateModified: 2026-10-07T17:03:11+00:00 --- # NCERT Solutions Class 7 Maths Chapter 1 Large Numbers Around Us This chapter introduces students to large numbers up to crores and arabs, comparing the Indian and American place value systems. It covers applications of these numbers through estimation, approximation, and operations like multiplication and division shortcuts. Free PDF (45 pages): https://www.swavid.com/api/seo/pdf/ncert/maths/class-7/swavid-ncert-solutions-class-7-maths-chapter-1-large-numbers-around-us-a3188ed1f1.pdf ## A Lakh Varieties! ### Question 1 *3 marks · Short answer* What do you think? Guess. **Solution** 1. State that one lakh is $1,00,000$ and a lifetime is $100$ years. 2. Calculate the total number of days in $100$ years as $365 \times 100 = 36,500$ days. 3. Conclude that since $36,500$ is less than $1,00,000$, a person trying a new variety each day cannot taste all varieties in $100$ years. **Answer:** No, a person cannot taste all $1,00,000$ varieties in a lifetime of $100$ years because there are only about $36,500$ days in $100$ years. > Common mistake: Confusing the total number of days in 100 years with the number of varieties. ## A Lakh Varieties! ### Question 1 *1 mark · Fill in the blank* Observe the pattern and fill in the boxes given below. The largest 3-digit number is 999 The smallest 4-digit number is The largest 4-digit number is The smallest 5-digit number is The largest 5-digit number is The smallest 6-digit number is 1,00,000 **Solution** 1. The smallest 4-digit number is 1,000. **Answer:** 1,000 > Common mistake: Confusing the smallest 4-digit number with the largest 3-digit number. ### Question 2 *3 marks · Short answer* What if a person ate 3 varieties of rice every day? Will they be able to taste all the lakh varieties in a 100 year lifetime? Find out. **Solution** 1. Given that a person eats 3 varieties of rice every day and lives for 100 years. 2. Number of days in 100 years (ignoring leap years) = $365 \times 100 = 36,500$ days. 3. Total varieties tasted in 100 years = $36,500 \times 3 = 1,09,500$ varieties. 4. Since $1,09,500$ is greater than $1,00,000$ (one lakh), the person will be able to taste all the lakh varieties. **Answer:** Yes, the person will be able to taste all the lakh varieties as they can taste 1,09,500 varieties in 100 years. > Common mistake: Forgetting to multiply the number of days by 3 varieties per day. ### Question 3 *3 marks · Short answer* Choose a number for $y$. How close to one lakh is the number of days in $y$ years, for the $y$ of your choice? **Solution** 1. Let us choose $y = 274$ years as suggested in the chapter. 2. The number of days in 274 years = $365 \times 274 = 1,00,010$ days. 3. Comparing this with one lakh ($1,00,000$), the number of days in 274 years is only 10 days more than one lakh. **Answer:** For $y = 274$ years, the number of days is 1,00,010, which is very close (just 10 days more) to one lakh. > Common mistake: Calculation error while multiplying 365 by 274. ## Figure it Out ### Question 1 *3 marks · Short answer* According to the 2011 Census, the population of the town of Chintamani was about 75,000. How much less than one lakh is 75,000? **Solution** 1. One lakh is written as $1,00,000$. 2. The population of Chintamani in 2011 was $75,000$. 3. Subtract $75,000$ from $1,00,000$ to find the difference: $1,00,000 - 75,000 = 25,000$. 4. Therefore, $75,000$ is $25,000$ less than one lakh. **Answer:** $25,000$ > Common mistake: Subtracting incorrectly by mixing up place values. ### Question 2 *3 marks · Short answer* The estimated population of Chintamani in the year 2024 is 1,06,000. How much more than one lakh is 1,06,000? **Solution** 1. One lakh is written as $1,00,000$. 2. The estimated population in 2024 is $1,06,000$. 3. Subtract $1,00,000$ from $1,06,000$ to find how much more it is: $1,06,000 - 1,00,000 = 6,000$. 4. Therefore, $1,06,000$ is $6,000$ more than one lakh. **Answer:** $6,000$ > Common mistake: Writing an incorrect number of zeros. ### Question 3 *3 marks · Short answer* By how much did the population of Chintamani increase from 2011 to 2024? **Solution** 1. The population in 2011 was $75,000$. 2. The estimated population in 2024 is $1,06,000$. 3. Calculate the increase by subtracting the 2011 population from the 2024 population: $1,06,000 - 75,000 = 31,000$. 4. Therefore, the population increased by $31,000$. **Answer:** $31,000$ > Common mistake: Subtracting the smaller population from itself or making a borrowing error. ## Getting a Feel of Large Numbers ### Question 1 *3 marks · Short answer* Which is taller — The Statue of Unity or this building? How much taller? ____________m. **Solution** 1. Height of the Statue of Unity = $180\text{ m}$ 2. Height of Somu's building = $40\text{ m}$ 3. Difference in height = $180\text{ m} - 40\text{ m} = 140\text{ m}$ **Answer:** The Statue of Unity is 140 m taller than the building. > Common mistake: Subtracting incorrect heights from the text or figure. ### Question 2 *3 marks · Short answer* How much taller is the Kunchikal waterfall than Somu's building? ____________m. **Solution** 1. Height of the Kunchikal waterfall is given as about $450\text{ m}$. 2. Height of Somu's building is $40\text{ m}$. 3. Difference in height = $450\text{ m} - 40\text{ m} = 410\text{ m}$. **Answer:** $410\text{ m}$ > Common mistake: Subtracting the wrong values or mixing up the height of Somu with the height of the building. ### Question 3 *3 marks · Short answer* How many floors should Somu’s building have to be as high as the waterfall? ____________ . **Solution** 1. Given that the height of the Kunchikal waterfall is $450\text{ m}$ and Somu's building has a height of $40\text{ m}$ (where each floor is approx $4\text{ m}$ based on Somu being $1\text{ m}$ tall and each floor being 4 times his height). 2. To find the number of floors required to match the waterfall's height, divide the waterfall's height by the height of one floor ($4\text{ m}$). 3. Calculation: $450 \div 4 = 112.5\text{ floors}$. 4. Rounding to the nearest whole number, approximately $113$ floors are needed. **Answer:** Approx 113 floors > Common mistake: Dividing by the total building height instead of the height per floor. ## Is One Lakh a Very Large Number? ### Question 1 *3 marks · Short answer* How do you view a lakh — is a lakh big or small? **Solution** 1. A lakh can be viewed as a large number when compared to everyday countable quantities like the number of varieties of rice or days in a human lifetime. 2. It can also be viewed as relatively small when considering packed spaces, such as a cricket stadium seating capacity or the number of hairs on a human head. 3. Thus, whether a lakh is big or small depends entirely on the context and the objects being counted. **Answer:** A lakh can be viewed as a large number in contexts like rice varieties or days, but as a small quantity when considering stadium crowds or hairs on the head. > Common mistake: Stating that a lakh is simply always big or always small without providing contextual examples. ## Reading and Writing Numbers ### Question 1 *3 marks · Short answer* Write each of the numbers given below in words: (a) 3,00,600 (b) 5,04,085 (c) 27,30,000 (d) 70,53,138 **Part (a)** 1. Observe the digits according to the Indian place value system: $3,00,600$. 2. Read the digits: Three lakh six hundred. Answer (a): Three lakh six hundred **Part (b)** 1. Observe the digits according to the Indian place value system: $5,04,085$. 2. Read the digits: Five lakh four thousand eighty-five. Answer (b): Five lakh four thousand eighty-five **Part (c)** 1. Observe the digits according to the Indian place value system: $27,30,000$. 2. Read the digits: Twenty-seven lakh thirty thousand. Answer (c): Twenty-seven lakh thirty thousand **Part (d)** 1. Observe the digits according to the Indian place value system: $70,53,138$. 2. Read the digits: Seventy lakh fifty-three thousand one hundred thirty-eight. Answer (d): Seventy lakh fifty-three thousand one hundred thirty-eight **Answer:** Refer to the parts for the number names in words. > Common mistake: Misplacing commas or misreading the place values of lakhs and thousands periods. ### Question 2 *3 marks · Short answer* Write the corresponding number in the Indian place value system for each of the following: (a) One lakh twenty three thousand four hundred and fifty six (b) Four lakh seven thousand seven hundred and four (c) Fifty lakhs five thousand and fifty (d) Ten lakhs two hundred and thirty five **Part (a)** 1. Identify the periods: One lakh is $1$, twenty-three thousand is $23$, four hundred and fifty-six is $456$. 2. Place the commas according to the Indian system: $1,23,456$. Answer (a): $1,23,456$ **Part (b)** 1. Identify the periods: Four lakh is $4$, seven thousand is $07$, seven hundred and four is $704$. 2. Place the commas according to the Indian system: $4,07,704$. Answer (b): $4,07,704$ **Part (c)** 1. Identify the periods: Fifty lakhs is $50$, five thousand is $05$, and fifty is $050$. 2. Place the commas according to the Indian system: $50,05,050$. Answer (c): $50,05,050$ **Part (d)** 1. Identify the periods: Ten lakhs is $10$, thousands period has no mention so it is $00$, two hundred and thirty-five is $235$. 2. Place the commas according to the Indian system: $10,00,235$. Answer (d): $10,00,235$ **Answer:** Refer to the parts for the corresponding numbers. > Common mistake: Forgetting to include place holder zeros for missing periods such as thousands. ## 1.2 Land of Tens ### Question 1 *3 marks · Short answer* The Thoughtful Thousands only has a +1000 button. How many times should it be pressed to show: (a) Three thousand? 3 times (b) 10,000? ____________ (c) Fifty three thousand? ________ __ (d) 90,000? ______________ (e) One Lakh? ________________ (f) ____________? 153 times (g) How many thousands are required to make one lakh? **Part (a)** 1. $3,000 \div 1,000 = 3$. Answer (a): 3 times **Part (b)** 1. $10,000 \div 1,000 = 10$. Answer (b): 10 times **Part (c)** 1. $53,000 \div 1,000 = 53$. Answer (c): 53 times **Part (d)** 1. $90,000 \div 1,000 = 90$. Answer (d): 90 times **Part (e)** 1. $1,00,000 \div 1,000 = 100$. Answer (e): 100 times **Part (f)** 1. $153 \times 1,000 = 1,53,000$. Answer (f): $1,53,000$ **Part (g)** 1. $1,00,000 \div 1,000 = 100$. Answer (g): 100 thousands **Answer:** The number of button presses for each part has been found. > Common mistake: Confusing the number of zeros when dividing by 1,000. ### Question 2 *3 marks · Short answer* The Tedious Tens only has a +10 button. How many times should it be pressed to show: (a) Five hundred? _____________ (b) 780? _________ (c) 1000? _________ (d) 3700? ________ (e) 10,000? ___________ (f) One lakh? _____________ (g) ____________? 435 times **Part (a)** 1. $500 \div 10 = 50$. Answer (a): 50 times **Part (b)** 1. $780 \div 10 = 78$. Answer (b): 78 times **Part (c)** 1. $1,000 \div 10 = 100$. Answer (c): 100 times **Part (d)** 1. $3,700 \div 10 = 370$. Answer (d): 370 times **Part (e)** 1. $10,000 \div 10 = 1,000$. Answer (e): $1,000$ times **Part (f)** 1. $1,00,000 \div 10 = 10,000$. Answer (f): $10,000$ times **Part (g)** 1. $435 \times 10 = 4,350$. Answer (g): $4,350$ **Answer:** The number of button presses for each part has been found. > Common mistake: Dropping or adding extra zeros during division by 10. ### Question 3 *3 marks · Short answer* The Handy Hundreds only has a +100 button. How many times should it be pressed to show: (a) Four hundred? ___________times (b) 3,700? __________ (c) 10,000? __________ (d) Fifty three thousand? __________ (e) 90,000? __________ (f) 97,600? __________ (g) 1,00,000? __________ (h) _________? 582 times (i) How many hundreds are required to make ten thousand? (j) How many hundreds are required to make one lakh? (k) Handy Hundreds says, “There are some numbers which Tedious Tens and Thoughtful Thousands can’t show but I can.” Is this statement true? Think and explore. **Part (a)** 1. $400 \div 100 = 4$. Answer (a): 4 times **Part (b)** 1. $3,700 \div 100 = 37$. Answer (b): 37 times **Part (c)** 1. $10,000 \div 100 = 100$. Answer (c): 100 times **Part (d)** 1. $53,000 \div 100 = 530$. Answer (d): 530 times **Part (e)** 1. $90,000 \div 100 = 900$. Answer (e): 900 times **Part (f)** 1. $97,600 \div 100 = 976$. Answer (f): 976 times **Part (g)** 1. $1,00,000 \div 100 = 1,000$. Answer (g): $1,000$ times **Part (h)** 1. $582 \times 100 = 58,200$. Answer (h): $58,200$ **Part (i)** 1. $10,000 \div 100 = 100$. Answer (i): 100 hundreds **Part (j)** 1. $1,00,000 \div 100 = 1,000$. Answer (j): $1,000$ hundreds **Part (k)** 1. Handy Hundreds can only show numbers that are exact multiples of 100, which can also be shown by Tedious Tens and Thoughtful Thousands. Answer (k): No, the statement is not true. **Answer:** The number of button presses and answers for each part have been found. > Common mistake: Miscalculating the number of hundreds in larger numbers. ### Question 4 *3 marks · Short answer* Find a different way to get 5072 and write an expression for the same. **Solution** 1. The standard way given in the table for 5072 is $(5 \times 1000) + (7 \times 10) + (2 \times 1)$. 2. We can regroup 1 thousand into 10 hundreds, so 5 thousands can be written as 4 thousands and 10 hundreds. 3. Alternatively, another expression is $(4 \times 1000) + (10 \times 100) + (7 \times 10) + (2 \times 1)$. 4. Other valid expressions are also possible by regrouping tens and hundreds. **Answer:** $(4 \times 1000) + (10 \times 100) + (7 \times 10) + (2 \times 1)$ > Common mistake: Writing an expression that does not evaluate to the exact number 5072. ## Figure it Out ### Question 1 *3 marks · Short answer* For each number given below, write expressions for at least two different ways to obtain the number through button clicks. Think like Chitti and be creative. (a) 8300 (b) 40629 (c) 56354 (d) 66666 (e) 367813 **Part (a)** 1. Expression 1 using thousands and hundreds: $(8 \times 1000) + (3 \times 100)$ 2. Expression 2 using alternative grouping: $(5 \times 1000) + (33 \times 100)$ Answer (a): $(8 \times 1000) + (3 \times 100)$ or $(5 \times 1000) + (33 \times 100)$ **Part (b)** 1. Expression 1 using place values: $(40 \times 1000) + (6 \times 100) + (2 \times 10) + (9 \times 1)$ 2. Expression 2 using alternative grouping: $(30 \times 1000) + (106 \times 100) + (29 \times 1)$ Answer (b): $(40 \times 1000) + (6 \times 100) + (2 \times 10) + (9 \times 1)$ or $(30 \times 1000) + (106 \times 100) + (29 \times 1)$ **Part (c)** 1. Expression 1 using place values: $(56 \times 1000) + (3 \times 100) + (5 \times 10) + (4 \times 1)$ 2. Expression 2 using alternative grouping: $(46 \times 1000) + (103 \times 100) + (54 \times 1)$ Answer (c): $(56 \times 1000) + (3 \times 100) + (5 \times 10) + (4 \times 1)$ or $(46 \times 1000) + (103 \times 100) + (54 \times 1)$ **Part (d)** 1. Expression 1 using thousands and ones: $(66 \times 1000) + (6 \times 100) + (66 \times 1)$ 2. Expression 2 using all place values: $(6 \times 10000) + (6 \times 1000) + (6 \times 100) + (6 \times 10) + (6 \times 1)$ Answer (d): $(66 \times 1000) + (6 \times 100) + (66 \times 1)$ or $(6 \times 10000) + (6 \times 1000) + (6 \times 100) + (6 \times 10) + (6 \times 1)$ **Part (e)** 1. Expression 1 using standard place values: $(3 \times 100000) + (6 \times 10000) + (7 \times 1000) + (8 \times 100) + (1 \times 10) + (3 \times 1)$ 2. Expression 2 using alternative grouping: $(36 \times 10000) + (78 \times 1000) + (13 \times 1)$ Answer (e): $(3 \times 100000) + (6 \times 10000) + (7 \times 1000) + (8 \times 100) + (1 \times 10) + (3 \times 1)$ or $(36 \times 10000) + (78 \times 1000) + (13 \times 1)$ **Answer:** Two different expressions for each number are provided in the sub-parts. > Common mistake: Forgetting to ensure that each expression correctly evaluates to the given number. ## Creative Chitti has some questions for you — ### Question 1 *3 marks · Short answer* (a) You have to make exactly 30 button presses. What is the largest 3-digit number you can make? What is the smallest 3-digit number you can make? (b) 997 can be made using 25 clicks. Can you make 997 with a different number of clicks? **Part (a)** 1. To get the largest 3-digit number with 30 button presses, use the largest possible digits in the hundreds and tens place, giving $993$ ($9 + 9 + 3 = 21$ clicks is not 30, so adjust to use 30 clicks: $900 + 90 + 3$ uses $9+9+12=30$, so $993$). 2. To get the smallest 3-digit number with 30 button presses, use the smallest possible digits, which is $102$ ($1 + 0 + 29 = 30$ clicks using $100 + 0 + 29$). Answer (a): The largest 3-digit number is 993 and the smallest 3-digit number is 102. **Part (b)** 1. Express 997 using different button clicks such as $(8 \times 100) + (19 \times 10) + (7 \times 1)$. 2. Calculate the total number of clicks as $8 + 19 + 7 = 34$ clicks. Answer (b): Yes, 997 can be made using 34 clicks. **Answer:** (a) Largest 3-digit number is 993, smallest is 102. (b) Yes, 997 can be made with 34 clicks. > Common mistake: Forgetting that the sum of the digits or multipliers must equal the exact number of button presses required. ## Systematic Sippy ### Question 1 *3 marks · Short answer* How can we get the numbers (a) 5072, (b) 8300 using as few button clicks as possible? **Part (a)** 1. Express 5072 using as few button clicks as possible: $5072 = (5 \times 1000) + (7 \times 10) + (2 \times 1)$. 2. Count the total number of button presses: $5 + 7 + 2 = 14$ clicks. Answer (a): $5072 = (5 \times 1000) + (7 \times 10) + (2 \times 1)$ using 14 clicks. **Part (b)** 1. Express 8300 using as few button clicks as possible: $8300 = (8 \times 1000) + (3 \times 100)$. 2. Count the total number of button presses: $8 + 3 = 11$ clicks. Answer (b): $8300 = (8 \times 1000) + (3 \times 100)$ using 11 clicks. **Answer:** For 5072: $(5 \times 1000) + (7 \times 10) + (2 \times 1)$ with 14 clicks. For 8300: $(8 \times 1000) + (3 \times 100)$ with 11 clicks. > Common mistake: Using smaller place value buttons than necessary, which increases the total number of button clicks. ## Figure it Out ### Question 1 *3 marks · Short answer* For the numbers in the previous exercise, find out how to get each number by making the smallest number of button clicks and write the expression. **Part (a)** 1. Write 8300 using the largest available button values with the fewest clicks. 2. $8300 = (8 \times 1000) + (3 \times 100)$ 3. Number of clicks = $8 + 3 = 11$ Answer (a): $8300 = (8 \times 1000) + (3 \times 100)$, Clicks: 11 **Part (b)** 1. Write 40,629 using place values for the least button clicks. 2. $40629 = (4 \times 10000) + (6 \times 100) + (2 \times 10) + (9 \times 1)$ 3. Number of clicks = $4 + 6 + 2 + 9 = 21$ Answer (b): $40629 = (4 \times 10000) + (6 \times 100) + (2 \times 10) + (9 \times 1)$, Clicks: 21 **Part (c)** 1. Write 56,354 using place values for the least button clicks. 2. $56354 = (5 \times 10000) + (6 \times 1000) + (3 \times 100) + (5 \times 10) + (4 \times 1)$ 3. Number of clicks = $5 + 6 + 3 + 5 + 4 = 23$ Answer (c): $56354 = (5 \times 10000) + (6 \times 1000) + (3 \times 100) + (5 \times 10) + (4 \times 1)$, Clicks: 23 **Answer:** Expressions using minimum button clicks for each number. > Common mistake: Using more buttons than necessary by not grouping into the highest place value buttons. ### Question 2 *3 marks · Short answer* Do you see any connection between each number and the corresponding smallest number of button clicks? **Solution** 1. The smallest number of button clicks corresponds to the sum of the digits of the number. 2. Each digit in the standard place-value notation represents the multiplier for a specific button like $+1$, $+10$, $+100$, $+1000$, $+10000$, or $+100000$. 3. Therefore, the minimum total number of button presses needed to form the number is equal to the sum of all its digits. **Answer:** The smallest number of button clicks for each number is equal to the sum of its digits. > Common mistake: Confusing the total value of the number with the individual face values of its digits when counting button presses. ### Question 3 *3 marks · Short answer* If you notice, the expressions for the least button clicks also give the Indian place value notation of the numbers. Think about why this is so. **Solution** 1. Systematic Sippy uses buttons corresponding to powers of 10 like $1, 10, 100, 1000, 10000$, and $100000$. 2. Writing a number as a sum of products of its digits and these powers of 10 is precisely the expanded form in the Indian place value system. 3. Minimising button clicks forces us to use the standard place value representation. **Answer:** The expressions represent numbers in expanded form based on the Indian place value system. > Common mistake: Ignoring place value expansion rules. ## 1.3 Of Crores and Crores! ### Question 1 *1 mark · Fill in the blank* How many zeros does a thousand lakh have? _____ **Solution** 1. A thousand lakh is written as $1,000 \times 1,00,000 = 10,00,00,000$ (10 crore). **Answer:** 8 zeros > Common mistake: Confusing lakhs and crores, leading to an incorrect number of zeros. ### Question 2 *1 mark · Fill in the blank* How many zeros does a hundred thousand have? _____ **Solution** 1. A hundred thousand is written as $1,00,000$, which has 5 zeros. **Answer:** 5 zeros > Common mistake: Writing 6 zeros by mistake for a million instead of a hundred thousand. ## Figure it Out ### Question 1 *3 marks · Short answer* Read the following numbers in Indian place value notation and write their number names in both the Indian and American systems: (a) 4050678 (b) 48121620 (c) 20022002 (d) 246813579 (e) 345000543 (f) 1020304050 **Part (a)** 1. Indian system notation: 40,50,678 2. Indian system number name: Forty lakh fifty thousand six hundred seventy-eight 3. American system number name: Four million fifty thousand six hundred seventy-eight Answer (a): 40,50,678; Forty lakh fifty thousand six hundred seventy-eight (Indian), Four million fifty thousand six hundred seventy-eight (American) **Part (b)** 1. Indian system notation: 4,81,21,620 2. Indian system number name: Four crore eighty-one lakh twenty-one thousand six hundred twenty 3. American system number name: Forty-eight million one hundred twenty-one thousand six hundred twenty Answer (b): 4,81,21,620; Four crore eighty-one lakh twenty-one thousand six hundred twenty (Indian), Forty-eight million one hundred twenty-one thousand six hundred twenty (American) **Part (c)** 1. Indian system notation: 2,00,22,002 2. Indian system number name: Two crore twenty-two thousand two 3. American system number name: Twenty million twenty-two thousand two Answer (c): 2,00,22,002; Two crore twenty-two thousand two (Indian), Twenty million twenty-two thousand two (American) **Part (d)** 1. Indian system notation: 24,68,13,579 2. Indian system number name: Twenty-four crore sixty-eight lakh thirteen thousand five hundred seventy-nine 3. American system number name: Two hundred forty-six million eight hundred thirteen thousand five hundred seventy-nine Answer (d): 24,68,13,579; Twenty-four crore sixty-eight lakh thirteen thousand five hundred seventy-nine (Indian), Two hundred forty-six million eight hundred thirteen thousand five hundred seventy-nine (American) **Part (e)** 1. Indian system notation: 34,50,00,543 2. Indian system number name: Thirty-four crore fifty lakh five hundred forty-three 3. American system number name: Three hundred forty-five million five hundred forty-three Answer (e): 34,50,00,543; Thirty-four crore fifty lakh five hundred forty-three (Indian), Three hundred forty-five million five hundred forty-three (American) **Part (f)** 1. Indian system notation: 1,02,03,04,050 2. Indian system number name: One arab two crore three lakh four thousand fifty 3. American system number name: One billion twenty million three hundred four thousand fifty Answer (f): 1,02,03,04,050; One arab two crore three lakh four thousand fifty (Indian), One billion twenty million three hundred four thousand fifty (American) **Answer:** Refer to individual parts for the Indian place value notation and number names in both systems. > Common mistake: Mixing up the comma placement between the Indian (3-2-2) and American (3-3-3) systems. ### Question 2 *3 marks · Short answer* Write the following numbers in Indian place value notation: (a) One crore one lakh one thousand ten (b) One billion one million one thousand one (c) Ten crore twenty lakh thirty thousand forty (d) Nine billion eighty million seven hundred thousand six hundred **Part (a)** 1. One crore = 1,00,00,000 2. One lakh = 1,00,000 3. One thousand ten = 1,010 4. Sum gives 1,01,01,010 Answer (a): 1,01,01,010 **Part (b)** 1. One billion equals 100 crore = 1,00,00,00,000 2. One million equals 10 lakh = 10,00,000 3. One thousand one = 1,001 4. Sum gives 1,00,10,01,001 Answer (b): 1,00,10,01,001 **Part (c)** 1. Ten crore = 10,00,00,000 2. Twenty lakh = 20,00,000 3. Thirty thousand = 30,000 4. Forty = 40 5. Sum gives 10,20,30,040 Answer (c): 10,20,30,040 **Part (d)** 1. Nine billion equals 900 crore = 9,00,00,00,000 2. Eighty million equals 8 crore = 8,00,00,000 3. Seven hundred thousand = 7,00,000 4. Six hundred = 600 5. Sum gives 9,08,07,00,600 Answer (d): 9,08,07,00,600 **Answer:** Refer to individual parts for the numbers written in Indian place value notation. > Common mistake: Incorrect conversion from millions and billions to Indian place value units like lakhs and crores. ### Question 3 *3 marks · Short answer* Compare and write ‘<’, ‘>’ or ‘=’: (a) 30 thousand ____ 3 lakhs (b) 500 lakhs ______ 5 million (c) 800 thousand ____ 8 million (d) 640 crore ______ 60 billion **Part (a)** 1. 30 thousand = 30,000 2. 3 lakhs = 3,00,000 3. Comparing 30,000 and 3,00,000, we get 30,000 < 3,00,000. Answer (a): 30 thousand < 3 lakhs **Part (b)** 1. 500 lakhs = 5 crore = 5,00,00,000 2. 5 million = 50 lakh = 50,00,000 3. Comparing 5,00,00,000 and 50,00,000, we get 5,00,00,000 > 50,00,000. Answer (b): 500 lakhs > 5 million **Part (c)** 1. 800 thousand = 8 lakh = 8,00,000 2. 8 million = 80 lakh = 80,00,000 3. Comparing 8,00,000 and 80,00,000, we get 8,00,000 < 80,00,000. Answer (c): 800 thousand < 8 million **Part (d)** 1. 640 crore = 6,40,00,00,000 2. 60 billion = 6,000 crore = 60,00,00,00,000 3. Comparing 640 crore and 60 billion, we get 640 crore < 60 billion. Answer (d): 640 crore < 60 billion **Answer:** Refer to individual parts for the comparison results. > Common mistake: Failing to convert both quantities into the same unit or place value system before comparing. ## 1.4 Exact and Approximate Values ### Question 1 *3 marks · Short answer* Think and share situations where it is appropriate to (a) round up, (b) round down, (c) either rounding up or rounding down is okay and (d) when exact numbers are needed. **Part (a)** 1. Rounding up is appropriate when we need to ensure enough resources are available, such as ordering items for a group. 2. Example: Ordering 750 sweets for 732 people so that everyone gets enough. Answer (a): Buying food or items for a group to ensure no one is short. **Part (b)** 1. Rounding down is appropriate when estimating costs or limits to be safe or convenient. 2. Example: Saying an item costs around ₹450 instead of ₹500 when its actual cost is ₹470. Answer (b): Estimating costs or time conservatively, like saying an item costs around ₹450 instead of ₹500. **Part (c)** 1. Either rounding up or rounding down is acceptable when a rough estimate is sufficient to give an idea of a quantity. 2. Example: Estimating the distance between far-off places or approximating a town's population. Answer (c): Estimating distances between far-off places or general population figures. **Part (d)** 1. Exact numbers are needed when precision is critical and errors cannot be tolerated. 2. Example: Handling financial transactions in bank accounts or recording official data. Answer (d): Handling money in bank accounts or official census records where exact counts are mandatory. **Answer:** Situations depend on whether we need to account for safety, efficiency, or exactness in calculations like transactions or measurements. > Common mistake: Confusing situations where rounding up is necessary for safety with situations where exact values are mandatory. ## Nearest Neighbours ### Question 1 *3 marks · Short answer* Similarly, write the five nearest neighbours for these numbers: (a) 3,87,69,957 (b) 29,05,32,481 **Part (a)** 1. For $3,87,69,957$, the nearest thousand is $3,87,70,000$. 2. The nearest ten thousand is $3,87,70,000$. 3. The nearest lakh is $3,88,00,000$, nearest ten lakh is $3,90,00,000$, and nearest crore is $4,00,00,000$. Answer (a): Nearest thousand: 3,87,70,000; Nearest ten thousand: 3,87,70,000; Nearest lakh: 3,88,00,000; Nearest ten lakh: 3,90,00,000; Nearest crore: 4,00,00,000 **Part (b)** 1. For $29,05,32,481$, the nearest thousand is $29,05,32,000$. 2. The nearest ten thousand is $29,05,30,000$. 3. The nearest lakh is $29,05,00,000$, nearest ten lakh is $29,10,00,000$, and nearest crore is $29,00,00,000$. Answer (b): Nearest thousand: 29,05,32,000; Nearest ten thousand: 29,05,30,000; Nearest lakh: 29,05,00,000; Nearest ten lakh: 29,10,00,000; Nearest crore: 29,00,00,000 **Answer:** Refer to parts (a) and (b) for the nearest neighbours. > Common mistake: Confusing rounding up and rounding down rules for large place values. ### Question 2 *3 marks · Short answer* I have a number for which all five nearest neighbours are 5,00,00,000. What could the number be? How many such numbers are there? **Solution** 1. The five nearest neighbours listed in the textbook are nearest thousand, nearest ten thousand, nearest lakh, nearest ten lakh, and nearest crore. 2. For all five nearest neighbours of a number to be equal to $5,00,00,000$, rounding the number to any of these places must yield $5,00,00,000$. 3. Thus, the number must be extremely close to $5,00,00,000$, specifically between $4,99,99,500$ and $5,00,00,500$ (excluding the boundary values where rounding differs). 4. Many such numbers exist, for example, $4,99,99,999$ or $5,00,00,001$. **Answer:** The number could be any integer very close to $5,00,00,000$ (such as $4,99,99,999$ or $5,00,00,001$). There are infinitely many such decimal or whole numbers depending on the range. > Common mistake: Assuming there is only one unique number like exactly 5,00,00,000. ## Roxie and Estu are estimating the values of simple expressions. ### Question 1 *3 marks · Short answer* 4,63,128 + 4,19,682, Roxie: “The sum is near 8,00,000 and is more than 8,00,000.” Estu: “The sum is near 9,00,000 and is less than 9,00,000.” (a) Are these estimates correct? Whose estimate is closer to the sum? (b) Will the sum be greater than 8,50,000 or less than 8,50,000? Why do you think so? (c) Will the sum be greater than 8,83,128 or less than 8,83,128? Why do you think so? (d) Exact value of 4,63,128 + 4,19,682 = ___________ **Part (a)** 1. The exact sum is 4,63,128 + 4,19,682 = 8,82,810. 2. Roxie's estimate (8,00,000) is too low, while Estu's estimate (9,00,000) is closer to the actual sum. Answer (a): Both are estimates, but Estu's estimate is closer to the sum. **Part (b)** 1. The sum 8,82,810 is greater than 8,50,000. 2. This is because 8,82,810 is closer to 9,00,000 than to 8,00,000. Answer (b): Greater than 8,50,000. **Part (c)** 1. The sum 8,82,810 is less than 8,83,128. 2. Comparing the numbers, 8,82,810 < 8,83,128. Answer (c): Less than 8,83,128. **Part (d)** 1. 4,63,128 + 4,19,682 = 8,82,810 Answer (d): 8,82,810 **Answer:** The exact sum is 8,82,810. > Common mistake: Confusing the place value while adding or comparing the sum with the given benchmarks. ### Question 2 *3 marks · Short answer* 14,63,128 – 4,90,020 Roxie: “The difference is near 10,00,000 and is less than 10,00,000.” Estu: “The difference is near 9,00,000 and is more than 9,00,000.” (a) Are these estimates correct? Whose estimate is closer to the difference? (b) Will the difference be greater than 9,50,000 or less than 9,50,000? Why do you think so? **Part (a)** 1. The exact difference is 14,63,128 - 4,90,020 = 9,73,108. 2. Roxie's estimate (10,00,000) is closer to 9,73,108 than Estu's estimate (9,00,000). Answer (a): Roxie's estimate is closer to the difference. **Part (b)** 1. The difference 9,73,108 is greater than 9,50,000. 2. This is because 9,73,108 is closer to 10,00,000 than to 9,00,000. Answer (b): Greater than 9,50,000. **Answer:** The exact difference is 9,73,108. > Common mistake: Incorrectly rounding the numbers or miscalculating the difference. ## Populations of Cities ### Question 1 *3 marks · Short answer* Will the difference be greater than 9,63,128 or less than 9,63,128? Why do you think so? **Solution** 1. The exact calculation of $14,63,128 - 4,90,020$ gives $9,73,108$. 2. Comparing the result, $9,73,108$ is greater than $9,63,128$. 3. This is because we are subtracting a number close to $4,90,000$ from $14,63,128$, which leaves more than $9,70,000$. **Answer:** The difference will be greater than 9,63,128 because the actual difference is 9,73,108. > Common mistake: Confusing whether subtracting a smaller rounded number makes the result larger or smaller. ### Question 2 *3 marks · Short answer* Exact value of 14,63,128 – 4,90,020 = __________ **Solution** 1. Write the numbers vertically aligned by their place values: $14,63,128 - 4,90,020$. 2. Subtract column by column from right to left: $8 - 0 = 8$, $2 - 2 = 0$, $1 - 0 = 1$, $3 - 0 = 3$, $16 - 9 = 7$, and $13 - 4 = 9$. 3. The exact value is $9,73,108$. **Answer:** $9,73,108$ > Common mistake: Borrowing errors during subtraction across lakhs and ten lakhs. ## From the information given in the table, answer the following questions by approximation: ### Question 1 *3 marks · Short answer* What is your general observation about this data? Share it with the class. **Solution** 1. Observe the population data for various Indian cities for the years 2001 and 2011 from the table in the textbook. 2. Compare the population values of each city between the two census years. 3. Note that most cities shown in the table have seen a significant rise in population from 2001 to 2011. **Answer:** Most cities shown in the table have seen a significant rise in population from 2001 to 2011. > Common mistake: Giving a specific numerical value instead of a general observation. ### Question 2 *3 marks · Short answer* What is an appropriate title for the above table? **Solution** 1. Examine the columns and rows of the table which display the rank, city name, and population for two different census years, 2011 and 2001. 2. Identify that the table summarises census data for major Indian cities over a decade. 3. Formulate a clear and descriptive title based on the table contents. **Answer:** Population of some Indian cities in 2001 and 2011. > Common mistake: Writing an incomplete title that mentions only one year. ### Question 3 *3 marks · Short answer* How much is the population of Pune in 2011? Approximately, by how much has it increased compared to 2001? **Solution** 1. Locate Pune in the population table and note its 2011 population as $31,15,431$ and its 2001 population as $25,38,473$. 2. Subtract the 2001 population from the 2011 population: $31,15,431 - 25,38,473 = 5,76,958$. 3. Approximate the increase to the nearest lakh, which is about $6$ lakh. **Answer:** The population of Pune in 2011 is $31,15,431$ and it has approximately increased by $6$ lakh. > Common mistake: Forgetting to round off or approximate the increase as requested. ### Question 4 *3 marks · Short answer* Which city’s population increased the most between 2001 and 2011? **Solution** 1. Calculate the difference in population between 2011 and 2001 for the major growing cities from the table. 2. Observe that Bengaluru's population in 2011 is $84,25,970$ and in 2001 it was $43,01,326$. 3. Compute the increase for Bengaluru: $84,25,970 - 43,01,326 = 41,24,644$, which is the highest increase among all cities listed. **Answer:** Bengaluru has shown the maximum increase in population which is $41,24,644$. > Common mistake: Confusing the highest population in 2011 with the highest population increase. ### Question 5 *3 marks · Short answer* Are there cities whose population has almost doubled? Which are they? **Solution** 1. Check the 2001 and 2011 populations for each city to see which ones have approximately doubled (i.e., where 2011 population is close to twice the 2001 population). 2. Observe Bengaluru ($43,01,326$ to $84,25,970$), Hyderabad ($36,37,483$ to $68,09,970$), Surat ($24,33,835$ to $44,67,797$), and Vadodara ($16,90,000$ to $35,52,371$). 3. Conclude that these cities have almost doubled their populations. **Answer:** Yes, they are Bengaluru, Hyderabad, Surat, and Vadodara. > Common mistake: Listing cities that grew significantly but did not double. ### Question 6 *3 marks · Short answer* By what number should we multiply Patna’s population to get a number/population close to that of Mumbai? **Solution** 1. Note Patna's population in 2011 from the table as $16,84,222$ and Mumbai's population as $1,24,42,373$. 2. Set up the division to find the multiplier: $1,24,42,373 \div 16,84,222$. 3. Calculate the quotient to get approximately $7.39$. **Answer:** We need to multiply Patna's population by $7.39$ to get a number close to the population of Mumbai. > Common mistake: Dividing Mumbai's population by the wrong city's population. ## 1.5 Patterns in Products ### Question 1 *3 marks · Short answer* Using the meaning of multiplication and division, can you explain why multiplying by 5 is the same as dividing by 2 and multiplying by 10? **Solution** 1. We know that the number 5 can be written as the fraction $\frac{10}{2}$. 2. Therefore, multiplying any number by 5 is equivalent to multiplying it by $\frac{10}{2}$. 3. This means we can first divide the number by 2 and then multiply the result by 10 to get the same product. **Answer:** Multiplying by 5 is the same as dividing by 2 and multiplying by 10 because $5 = \frac{10}{2}$. > Common mistake: Students might forget that division and multiplication can be performed in any order when dealing with fractions like 10/2. ## Figure it Out ### Question 1 *3 marks · Short answer* Find quick ways to calculate these products: (a) 2 × 1768 × 50 (b) 72 × 125 [Hint: $125 = \frac{1000}{8}$] (c) 125 × 40 × 8 × 25 **Part (a)** 1. Rearrange the terms as $2 \times 50 \times 1768$. 2. Multiply $2 \times 50$ to get $100$. 3. Multiply $100 \times 1768$ to get $1,76,800$. Answer (a): $1,76,800$ **Part (b)** 1. Use the hint $125 = \frac{1000}{8}$ to rewrite the expression as $72 \times \frac{1000}{8}$. 2. Divide $72$ by $8$ to get $9$. 3. Multiply $9 \times 1000$ to get $9,000$. Answer (b): $9,000$ **Part (c)** 1. Regroup the terms as $(125 \times 8) \times (40 \times 25)$. 2. Calculate $125 \times 8 = 1000$ and $40 \times 25 = 1000$. 3. Multiply $1000 \times 1000$ to get $10,00,000$. Answer (c): $10,00,000$ **Answer:** (a) $1,76,800$, (b) $9,000$, (c) $10,00,000$ > Common mistake: Multiplying directly without grouping to form powers of 10 first, leading to calculation errors. ### Question 2 *3 marks · Short answer* Calculate these products quickly. (a) 25 × 12 = _____________ (b) 25 × 240 = _____________ (c) 250 × 120 = _____________ (d) 2500 × 12 =_____________ (e) ______×______= 120000000 **Part (a)** 1. Write $25 \times 12$ as $\frac{100}{4} \times 12$. 2. Simplify as $100 \times \frac{12}{4} = 100 \times 3$. Answer (a): $300$ **Part (b)** 1. Write $25 \times 240$ as $\frac{100}{4} \times 240$. 2. Simplify as $100 \times \frac{240}{4} = 100 \times 60$. Answer (b): $6,000$ **Part (c)** 1. Write $250 \times 120$ as $\frac{1000}{4} \times 120$. 2. Simplify as $1000 \times \frac{120}{4} = 1000 \times 30$. Answer (c): $30,000$ **Part (d)** 1. Write $2500 \times 12$ as $\frac{10000}{4} \times 12$. 2. Simplify as $10000 \times \frac{12}{4} = 10000 \times 3$. Answer (d): $30,000$ **Part (e)** 1. Find two numbers whose product is $12,00,00,000$. 2. Using $1200 \times 1,00,000 = 12,00,00,000$. Answer (e): $1200 \times 1,00,000$ **Answer:** (a) $300$, (b) $6,000$, (c) $30,000$, (d) $30,000$, (e) $1200 \times 1,00,000 = 12,00,00,000$ > Common mistake: Misplacing zeros when multiplying by powers of 10. ## How Long is the Product? ### Question 1 *3 marks · Short answer* Observe the number of digits in the two numbers being multiplied and their product in each case. Is there any connection between the numbers being multiplied and the number of digits in their product? **Solution** 1. The maximum digits in the product of two numbers is equal to the sum of the digits of the two numbers. 2. The minimum number of digits in the product is one less than the sum of the digits. **Answer:** The maximum digits in the product is the sum of the digits, and the minimum digits is one less than their sum. > Common mistake: Confusing maximum digits with minimum digits. ### Question 2 *1 mark · True or false* Roxie says that the product of two 2-digit numbers can only be a 3- or a 4-digit number. Is she correct? **Solution** 1. The smallest 2-digit number is $10$ and the largest is $99$. 2. Multiplying smallest gives $10 \times 10 = 100$ (3 digits) and multiplying largest gives approximately less than $10,000$ (4 digits). 3. Thus, Roxie is correct. **Answer:** True > Common mistake: Stating False without checking boundary values. ### Question 3 *3 marks · Short answer* Should we try all possible multiplications with 2-digit numbers to tell whether Roxie’s claim is true? Or is there a better way to find out? **Solution** 1. We do not need to try all possible multiplications. 2. We can find the minimum and maximum possible products using the smallest and greatest numbers of those digits. **Answer:** No, we can use the minimum and maximum possible products instead. > Common mistake: Assuming checking all combinations is the only way. ### Question 4 *1 mark · True or false* Can multiplying a 3-digit number with another 3-digit number give a 4-digit number? **Solution** 1. False, the smallest product of two 3-digit numbers is $100 \times 100 = 10,000$, which is a 5-digit number. 2. Thus, the product of two 3-digit numbers can never be a 4-digit number. **Answer:** False > Common mistake: Thinking that multiplying two 3-digit numbers can yield a 4-digit number. ### Question 5 *1 mark · True or false* Can multiplying a 4-digit number with a 2-digit number give a 5-digit number? **Solution** 1. The smallest 4-digit number is $1000$ and smallest 2-digit number is $10$. 2. Their product is $1000 \times 10 = 10,000$, which is a 5-digit number. **Answer:** True > Common mistake: Assuming the product is always 6 digits. ### Question 6 *1 mark · Fill in the blank* Observe the multiplication statements below. Do you notice any patterns? See if this pattern extends for other numbers as well. 5-digit × 5-digit = ______ or ______ 8-digit × 3-digit = ______ or ______ 12-digit × 13-digit = ______ or ______ **Part (a)** 1. The minimum number of digits in the product of a 5-digit number and a 5-digit number is $5 + 5 - 1 = 9$ digits. 2. The maximum number of digits in the product is $5 + 5 = 10$ digits. Answer (a): 9-digit or 10-digit **Part (b)** 1. The minimum number of digits in the product of an 8-digit number and a 3-digit number is $8 + 3 - 1 = 10$ digits. 2. The maximum number of digits in the product is $8 + 3 = 11$ digits. Answer (b): 10-digit or 11-digit **Part (c)** 1. The minimum number of digits in the product of a 12-digit number and a 13-digit number is $12 + 13 - 1 = 24$ digits. 2. The maximum number of digits in the product is $12 + 13 = 25$ digits. Answer (c): 24-digit or 25-digit **Answer:** 9-digit or 10-digit > Common mistake: Confusing the minimum number of digits with the sum of digits without subtracting one. ## Fascinating Facts about Large Numbers ### Question 1 *3 marks · Short answer* 1250 × 380 ______________ is the number of kīrtanas composed by Purandaradāsa according to legends. **Solution** 1. Given product: $1250 \times 380$ 2. Calculate the product: $1250 \times 380 = 4,75,000$ 3. The number of kīrtanas composed by Purandaradāsa is 4,75,000 (four lakh seventy-five thousand). **Answer:** 4,75,000 > Common mistake: Errors in counting zeroes while multiplying numbers ending in zero. ### Question 2 *3 marks · Short answer* How many years did he live to compose so many songs? At what age did he start composing songs? **Solution** 1. According to historical accounts, Purandaradāsa lived from 1484 to 1564, which is a lifespan of about 80 years. 2. Legends state that he started composing songs during his later life or after dedicating himself to music and spirituality. 3. Exact ages vary by legend, but he lived for approximately 80 years. **Answer:** He lived for about 80 years. > Common mistake: Guessing random numbers without historical context. ### Question 3 *3 marks · Short answer* If he composed 4,75,000 songs, how many songs per year did he have to compose? **Solution** 1. Given total number of songs = 4,75,000 2. Assuming he composed songs over an active composing period of about 40 to 50 years, divide the total songs by the number of years. 3. For example, if divided over 47.5 years, he would have to compose 10,000 songs per year. **Answer:** About 10,000 songs per year (assuming a 47.5-year composing span). > Common mistake: Dividing by total lifespan instead of the active composing years. ### Question 4 *3 marks · Short answer* 2100 × 70,000 _______________ is the approximate distance in kilometers, between the Earth and the Sun. **Solution** 1. Given product: $2100 \times 70,000$ 2. Calculate the product: $21 \times 7 = 147$, followed by $2 + 4 = 6$ zeroes. 3. Result: $14,70,00,000$ kilometres (14 crore 70 lakh kilometres). **Answer:** 14,70,00,000 km > Common mistake: Incorrect placement of commas or miscounting zeroes. ## As you did before, divide the given numbers to uncover interesting facts about division. ### Question 1 *3 marks · Short answer* 6400 × 62,500 _________________ is the average number of litres of water the Amazon river discharges into the Atlantic Ocean every second. **Solution** 1. Write the given product: $6400 \times 62,500$. 2. Regroup the numbers as $(64 \times 625) \times 100 \times 100$. 3. Calculate the product to get $40,00,00,000$ litres. **Answer:** $40,00,00,000$ litres (or 40 crore litres) > Common mistake: Errors in counting the number of zeroes in the product. ### Question 2 *3 marks · Short answer* 13,95,000 ÷ 150 _________________ is the distance (in kms) of the longest single-train journey in the world. **Solution** 1. Write the given division: $13,95,000 \div 150$. 2. Cancel the trailing zero: $1,39,500 \div 15$. 3. Perform the division to get $9300$ kilometers. **Answer:** $9300$ km > Common mistake: Mistakes in handling trailing zeros during division. ## 1.6 Did You Ever Wonder…? ### Question 1 *1 mark · True or false* The RMS Titanic ship carried about 2500 passengers. Can the population of Mumbai fit into 5000 such ships? **Solution** 1. Capacity of 5000 ships with 2500 passengers each = $5000 \times 2500 = 1,25,00,000$ (1 crore 25 lakhs). 2. The population of Mumbai is more than 1 crore 24 lakhs, but let us check accurately using the textbook table where Mumbai's population in 2011 is 1,24,42,373. 3. Since 1,25,00,000 is greater than 1,24,42,373, the entire population of Mumbai can fit into 5000 such ships. **Answer:** True > Common mistake: Comparing the passenger capacity incorrectly without calculating the total. ### Question 2 *3 marks · Short answer* Find out if you can reach the Sun in a lifetime, if you travel 1000 kilometers every day. (You had written down the distance between the Earth and the Sun in a previous exercise.) **Solution** 1. The approximate distance between the Earth and the Sun is $2100 \times 70,000 = 14,70,00,000$ km (or about 14.7 crore km). 2. Distance travelled in 1 year at 1000 km per day = $1000 \times 365 = 3,65,000$ km. 3. Number of years to reach the Sun = $14,70,00,000 \div 3,65,000 \approx 402.7$ years. 4. Since a human lifetime is around 100 years, we cannot reach the Sun in a lifetime. **Answer:** We cannot reach the Sun in a lifetime because it takes approximately 403 years. > Common mistake: Forgetting to multiply the daily distance by 365 days to find the yearly distance. ### Question 3 *3 marks · Short answer* Make necessary reasonable assumptions and answer the questions below: (a) If a single sheet of paper weighs 5 grams, could you lift one lakh sheets of paper together at the same time? (b) If 250 babies are born every minute across the world, will a million babies be born in a day? (c) Can you count 1 million coins in a day? Assume you can count 1 coin every second. **Part (a)** 1. Weight of one single sheet of paper = $5\text{ g}$. 2. Weight of one lakh ($1,00,000$) sheets = $1,00,000 \times 5\text{ g} = 5,00,000\text{ g} = 500\text{ kg}$. 3. Since a person cannot lift $500\text{ kg}$ at once, you cannot lift one lakh sheets of paper together. Answer (a): No, because one lakh sheets weigh $500\text{ kg}$, which is too heavy to lift. **Part (b)** 1. Number of minutes in a day = $24 \times 60 = 1,440\text{ minutes}$. 2. Babies born in a day at the rate of $250$ per minute = $1,440 \times 250 = 3,60,000\text{ babies}$. 3. Since $3,60,000$ is less than one million ($10,00,000$), a million babies will not be born in a day. Answer (b): No, only $3,60,000$ babies are born in a day, which is less than a million. **Part (c)** 1. Number of seconds in a day = $24 \times 60 \times 60 = 86,400\text{ seconds}$. 2. At the rate of $1$ coin per second, the maximum coins counted in a day = $86,400\text{ coins}$. 3. Since $86,400$ is far less than $1,000,000$, you cannot count $1$ million coins in a day. Answer (c): No, because you can only count $86,400$ coins in a day at $1$ coin per second. **Answer:** Refer to individual parts. > Common mistake: Calculation errors while converting grams to kilograms or minutes to hours and days. ## Figure it Out ### Question 1 *3 marks · Short answer* Using all digits from 0–9 exactly once (the first digit cannot be 0) to create a 10-digit number, write the — (a) Largest multiple of 5 (b) Smallest even number **Solution** 1. To find the largest multiple of 5 using all digits 0-9 exactly once, we arrange the digits in descending order and ensure the last digit is 0: 9876543210. 2. To find the smallest even number using all digits 0-9 exactly once, we arrange the digits in ascending order, ensuring the first digit is not 0 and the last digit is even: 1023456798. **Answer:** (a) 9876543210, (b) 1023456798 > Common mistake: Forgetting that the first digit cannot be 0 in the smallest number. ### Question 2 *3 marks · Short answer* The number 10,30,285 in words is Ten lakhs thirty thousand two hundred eighty five, which has 42 letters. Give a 7-digit number name which has the maximum number of letters. **Solution** 1. To get the maximum number of letters in a 7-digit number name, choose digits whose number names have the highest number of letters. 2. The digit 7 ("Seventy" + "Seven" = 7 + 5 = 12 letters) and digit 8 ("Eighty" + "Eight" = 6 + 5 = 11 letters) have long names. 3. The number 78,78,773 has seventy-eight lakh seventy-eight thousand seven hundred seventy-three, which gives the maximum number of letters. **Answer:** 78,78,773 > Common mistake: Choosing numbers with shorter words like one, two, or three. ### Question 3 *3 marks · Short answer* Write a 9-digit number where exchanging any two digits results in a bigger number. How many such numbers exist? **Solution** 1. A 9-digit number where exchanging any two digits results in a bigger number must have its digits in strictly increasing order. 2. The only such number is 123456789. 3. Since the digits must be in strictly increasing order, only one such number exists. **Answer:** 123456789; only one such number exists. > Common mistake: Assuming multiple such numbers exist. ### Question 4 *3 marks · Short answer* Strike out 10 digits from the number 12345123451234512345 so that the remaining number is as large as possible. **Solution** 1. Given number is 12345123451234512345, which consists of four blocks of 12345. 2. To make the remaining 10-digit number as large as possible, we should retain the largest possible digits starting from the leftmost side and compare values. 3. Comparing digits and greedily keeping the largest digits gives the remaining number as 5534512345. **Answer:** 5534512345 > Common mistake: Striking out digits without maintaining the order of the remaining digits. ### Question 5 *3 marks · Short answer* The words ‘zero’ and ‘one’ share letters ‘e’ and ‘o’. The words ‘one’ and ‘two’ share a letter ‘o’, and the words ‘two’ and ‘three’ also share a letter ‘t’. How far do you have to count to find two consecutive numbers which do not share an English letter in common? **Solution** 1. We check consecutive numbers for common English letters. 2. The words for numbers share letters frequently (e.g., 'one' and 'two' share 'o'). 3. This is a linguistic puzzle requiring checking the spelling of consecutive numbers. **Answer:** This is a linguistic puzzle; one must check the spelling of consecutive numbers to find a pair with no common letters. > Common mistake: Misspelling the number names. ### Question 6 *3 marks · Short answer* Suppose you write down all the numbers 1, 2, 3, 4, …, 9, 10, 11, ... The tenth digit you write is ‘1’ and the eleventh digit is ‘0’, as part of the number 10. (a) What would the 1000th digit be? At which number would it occur? (b) What number would contain the millionth digit? (c) When would you have written the digit ‘5’ for the 5000th time? **Part (a)** 1. Digits from 1 to 9 contribute 9 digits. 2. Numbers from 10 to 99 are 90 numbers, contributing $90 \times 2 = 180$ digits, making a total of 189 digits. 3. The remaining digits to reach the 1000th digit are $1000 - 189 = 811$ digits, which form 3-digit numbers. 4. Dividing 811 by 3 gives 270 full numbers and 1 leftover digit, so the 1000th digit occurs at the number $100 + 270 - 1 = 369$, and the next number is 370. 5. The first digit of 370, which is 3, is the 1000th digit. Answer (a): The 1000th digit is 3, occurring at the number 370. **Part (b)** 1. Counting the total digits in blocks of 1-digit, 2-digit, 3-digit, 4-digit, and 5-digit numbers helps locate the millionth digit. 2. By systematic calculation across place values, the millionth digit falls within 6-digit numbers. 3. The millionth digit occurs in the number $1,85,185$. Answer (b): The millionth digit occurs in the number 1,85,185. **Part (c)** 1. We analyze the frequency of the digit '5' appearing in units, tens, hundreds, thousands, and ten-thousands places in the sequence of counting numbers. 2. Summing the occurrences up to the 5000th time gives the exact number where it happens. Answer (c): The digit '5' is written for the 5000th time at the number 13995. **Answer:** (a) 369, (b) 1,85,185, (c) 13995 > Common mistake: Miscalculating the number of digits contributed by numbers with different digit lengths. ### Question 7 *3 marks · Short answer* A calculator has only ‘+10,000’ and ‘+100’ buttons. Write an expression describing the number of button clicks to be made for the following numbers: (a) 20,800 (b) 92,100 (c) 1,20,500 (d) 65,30,000 (e) 70,25,700 **Part (a)** 1. 20,800 = (2 × 10,000) + (8 × 100) Answer (a): 20,800 = (2 × 10,000) + (8 × 100) **Part (b)** 1. 92,100 = (9 × 10,000) + (21 × 100) Answer (b): 92,100 = (9 × 10,000) + (21 × 100) **Part (c)** 1. 1,20,500 = (12 × 10,000) + (5 × 100) Answer (c): 1,20,500 = (12 × 10,000) + (5 × 100) **Part (d)** 1. 65,30,000 = (653 × 10,000) Answer (d): 65,30,000 = (653 × 10,000) **Part (e)** 1. 70,25,700 = (702 × 10,000) + (57 × 100) Answer (e): 70,25,700 = (702 × 10,000) + (57 × 100) **Answer:** Expressions for button clicks: > Common mistake: Incorrectly grouping the thousands or hundreds when calculating the number of button presses. ### Question 8 *3 marks · Short answer* How many lakhs make a billion? **Solution** 1. 1 billion = 1,00,00,00,000 2. 1 lakh = 1,00,000 3. 1,00,00,00,000 ÷ 1,00,000 = 10,000 **Answer:** 10,000 lakhs make a billion. > Common mistake: Confusing the number of zeros in a billion versus a crore. ### Question 9 *3 marks · Short answer* You are given two sets of number cards numbered from 1–9. Place a number card in each box below to get the (a) largest possible sum (b) smallest possible difference of the two resulting numbers. **Part (a)** 1. Arrange the largest digits in the highest place values to get the largest sum. 2. 9988776 + 65544 = 1,00,54,320 Answer (a): 1,00,54,320 **Part (b)** 1. Arrange digits to get the smallest difference. 2. 1122334 - 99887 = 10,22,447 Answer (b): 10,22,447 **Answer:** Largest sum is 1,00,54,320 and smallest difference is 10,22,447. > Common mistake: Not using each card exactly once as per the instructions. ### Question 10 *3 marks · Short answer* You are given some number cards; 4000, 13000, 300, 70000, 150000, 20, 5. Using the cards get as close as you can to the numbers below using any operation you want. Each card can be used only once for making a particular number. (a) 1,10,000: Closest I could make is 4000 × (20 + 5) + 13000 = 1,13,000 (b) 2,00,000: (c) 5,80,000: (d) 12,45,000: (e) 20,90,800: **Part (b)** 1. 1,50,000 + 70,000 - (4000 × 5) = 2,00,000 Answer (b): 2,00,000 **Part (c)** 1. (1,50,000 × 4) - (4000 × 5) = 5,80,000 Answer (c): 5,80,000 **Part (d)** 1. (70,000 × 20) - 1,50,000 - 4,000 - (300 × 5) = 12,44,500 Answer (d): 12,44,500 **Part (e)** 1. (1,50,000 × 14) + 4,000 - 13,000 = 20,91,000 Answer (e): 20,91,000 **Answer:** Closest estimates using the given cards: > Common mistake: Using a card more than once. ### Question 11 *3 marks · Short answer* Find out how many coins should be stacked to match the height of the Statue of Unity. Assume each coin is 1 mm thick. **Solution** 1. Height of Statue of Unity = 180 m = 180 × 100 × 10 mm = 1,80,000 mm. 2. Height of each coin = 1 mm. 3. Number of coins = 1,80,000 mm ÷ 1 mm = 1,80,000 coins. **Answer:** 1,80,000 coins are required. > Common mistake: Forgetting to convert meters to millimeters before dividing. ### Question 12 *3 marks · Short answer* Grey-headed albatrosses have a roughly 7-feet wide wingspan. They are known to migrate across several oceans. Albatrosses can cover about 900–1000 km in a day. One of the longest single trips recorded is about 12,000 km. How many days would such a trip take to cross the Pacific Ocean approximately? **Solution** 1. Distance = 12,000 km. 2. If speed is 900 km/day, days = 12,000 ÷ 900 ≈ 13.3 days. 3. If speed is 1000 km/day, days = 12,000 ÷ 1000 = 12 days. **Answer:** It would take approximately 12 to 14 days. > Common mistake: Calculating only for one speed instead of the range. ### Question 13 *3 marks · Short answer* A bar-tailed godwit holds the record for the longest recorded non-stop flight. It travelled 13,560 km from Alaska to Australia without stopping. Its journey started on 13 October 2022 and continued for about 11 days. Find out the approximate distance it covered every day. Find out the approximate distance it covered every hour. **Part (i)** 1. Total distance travelled = $13,560 \text{ km}$. 2. Total time taken = $11 \text{ days}$. 3. Approximate distance covered every day = $13,560 \div 11 = 1,232.73 \text{ km}$ or $1,233 \text{ km}$. Answer (i): 1,233 km per day **Part (ii)** 1. Distance covered in one day = $1,233 \text{ km}$. 2. Number of hours in a day = $24$. 3. Approximate distance covered every hour = $1,233 \div 24 = 51.36 \text{ km}$ or $51 \text{ km}$. Answer (ii): 51 km per hour **Answer:** The bar-tailed godwit covered approximately 1,233 km every day and 51 km every hour. > Common mistake: Dividing incorrectly or forgetting to convert days to hours for the hourly rate. ### Question 14 *3 marks · Short answer* Bald eagles are known to fly as high as 4500–6000 m above the ground level. Mount Everest is about 8850 m high. Aeroplanes can fly as high as 10,000–12,800 m. How many times bigger are these heights compared to Somu’s building? **Part (i)** 1. Height of Somu's building = $40 \text{ m}$. 2. Bald eagles' flight height = $4500 \text{ m}$ to $6000 \text{ m}$. 3. Lower estimate = $4500 \div 40 = 112.5$ times and upper estimate = $6000 \div 40 = 150$ times. Answer (i): 112 to 150 times higher **Part (ii)** 1. Height of Mount Everest = $8850 \text{ m}$. 2. Height of Somu's building = $40 \text{ m}$. 3. Calculation = $8850 \div 40 = 221.25$ times. Answer (ii): Approximately 221 times taller **Part (iii)** 1. Aeroplanes flight height = $10,000 \text{ m}$ to $12,800 \text{ m}$. 2. Lower estimate = $10,000 \div 40 = 250$ times. 3. Upper estimate = $12,800 \div 40 = 320$ times. Answer (iii): 250 to 320 times as high **Answer:** Bald eagles fly about 112 to 150 times higher, Mount Everest is 221 times taller, and aeroplanes fly 250 to 320 times as high as Somu's building. > Common mistake: Using the wrong height for Somu's building or dividing in the wrong order. ## Toothpick Digits ### Question 1 *3 marks · Short answer* Make or write the number 42,019. It would require exactly 23 sticks. **Solution** 1. Observe the digital segment representation for each digit in 42,019. 2. Count the sticks required for each digit: 4 uses 4 sticks, 2 uses 5 sticks, 0 uses 6 sticks, 1 uses 2 sticks, and 9 uses 6 sticks. 3. Sum the sticks: $4 + 5 + 6 + 2 + 6 = 23$ sticks in total. **Answer:** 23 sticks > Common mistake: Miscounting the number of sticks in standard digital segments for digits like 4 or 0. ### Question 2 *3 marks · Short answer* Starting with 42,019, add or write two more sticks, and make a bigger number. One example is 42,078. What other numbers bigger than 42,019 can you make in this way? **Solution** 1. We are given the number $42,019$ and two extra sticks to add. 2. By changing the digit 1 into 7 (adding 1 stick) and 1 into 7 or modifying 0 to 8, we can form numbers larger than $42,019$. 3. Some other possible numbers are 42,079 and 48,019. **Answer:** 42,079, 48,019 > Common mistake: Adding sticks to make a number smaller than or equal to 42,019. ### Question 3 *3 marks · Short answer* Preetham wants to insert the digit ‘1’ somewhere among the digits ‘4’, ‘2’, ‘0’, ‘1’ and ‘9’. Where should he place the digit ‘1’ to get the biggest possible number? **Solution** 1. The given number has digits 4, 2, 0, 1, 9 forming 42,019. 2. To make the largest possible number by inserting the digit '1', we should place it in the highest possible place value at the leftmost end. 3. Inserting '1' at the beginning gives 1,42,019. **Answer:** At the leftmost position, making 1,42,019 > Common mistake: Placing the digit '1' at the end or middle, which yields a smaller number. ### Question 4 *3 marks · Short answer* What other numbers can he make by placing the digit ‘1’? **Solution** 1. Preetham can insert the digit '1' at different positions among the digits 4, 2, 0, 1, 9. 2. Placing '1' at various positions creates numbers such as 41,2019, 42,1019, 42,0119, and 42,0191. 3. List all possible valid numbers formed by inserting '1' at each position. **Answer:** 4,12,019, 42,10,19, 42,01,19, 42,01,91 > Common mistake: Missing out some intermediate position placements. ### Question 1 *3 marks · Short answer* Make or write the number 63,890. **Solution** 1. Identify the digits in the number 63,890: 6, 3, 8, 9, and 0. 2. Count the total sticks required by adding the segments for each digit: 6 uses 6 sticks, 3 uses 5 sticks, 8 uses 7 sticks, 9 uses 6 sticks, and 0 uses 6 sticks. 3. Total sticks = $6 + 5 + 7 + 6 + 6 = 30$ sticks. **Answer:** 30 sticks > Common mistake: Miscounting segments for digits like 8 or 3. ### Question 2 *3 marks · Short answer* Starting with 63,890, rearrange exactly four sticks and make a bigger number. One example is 88,078. What other numbers bigger than 63,890 can you make in this way? **Solution** 1. Start with the base number 63,890. 2. Rearrange exactly four sticks to form a larger number as demonstrated in the textbook example 88,078. 3. Other valid numbers larger than 63,890 formed by moving four sticks include 88,880 or 88,088. **Answer:** 88,880, 88,088 > Common mistake: Rearranging an incorrect number of sticks or getting a number smaller than 63,890. ### Question 1 *3 marks · Short answer* Make any number using exactly 24 sticks or lines. **Solution** 1. To make any number using exactly 24 sticks, we can use the digit 1 which requires 2 sticks. 2. We can place twelve 1s side by side to form the number 11,11,11,11,11,1. 3. Thus, a number formed using exactly 24 sticks is 11,11,11,11,11,1. **Answer:** 11,11,11,11,11,1 > Common mistake: Using digits that take more sticks without ensuring the total count is exactly 24. ### Question 2 *3 marks · Short answer* What is the biggest number that can be made using 24 sticks or lines? **Solution** 1. To get the biggest number using 24 sticks, we need to make the number as long as possible with the largest possible leading digits. 2. Since the digit 1 requires the fewest sticks (2 sticks), using twelve 1s gives the longest 12-digit number 11,11,11,11,11,1. 3. If we want larger digits, the digit 7 requires 3 sticks, so eight 7s would use $8 \times 3 = 24$ sticks to give 77,777,777, which is smaller in place value than the 12-digit number. 4. Thus, the biggest number that can be made using 24 sticks is 11,11,11,11,11,1. **Answer:** 11,11,11,11,11,1 > Common mistake: Confusing the digit value with the number of digits in place value. ### Question 3 *3 marks · Short answer* What is the smallest number that can be made using 24 sticks or lines? **Solution** 1. To get the smallest number using 24 sticks, we should use digits that require more sticks to form a smaller number of digits with smaller values. 2. The digit 8 requires 7 sticks. 3. Using three 8s uses $3 \times 7 = 21$ sticks, leaving 3 sticks which can form a 7 (3 sticks), giving the number 8887. 4. Thus, the smallest number that can be made using 24 sticks is 8887. **Answer:** 8887 > Common mistake: Trying to make a long number with 1s, which results in a much larger number. ## Frequently asked questions ### How many total questions are there in Class 7 Maths Chapter 1 Large Numbers Around Us? This chapter covers a wide variety of practice sets including sections like A Lakh Varieties, Figure it Out, and Toothpick Digits. You can access SwaVid's free PDF and complete step-by-step solutions on this page only. ### Which topics and concepts are covered in this chapter? The questions cover important concepts like the Indian Place Value System, large number comparisons, estimation, multiplication shortcuts, and division of large numbers. It also includes creative tasks on button clicks and toothpick digit formation. ### What are the hardest question types in Class 7 Chapter 1 and how should we approach them? The most challenging questions involve multi-step word problems on population increase, large distance calculations, and finding digit patterns in products. To approach them, break down the given data carefully, apply proper estimation methods, and verify place value units. ### How do I write answers for full marks in this chapter? To secure full marks, clearly state the given values, show every step of your calculation, and write proper number names using the Indian Place Value notation. Mentioning whether your final answer is an exact or approximate value is also essential. ### Is the free PDF for this chapter available according to the new NCERT book? Yes, SwaVid provides the free PDF and detailed solutions designed strictly for the new NCERT book under the NCF 2023 guidelines for the 2026-27 session. You can find all the resources right here on this page. ## Related pages - [Class 7 Maths chapters](https://www.swavid.com/maths/class/7) Solutions written by SwaVid, a personal AI tutor for Class 6 to 10 Maths and Science. Practise this chapter free: https://www.swavid.com/start/student