Large Numbers Around Us Class 7 Solutions Ganita Prakash Maths Chapter 1

Welcome to our comprehensive solution guide for Chapter 1: Large Numbers Around Us. Whether you are a student double-checking your work or a parent helping with homework, we have extracted and solved every question from the chapter, organized clearly by page number. Let’s dive in!

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Page 1

Question: What do you think? Guess. (Regarding tasting 1 lakh varieties of rice in a 100-year lifetime by trying 3 varieties each day)

Answer: Yes! If you ate 3 varieties every single day, you would actually consume over 1 lakh varieties. (365 × 100 × 3 = 1,09,500).

Page 2

Question: Observe the pattern and fill in the boxes given below.

Answer:

• The largest 3-digit number is 999. + 1 = The smallest 4-digit number is 1,000.
• The largest 4-digit number is 9,999. + 1 = The smallest 5-digit number is 10,000.
• The largest 5-digit number is 99,999. + 1 = The smallest 6-digit number is 1,00,000.

Question: 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.

Answer: Number of days in 100 years = 365 × 100 = 36,500 days (ignoring leap years). 36,500 days × 3 varieties/day = 1,09,500 varieties. Yes, they would be able to taste all 1 lakh varieties.

Page 3

Question: Choose a number for y. How close to one lakh is the number of days in y years, for the y of your choice?

Answer: Let y = 274. The number of days is 274 × 365 = 1,00,010 days. This is exceptionally close to one lakh.

Figure it Out

1. How much less than one lakh is 75,000?
Answer: 1,00,000 – 75,000 = 25,000 less.

2. How much more than one lakh is 1,06,000?
Answer: 1,06,000 – 1,00,000 = 6,000 more.

3. By how much did the population of Chintamani increase from 2011 to 2024?
Answer: 1,06,000 – 75,000 = 31,000.

Question: Which is taller – The Statue of Unity or this building? How much taller?

Answer: The Statue of Unity is 180 meters tall. The book states the statue is “close to 4 times the height of Somu’s building”, meaning the building is roughly 180 ÷ 4 = 45 meters. The Statue is 180 – 45 = 135 meters taller.

Question: How much taller is the Kunchikal waterfall than Somu’s building?

Answer: The waterfall is 450 meters. 450 – 45 = 405 meters taller.

Question: How many floors should Somu’s building have to be as high as the waterfall?

Answer: If one floor is 4 times Somu’s height (4 meters), the building needs 450 ÷ 4 = 112.5 (or 113) floors.

Page 4

Question: How do you view a lakh – is a lakh big or small?

Answer: It depends on the context! One lakh is huge when counting days lived (274 years), but very small when counting the hairs on a human head.

Question: Write each of the numbers given below in words:

Answer:

• (a) 3,00,600: Three lakh six hundred
• (b) 5,04,085: Five lakh four thousand eighty five
• (c) 27,30,000: Twenty seven lakh thirty thousand
• (d) 70,53,138: Seventy lakh fifty three thousand one hundred thirty eight

Page 5

Question: Write the corresponding number in the Indian place value system…

Answer:

• (a) One lakh twenty three thousand four hundred and fifty six: 1,23,456
• (b) Four lakh seven thousand seven hundred and four: 4,07,704
• (c) Fifty lakhs five thousand and fifty: 50,05,050
• (d) Ten lakhs two hundred and thirty five: 10,00,235

Question: 1. The Thoughtful Thousands only has a +1000 button. How many times should it be pressed to show:

Answer:

• (a) Three thousand? 3 times
• (b) 10,000? 10 times
• (c) Fifty three thousand? 53 times
• (d) 90,000? 90 times
• (e) One Lakh? 100 times
• (f) ? 153 times → 1,53,000
• (g) How many thousands are required to make one lakh? 100

Question: 2. The Tedious Tens only has a +10 button. How many times should it be pressed to show:

Answer:

• (a) Five hundred? 50 times
• (b) 780? 78 times
• (c) 1000? 100 times
• (d) 3700? 370 times
• (e) 10,000? 1,000 times
• (f) One lakh? 10,000 times
• (g) ? 435 times → 4,350

Question: 3. The Handy Hundreds only has a +100 button. How many times should it be pressed to show:

Answer:

• (a) Four hundred? 4 times
• (b) 3,700? 37 times

Page 6

Question: 3. (Handy Hundreds continued) How many times should it be pressed to show:

Answer:

• (c) 10,000? 100 times
• (d) Fifty three thousand? 530 times
• (e) 90,000? 900 times
• (f) 97,600? 976 times
• (g) 1,00,000? 1000 times
• (h) ? 582 times → 58,200
• (i) How many hundreds are required to make ten thousand? 100
• (j) How many hundreds are required to make one lakh? 1000
• (k) Is this statement true? (Handy Hundreds claims it can show numbers Tens and Thousands can’t).
  False. Any number Handy Hundreds can show (multiples of 100) can also be shown by Tedious Tens (multiples of 10).

Question: 4. Creative Chitti: To get 321, it presses +10 thirty two times and +1 once. Will it get 321?

Answer: Yes, (32 × 10) + 1 = 321.

Question: 5. Find a different way to get 5072 and write an expression for the same.

Answer: You can use 50 hundreds and 72 ones: (50 × 100) + (72 × 1) = 5072.

Figure it Out

For each number given below, write expressions for at least two different ways to obtain the number:

• (a) 8300: (8 × 1000) + (3 × 100) OR (83 × 100)
• (b) 40629: (4 × 10000) + (6 × 100) + (2 × 10) + (9 × 1) OR (406 × 100) + (29 × 1)
• (c) 56354: (5 × 10000) + (6 × 1000) + (3 × 100) + (5 × 10) + (4 × 1) OR (563 × 100) + (54 × 1)

Page 7

Figure it Out (Continued)

• (d) 66666: (6 × 10000) + (66 × 100) + (66 × 1) OR (66 × 1000) + (666 × 1)
• (e) 367813: (3 × 100000) + (67 × 1000) + (813 × 1) OR (36 × 10000) + (78 × 100) + (13 × 1)

Question: (a) You have to make exactly 30 button presses. What is the largest 3-digit number you can make? What is the smallest?

Largest Answer: To maximize a 3-digit number, prioritize the +100 button without going over 30 presses or 999. Use 9 presses of +100 (900). You have 21 presses left. Use 8 presses of +10 (80) and 13 presses of +1 (13). Sum = 900 + 80 + 13 = 993.

Smallest Answer: To minimize a 3-digit number, we want the lowest value over 100 using 30 clicks. Use 0 presses of +100. Use 8 presses of +10 (80) and 22 presses of +1 (22). Sum = 80 + 22 = 102.

Question: (b) 997 can be made using 25 clicks. Can you make 997 with a different number of clicks?

Answer: Yes. (8 × 100) + (19 × 10) + (7 × 1) = 997 (which is 34 clicks).

Question: How can we get the numbers (a) 5072, (b) 8300 using as few button clicks as possible?

Answer:
5072: 5 thousands, 0 hundreds, 7 tens, 2 ones (5 + 0 + 7 + 2 = 14 clicks).
8300: 8 thousands, 3 hundreds (8 + 3 = 11 clicks).

Figure it Out: Smallest Button Clicks

1. Find out how to get each number by making the smallest number of button clicks…

• 40629: 4 + 0 + 6 + 2 + 9 = 21 clicks.
• 56354: 5 + 6 + 3 + 5 + 4 = 23 clicks.
• 66666: 6 + 6 + 6 + 6 + 6 = 30 clicks.
• 367813: 3 + 6 + 7 + 8 + 1 + 3 = 28 clicks.

Question: 2. Do you see any connection between each number and the corresponding smallest number of button clicks?

Answer: Yes, the smallest number of clicks is simply the sum of all the digits in the number.

Question: 3. Why is this so?

Answer: Because the most efficient way to build a number base-10 is to match the face value of the digit at each place value.

Page 8

Question: What if we press the +10,00,000 button ten times? What number will come up? How many zeroes will it have?

Answer: 1,00,00,000 (1 crore). It has 7 zeroes.

Question: How many zeros does a thousand lakh have?

Answer: 1000 × 1,00,000 = 10,00,00,000. It has 8 zeroes.

Page 9

Question: How many zeros does a hundred thousand have?

Answer: 100,000 has 5 zeros.

Figure it Out

1. Read the following numbers in Indian place value notation and write their number names in both the Indian and American systems:

• (a) 4050678: Indian: 40,50,678 (Forty lakh fifty thousand six hundred seventy eight). American: 4,050,678 (Four million fifty thousand six hundred seventy eight).
• (b) 48121620: Indian: 4,81,21,620 (Four crore eighty one lakh twenty one thousand six hundred twenty). American: 48,121,620 (Forty eight million one hundred twenty one thousand six hundred twenty).
• (c) 20022002: Indian: 2,00,22,002 (Two crore twenty two thousand two). American: 20,022,002 (Twenty million twenty two thousand two).
• (d) 246813579: Indian: 24,68,13,579 (Twenty four crore sixty eight lakh thirteen thousand five hundred seventy nine). American: 246,813,579 (Two hundred forty six million eight hundred thirteen thousand five hundred seventy nine).
• (e) 345000543: Indian: 34,50,00,543 (Thirty four crore fifty lakh five hundred forty three). American: 345,000,543 (Three hundred forty five million five hundred forty three).
• (f) 1020304050: Indian: 1,02,03,04,050 (One arab two crore three lakh four thousand fifty). American: 1,020,304,050 (One billion twenty million three hundred four thousand fifty).

2. Write the following numbers in Indian place value notation:

• (a) One crore one lakh one thousand ten: 1,01,01,010
• (b) One billion one million one thousand one: 1,00,10,01,001
• (c) Ten crore twenty lakh thirty thousand forty: 10,20,30,040
• (d) Nine billion eighty million seven hundred thousand six hundred: 9,08,07,00,600

3. Compare and write ‘<‘, ‘>’ or ‘=’:

• (a) 30 thousand < 3 lakhs
• (b) 500 lakhs > 5 million (5 million is 50 lakhs)
• (c) 800 thousand < 8 million
• (d) 640 crore < 60 billion (60 billion is 6000 crore)

Page 10

Question: 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.

• (a) Round up: Budgeting money for an upcoming trip, or ordering catering for a party to ensure enough supplies.
• (b) Round down: Calculating how much fuel you have left in a car tank (to be safe), or claiming business expenses.
• (c) Either: Discussing the population of your town in casual conversation, or stating how many followers a YouTube channel has.
• (d) Exact: Dialing emergency numbers, administering medication dosages, or tracking bank account balances.

Page 11

Question: Write the five nearest neighbours for these numbers:

(a) 3,87,69,957

• Nearest 1,000: 3,87,70,000
• Nearest 10,000: 3,87,70,000
• Nearest Lakh: 3,88,00,000
• Nearest 10 Lakh: 3,90,00,000
• Nearest Crore: 4,00,00,000

(b) 29,05,32,481

• Nearest 1,000: 29,05,32,000
• Nearest 10,000: 29,05,30,000
• Nearest Lakh: 29,05,00,000
• Nearest 10 Lakh: 29,10,00,000
• Nearest Crore: 29,00,00,000

Question: 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?

Answer: For the tightest condition (nearest 1,000) to round to 5,00,00,000, the number must be between 4,99,99,500 and 5,00,00,499. There are exactly 1,000 such whole numbers.

Math Talk

1. 4,63,128 + 4,19,682

• (a) Are these estimates correct? Whose estimate is closer to the sum? Yes, both bound the sum. Estu’s estimate (9,00,000) is closer than Roxie’s (8,00,000) because the actual sum is ∼8.82 lakhs.
• (b) Will the sum be greater than 8,50,000 or less than 8,50,000? Why do you think so? Greater. 4.6 lakhs + 4.1 lakhs is already 8.7 lakhs, which is larger than 8.5.
• (c) Will the sum be greater than 8,83,128 or less than 8,83,128? Why? Less. Because 4,63,128 + 4,20,000 = 8,83,128. Since we are adding less than 4,20,000 (only 4,19,682), the sum must be less.
• (d) Exact value: 8,82,810

2. 14,63,128 – 4,90,020

• (a) Are these estimates correct? Whose estimate is closer to the difference? Yes, they bound it. The exact difference is roughly 9.73 lakhs, so Roxie’s estimate (10,00,000) is closer than Estu’s (9,00,000).
• (b) Will the difference be greater than 9,50,000 or less than 9,50,000? Why? Greater. 14.6 – 4.9 = 9.7, which is > 9.5.

Page 12

(Math Talk Continued)

• (c) Will the difference be greater than 9,63,128 or less than 9,63,128? Why do you think so? Greater. If we subtracted exactly 5,00,000, we’d get 9,63,128. Since we are subtracting a smaller number (4,90,020), the resulting difference will be larger.
• (d) Exact value: 9,73,108

Page 13

Question: From the information given in the table, answer the following questions:

1. What is your general observation about this data? Most top-tier Indian cities experienced significant, rapid population growth between 2001 and 2011.
2. What is an appropriate title for the above table? “Population Growth of the Top 20 Largest Indian Cities (2001–2011)”.
3. How much is the population of Pune in 2011? Approximately, by how much has it increased compared to 2001? 2011 population: 31,15,431. It increased by approximately 6 lakhs (exact: 5,76,958).
4. Which city’s population increased the most between 2001 and 2011? Bengaluru (an increase of 41,24,644).
5. Are there cities whose population has almost doubled? Which are they? Yes: Bengaluru, Surat, and Vadodara (which more than doubled).
6. By what number should we multiply Patna’s population to get a number/population close to that of Mumbai? 1,24,42,373 ÷ 16,84,222 ≈ 7.38. So you would multiply it by roughly 7.4.

Question: 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?

Answer: Because the fraction 10/2 is mathematically equal to 5. So computing x × 5 is functionally identical to x × (10 ÷ 2).

Page 14

Figure it Out

1. Find quick ways to calculate these products:

• (a) 2 × 1768 × 50 → (2 × 50) × 1768 → 100 × 1768 = 1,76,800
• (b) 72 × 125 → 72 × (1000 / 8) → (72 / 8) × 1000 → 9 × 1000 = 9,000
• (c) 125 × 40 × 8 × 25 → (125 × 8) × (40 × 25) → 1000 × 1000 = 10,00,000

2. Calculate these products quickly:

• (a) 25 × 12 → 12 × 100 / 4 = 300
• (b) 25 × 240 → 240 × 100 / 4 = 6000
• (c) 250 × 120 → 120 × 1000 / 4 = 30,000
• (d) 2500 × 12 → 12 × 10000 / 4 = 30,000
• (e) _ × _ = 120000000 → For example, 12000 × 10000.

How Long is the Product? Evaluate to find the pattern:

• 11 × 11 = 121
• 111 × 111 = 12321
• 1111 × 1111 = 1234321
• 66 × 61 = 4026
• 666 × 661 = 440226
• 6666 × 6661 = 44402226
• 3 × 5 = 15
• 33 × 35 = 1155
• 333 × 335 = 111555
• 101 × 101 = 10201
• 102 × 102 = 10404
• 103 × 103 = 10609

Page 15

Question: Is there any connection between the numbers being multiplied and the number of digits in their product?

Answer: Yes. If you multiply a number with D1 digits by a number with D2 digits, the product will always have either (D1 + D2) digits or (D1 + D2 – 1) digits.

Question: Roxie says that the product of two 2-digit numbers can only be a 3- or a 4-digit number. Is she correct?

Answer: Yes. The smallest product (10 × 10) is 100 (3 digits). The largest product (99 × 99) is 9801 (4 digits).

Question: Should we try all possible multiplications… Or is there a better way?

Answer: Finding the boundaries (the smallest and largest possible products) like Roxie did is the much better, logical way.

Question: Can multiplying a 3-digit number with another 3-digit number give a 4-digit number?

Answer: No. The smallest possible is 100 × 100 = 10,000 (which is 5 digits).

Question: Can multiplying a 4-digit number with a 2-digit number give a 5-digit number?

Answer: Yes. Example: 1000 × 10 = 10,000 (5 digits).

Complete the Multiplication Pattern Table:

• 5-digit × 5-digit = 9-digit or 10-digit
• 8-digit × 3-digit = 10-digit or 11-digit
• 12-digit × 13-digit = 24-digit or 25-digit

Page 16

Question: Calculate the product: 1250 × 380

Answer: 4,75,000. (Indian: Four lakh seventy five thousand. American: Four hundred seventy five thousand).

Question: How many years did he live to compose so many songs? At what age did he start composing? If he composed 4,75,000 songs, how many songs per year did he have to compose?

Answer: He would have had to compose an impossibly high number! If he composed for an incredible 60 years, 4,75,000 ÷ 60 ≈ 7,916 songs a year (which is over 21 songs every single day).

Question: Calculate the product: 2100 × 70,000

Answer: 14,70,00,000. (Indian: Fourteen crore seventy lakhs. American: One hundred forty seven million).

Page 17

Question: Calculate the product: 6400 × 62,500

Answer: 40,00,00,00,000. (Indian: Forty crore. American: Four hundred million).

Question: Calculate the quotient: 13,95,000 ÷ 150

Answer: 9,300.

Page 18

Question: Calculate the quotient: 10,50,00,000 ÷ 700

Answer: 1,50,000. (Indian: One lakh fifty thousand. American: One hundred fifty thousand).

Question: Calculate the quotient: 52,00,00,00,000 ÷ 130

Answer: 40,00,00,00,000. (Indian: Forty crore. American: Four hundred million).

Page 19

Question: Could the entire population of Mumbai fit into 1 lakh buses?

Answer: Assuming 50 people per bus, 1 lakh buses hold 50 lakh people. Mumbai’s population is over 1.24 crore, so no, they could not fit.

Question: The RMS Titanic ship carried about 2500 passengers. Can the population of Mumbai fit into 5000 such ships?

Answer: 5000 × 2500 = 1,25,00,000 (1.25 crore). Yes! 1.25 crore is slightly greater than 1.24 crore.

Question: If I could travel 100 kilometers every day, could I reach the Moon in 10 years?

Answer: 100 × 365 × 10 = 3,65,000 km. Since the moon is 3,84,400 km away, no, you would fall short by almost 20,000 km.

Question: Find out if you can reach the Sun in a lifetime, if you travel 1000 kilometers every day.

Answer: The sun is roughly 14,70,00,000 km away. 14,70,00,000 ÷ (1000 × 365) ≈ 402 years. No, it would take multiple lifetimes.

Question: (a) If a single sheet of paper weighs 5 grams, could you lift one lakh sheets of paper together at the same time?

Answer: 1,00,000 × 5g = 5,00,000g = 500kg. No, a human cannot safely lift 500 kg.

Question: (b) If 250 babies are born every minute across the world, will a million babies be born in a day?

Answer: 250 × 60 min × 24 hr = 3,60,000 babies. No, it falls short of a million.

Question: (c) Can you count 1 million coins in a day? Assume you can count 1 coin every second.

Answer: Seconds in a day: 60 × 60 × 24 = 86,400 seconds. No.

Figure it Out

1. Using all digits from 0-9 exactly once (the first digit cannot be 0) to create a 10-digit number…

• (a) Largest multiple of 5: 9876543210
• (b) Smallest even number: 1023456798

Page 20

Question: 2. Give a 7-digit number name which has the maximum number of letters.

Answer: 77,77,777 (Seventy seven lakh seventy seven thousand seven hundred seventy seven – 66 letters).

Question: 3. Write a 9-digit number where exchanging any two digits results in a bigger number. How many such numbers exist?

Answer: The digits must be strictly ascending. 123456789. Only 1 such number exists.

Question: 4. Strike out 10 digits from the number 12345123451234512345 so that the remaining number is as large as possible.

Answer: You want to leave the highest possible numbers starting from the left. Strike ‘1234’ (leaves 5). Strike ‘1234’ (leaves 5). Strike ’12’ (leaves 3). Leftover: 5534512345.

Question: 5. How far do you have to count to find two consecutive numbers which do not share an English letter in common?

Answer: You must count to 5 and 6 (“five” and “six” share no letters).

Question: 6. (Sequence 1234567891011…)

Answer: (a) What would the 1000th digit be? → Single digits take 9 places. Double digits take 180 places. Total so far = 189. We need 1000 – 189 = 811 more digits. 811 ÷ 3 = 270 full 3-digit numbers with 1 remainder. So, 100 + 269 = 369. The next number is 370. The first digit of 370 is 3.

Question: 7. A calculator has only ‘+10,000’ and ‘+100’ buttons. Write an expression describing the number of button clicks…

• (a) 20,800: (2 × 10,000) + (8 × 100)
• (b) 92,100: (9 × 10,000) + (21 × 100)
• (c) 1,20,500: (12 × 10,000) + (5 × 100)
• (d) 65,30,000: (653 × 10,000)
• (e) 70,25,700: (702 × 10,000) + (57 × 100)

Question: 8. How many lakhs make a billion?

Answer: 1,00,00,00,000 ÷ 1,00,000 = 10,000 lakhs.

Question: 9. Place a number card in each box below (Cards 1-9 used once).

• (a) Largest sum: 97531 + 8642
• (b) Smallest difference: 51234 – 49876

Page 21

Question: 10. You are given some number cards… (4000, 13000, 300, 70000, 150000, 20, 5). Get as close as you can to the numbers below using any operation.

• (b) 2,00,000: 150000 + 70000 – 13000 – 4000 – 3000 = 203000
• (c) 5,80,000: (70000 × 5) + 150000 = 500000
• (d) 12,45,000: (150000 × 5) + (70000 × 4000 ÷ 300) …

Question: 11. Find out how many coins should be stacked to match the height of the Statue of Unity. Assume each coin is 1 mm thick.

Answer: 180 meters = 180,000 mm. 1,80,000 coins.

Question: 12. How many days would such a trip take to cross the Pacific Ocean approximately?

Answer: 12,000 km ÷ 1,000 km/day = 12 days.

Question: 13. Find out the approximate distance it covered every day. Find out the approximate distance it covered every hour.

Answer: 13,560 km ÷ 11 days = 1,232 km/day. 1,232 ÷ 24 ≈ 51 km/hour.

Question: 14. How many times bigger are these heights compared to Somu’s building?

Answer: Using 45m for the building:

• Eagles (6000 ÷ 45): ∼133 times
• Everest (8850 ÷ 45): ∼196 times
• Planes (12800 ÷ 45): ∼284 times

Page 23

Question: 1. Make or write the number 42,019. It would require exactly 23 sticks. 2. Starting with 42,019, add or write two more sticks, and make a bigger number. One example is 42,078. What other numbers… can you make in this way?

Answer: You can place a ‘1’ anywhere (it uses 2 sticks). E.g., 142,019 or 421,019.

Question: 3. 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?

Answer: Between the 4 and the 2. 412019 (assuming we insert a stick character). But mathematically inserting 1 anywhere, 421019 is greater.

Question: 5. Starting with 63,890, rearrange exactly four sticks and make a bigger number.

Answer: Moving a stick from the 6 to make a 5, building an 8 out of the 3… 88,078 is given as an example, but you could also do 93,890 by re-arranging the sticks in the first digit.

Question: 1. Make any number using exactly 24 sticks. 2. What is the biggest number that can be made using 24 sticks? 3. What is the smallest?

Answer:

• Biggest: Use the digit ‘1’ (it uses the fewest sticks – 2). So, 24 ÷ 2 = 12 ones. 111,111,111,111.
• Smallest: Use the digit ‘8’ (it uses the most sticks – 7) to keep the number short. Three ‘8’s = 21 sticks. You have 3 sticks left, which makes a ‘7’. Arranged small: 7888.

• Question: Choose a number for y. How close to one lakh is the number of days in y years, for the y of your choice?
  • Answer: Let $y = 274$. The number of days is $274 \times 365 = 1,00,010$ days. This is exceptionally close to one lakh.
• Figure it Out
  • 1. How much less than one lakh is 75,000?
    • Answer: $1,00,000 – 75,000 = 25,000$ less.
  • 2. How much more than one lakh is 1,06,000?
    • Answer: $1,06,000 – 1,00,000 = 6,000$ more.
  • 3. By how much did the population of Chintamani increase from 2011 to 2024?
    • Answer: $1,06,000 – 75,000 = 31,000$.
• Question: Which is taller – The Statue of Unity or this building? How much taller?
  • Answer: The Statue of Unity is 180 meters tall. The book states the statue is “close to 4 times the height of Somu’s building”, meaning the building is roughly $180 \div 4 = 45$ meters. The Statue is $180 – 45 = 135$ meters taller.
• Question: How much taller is the Kunchikal waterfall than Somu’s building?
  • Answer: The waterfall is 450 meters. $450 – 45 = 405$ meters taller.
• Question: How many floors should Somu’s building have to be as high as the waterfall?
  • Answer: If one floor is 4 times Somu’s height (4 meters), the building needs $450 \div 4 = 112.5$ (or 113) floors.

Page 4

• Question: How do you view a lakh – is a lakh big or small?
  • Answer: It depends on the context! One lakh is huge when counting days lived (274 years) , but very small when counting the hairs on a human head.
• Question: Write each of the numbers given below in words:
  • (a) 3,00,600: Three lakh six hundred
  • (b) 5,04,085: Five lakh four thousand eighty five
  • (c) 27,30,000: Twenty seven lakh thirty thousand
  • (d) 70,53,138: Seventy lakh fifty three thousand one hundred thirty eight

Page 5

• Question: Write the corresponding number in the Indian place value system…
  • (a) One lakh twenty three thousand four hundred and fifty six: 1,23,456
  • (b) Four lakh seven thousand seven hundred and four: 4,07,704
  • (c) Fifty lakhs five thousand and fifty: 50,05,050
  • (d) Ten lakhs two hundred and thirty five: 10,00,235
• Question: 1. 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? 10 times
  • (c) Fifty three thousand? 53 times
  • (d) 90,000? 90 times
  • (e) One Lakh? 100 times
  • (f) ? 153 times -> 1,53,000
  • (g) How many thousands are required to make one lakh? 100
• Question: 2. The Tedious Tens only has a +10 button. How many times should it be pressed to show:
  • (a) Five hundred? 50 times
  • (b) 780? 78 times
  • (c) 1000? 100 times
  • (d) 3700? 370 times
  • (e) 10,000? 1,000 times
  • (f) One lakh? 10,000 times
  • (g) ? 435 times -> 4,350
• Question: 3. The Handy Hundreds only has a +100 button. How many times should it be pressed to show:
  • (a) Four hundred? 4 times
  • (b) 3,700? 37 times

Page 6

• (Handy Hundreds continued)
  • (c) 10,000? 100 times
  • (d) Fifty three thousand? 530 times
  • (e) 90,000? 900 times
  • (f) 97,600? 976 times
  • (g) 1,00,000? 1000 times
  • (h) ? 582 times -> 58,200
  • (i) How many hundreds are required to make ten thousand? 100
  • (j) How many hundreds are required to make one lakh? 1000
  • (k) Is this statement true? (Handy Hundreds claims it can show numbers Tens and Thousands can’t).
    • Answer: False. Any number Handy Hundreds can show (multiples of 100) can also be shown by Tedious Tens (multiples of 10).
• Question: 4. Creative Chitti: To get 321, it presses +10 thirty two times and +1 once. Will it get 321?
  • Answer: Yes, $(32 \times 10) + 1 = 321$.
• Question: 5. Find a different way to get 5072 and write an expression for the same.
  • Answer: You can use 50 hundreds and 72 ones: $(50 \times 100) + (72 \times 1) = 5072$.
• Figure it Out
  • For each number given below, write expressions for at least two different ways to obtain the number:
    • (a) 8300: $(8 \times 1000) + (3 \times 100)$ OR $(83 \times 100)$
    • (b) 40629: $(4 \times 10000) + (6 \times 100) + (2 \times 10) + (9 \times 1)$ OR $(406 \times 100) + (29 \times 1)$
    • (c) 56354: $(5 \times 10000) + (6 \times 1000) + (3 \times 100) + (5 \times 10) + (4 \times 1)$ OR $(563 \times 100) + (54 \times 1)$

Page 7

• (Figure it Out Continued)
  • (d) 66666: $(6 \times 10000) + (66 \times 100) + (66 \times 1)$ OR $(66 \times 1000) + (666 \times 1)$
  • (e) 367813: $(3 \times 100000) + (67 \times 1000) + (813 \times 1)$ OR $(36 \times 10000) + (78 \times 100) + (13 \times 1)$
• Question: (a) You have to make exactly 30 button presses. What is the largest 3-digit number you can make? What is the smallest?
  • Largest Answer: To maximize a 3-digit number, prioritize the $+100$ button without going over 30 presses or 999. Use 9 presses of $+100$ (900). You have 21 presses left. Use 8 presses of $+10$ (80) and 13 presses of $+1$ (13). Sum = $900 + 80 + 13 =$ 993.
  • Smallest Answer: To minimize a 3-digit number, we want the lowest value over 100 using 30 clicks. Use 0 presses of $+100$. Use 8 presses of $+10$ (80) and 22 presses of $+1$ (22). Sum = $80 + 22 =$ 102.
• Question: (b) 997 can be made using 25 clicks. Can you make 997 with a different number of clicks?
  • Answer: Yes. $(8 \times 100) + (19 \times 10) + (7 \times 1) = 997$ (which is 34 clicks).
• Question: How can we get the numbers (a) 5072, (b) 8300 using as few button clicks as possible?
  • Answer: * 5072: 5 thousands, 0 hundreds, 7 tens, 2 ones ($5+0+7+2 = 14$ clicks).
    • 8300: 8 thousands, 3 hundreds ($8+3 = 11$ clicks).
• Figure it Out: 1. Find out how to get each number by making the smallest number of button clicks…
  • 40629: $4+0+6+2+9 =$ 21 clicks.
  • 56354: $5+6+3+5+4 =$ 23 clicks.
  • 66666: $6+6+6+6+6 =$ 30 clicks.
  • 367813: $3+6+7+8+1+3 =$ 28 clicks.
• Question: 2. Do you see any connection between each number and the corresponding smallest number of button clicks?
  • Answer: Yes, the smallest number of clicks is simply the sum of all the digits in the number.
• Question: 3. Why is this so?
  • Answer: Because the most efficient way to build a number base-10 is to match the face value of the digit at each place value.

Page 8

• Question: What if we press the +10,00,000 button ten times? What number will come up? How many zeroes will it have?
  • Answer: 1,00,00,000 (1 crore). It has 7 zeroes.
• Question: How many zeros does a thousand lakh have?
  • Answer: $1000 \times 1,00,000 = 10,00,00,000$. It has 8 zeroes.

Page 9

• Question: How many zeros does a hundred thousand have?
  • Answer: $100,000$ has 5 zeros.
• Figure it Out
  • 1. Read the following numbers in Indian place value notation and write their number names in both the Indian and American systems:
    • (a) 4050678: Indian: 40,50,678 (Forty lakh fifty thousand six hundred seventy eight). American: 4,050,678 (Four million fifty thousand six hundred seventy eight).
    • (b) 48121620: Indian: 4,81,21,620 (Four crore eighty one lakh twenty one thousand six hundred twenty). American: 48,121,620 (Forty eight million one hundred twenty one thousand six hundred twenty).
    • (c) 20022002: Indian: 2,00,22,002 (Two crore twenty two thousand two). American: 20,022,002 (Twenty million twenty two thousand two).
    • (d) 246813579: Indian: 24,68,13,579 (Twenty four crore sixty eight lakh thirteen thousand five hundred seventy nine). American: 246,813,579 (Two hundred forty six million eight hundred thirteen thousand five hundred seventy nine).
    • (e) 345000543: Indian: 34,50,00,543 (Thirty four crore fifty lakh five hundred forty three). American: 345,000,543 (Three hundred forty five million five hundred forty three).
    • (f) 1020304050: Indian: 1,02,03,04,050 (One arab two crore three lakh four thousand fifty). American: 1,020,304,050 (One billion twenty million three hundred four thousand fifty).
  • 2. Write the following numbers in Indian place value notation:
    • (a) One crore one lakh one thousand ten: 1,01,01,010
    • (b) One billion one million one thousand one: 1,00,10,01,001
    • (c) Ten crore twenty lakh thirty thousand forty: 10,20,30,040
    • (d) Nine billion eighty million seven hundred thousand six hundred: 9,08,07,00,600
  • 3. Compare and write ‘<‘, ‘>’ or ‘=’:
    • (a) 30 thousand < 3 lakhs
    • (b) 500 lakhs > 5 million (5 million is 50 lakhs)
    • (c) 800 thousand < 8 million
    • (d) 640 crore < 60 billion (60 billion is 6000 crore)

Page 10

• Question: 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.
  • (a) Round up: Budgeting money for an upcoming trip, or ordering catering for a party to ensure enough supplies.
  • (b) Round down: Calculating how much fuel you have left in a car tank (to be safe), or claiming business expenses.
  • (c) Either: Discussing the population of your town in casual conversation, or stating how many followers a YouTube channel has.
  • (d) Exact: Dialing emergency numbers, administering medication dosages, or tracking bank account balances.

Page 11

• Question: Write the five nearest neighbours for these numbers:
  • (a) 3,87,69,957
    • Nearest 1,000: 3,87,70,000
    • Nearest 10,000: 3,87,70,000
    • Nearest Lakh: 3,88,00,000
    • Nearest 10 Lakh: 3,90,00,000
    • Nearest Crore: 4,00,00,000
  • (b) 29,05,32,481
    • Nearest 1,000: 29,05,32,000
    • Nearest 10,000: 29,05,30,000
    • Nearest Lakh: 29,05,00,000
    • Nearest 10 Lakh: 29,10,00,000
    • Nearest Crore: 29,00,00,000
• Question: 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?
  • Answer: For the tightest condition (nearest 1,000) to round to 5,00,00,000, the number must be between 4,99,99,500 and 5,00,00,499. There are exactly 1,000 such whole numbers.
• Math Talk: 1. 4,63,128 + 4,19,682
  • (a) Are these estimates correct? Whose estimate is closer to the sum? Yes, both bound the sum. Estu’s estimate (9,00,000) is closer than Roxie’s (8,00,000) because the actual sum is $\sim 8.82$ lakhs.
  • (b) Will the sum be greater than 8,50,000 or less than 8,50,000? Why do you think so? Greater. $4.6 \text{ lakhs} + 4.1 \text{ lakhs}$ is already $8.7 \text{ lakhs}$, which is larger than $8.5$.
  • (c) Will the sum be greater than 8,83,128 or less than 8,83,128? Why? Less. Because $4,63,128 + 4,20,000 = 8,83,128$. Since we are adding less than 4,20,000 (only 4,19,682), the sum must be less.
  • (d) Exact value: 8,82,810
• Math Talk: 2. 14,63,128 – 4,90,020
  • (a) Are these estimates correct? Whose estimate is closer to the difference? Yes, they bound it. The exact difference is roughly $9.73$ lakhs, so Roxie’s estimate (10,00,000) is closer than Estu’s (9,00,000).
  • (b) Will the difference be greater than 9,50,000 or less than 9,50,000? Why? Greater. $14.6 – 4.9 = 9.7$, which is $> 9.5$.

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• (Math Talk Continued) * (c) Will the difference be greater than 9,63,128 or less than 9,63,128? Why do you think so? Greater. If we subtracted exactly 5,00,000, we’d get 9,63,128. Since we are subtracting a smaller number (4,90,020), the resulting difference will be larger.
  • (d) Exact value: 9,73,108

Page 13

• Question: From the information given in the table, answer the following questions:
  • 1. What is your general observation about this data? Most top-tier Indian cities experienced significant, rapid population growth between 2001 and 2011.
  • 2. What is an appropriate title for the above table? “Population Growth of the Top 20 Largest Indian Cities (2001–2011)”.
  • 3. How much is the population of Pune in 2011? Approximately, by how much has it increased compared to 2001? 2011 population: 31,15,431. It increased by approximately 6 lakhs (exact: $5,76,958$).
  • 4. Which city’s population increased the most between 2001 and 2011? Bengaluru (an increase of $41,24,644$).
  • 5. Are there cities whose population has almost doubled? Which are they? Yes: Bengaluru, Surat, and Vadodara (which more than doubled).
  • 6. By what number should we multiply Patna’s population to get a number/population close to that of Mumbai? $1,24,42,373 \div 16,84,222 \approx 7.38$. So you would multiply it by roughly 7.4.
• Question: 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?
  • Answer: Because the fraction $10/2$ is mathematically equal to 5. So computing $x \times 5$ is functionally identical to $x \times (10 \div 2)$.

Page 14

• Figure it Out: 1. Find quick ways to calculate these products:
  • (a) $2 \times 1768 \times 50 \implies (2 \times 50) \times 1768 \implies 100 \times 1768 =$ 1,76,800
  • (b) $72 \times 125 \implies 72 \times (1000/8) \implies (72/8) \times 1000 \implies 9 \times 1000 =$ 9,000
  • (c) $125 \times 40 \times 8 \times 25 \implies (125 \times 8) \times (40 \times 25) \implies 1000 \times 1000 =$ 10,00,000
• Figure it Out: 2. Calculate these products quickly:
  • (a) $25 \times 12 \implies 12 \times 100 \div 4 =$ 300
  • (b) $25 \times 240 \implies 240 \times 100 \div 4 =$ 6000
  • (c) $250 \times 120 \implies 120 \times 1000 \div 4 =$ 30,000
  • (d) $2500 \times 12 \implies 12 \times 10000 \div 4 =$ 30,000
  • (e) $\_ \times \_ = 120000000 \implies$ For example, 12000 $\times$ 10000.
• How Long is the Product? Evaluate to find the pattern:
  • $11 \times 11 =$ 121
  • $111 \times 111 =$ 12321
  • $1111 \times 1111 =$ 1234321
  • $66 \times 61 =$ 4026
  • $666 \times 661 =$ 440226
  • $6666 \times 6661 =$ 44402226
  • $3 \times 5 =$ 15
  • $33 \times 35 =$ 1155
  • $333 \times 335 =$ 111555
  • $101 \times 101 =$ 10201
  • $102 \times 102 =$ 10404
  • $103 \times 103 =$ 10609

Page 15

• Question: Is there any connection between the numbers being multiplied and the number of digits in their product?
  • Answer: Yes. If you multiply a number with $D_1$ digits by a number with $D_2$ digits, the product will always have either $(D_1 + D_2)$ digits or $(D_1 + D_2 – 1)$ digits.
• Question: Roxie says that the product of two 2-digit numbers can only be a 3- or a 4-digit number. Is she correct?
  • Answer: Yes. The smallest product ($10 \times 10$) is 100 (3 digits). The largest product ($99 \times 99$) is 9801 (4 digits).
• Question: Should we try all possible multiplications… Or is there a better way?
  • Answer: Finding the boundaries (the smallest and largest possible products) like Roxie did is the much better, logical way.
• Question: Can multiplying a 3-digit number with another 3-digit number give a 4-digit number?
  • Answer: No. The smallest possible is $100 \times 100 = 10,000$ (which is 5 digits).
• Question: Can multiplying a 4-digit number with a 2-digit number give a 5-digit number?
  • Answer: Yes. Example: $1000 \times 10 = 10,000$ (5 digits).
• Complete the Multiplication Pattern Table:
  • 5-digit $\times$ 5-digit = 9-digit or 10-digit
  • 8-digit $\times$ 3-digit = 10-digit or 11-digit
  • 12-digit $\times$ 13-digit = 24-digit or 25-digit

Page 16

• Question: Calculate the product: $1250 \times 380$
  • Answer: 4,75,000. (Indian: Four lakh seventy five thousand. American: Four hundred seventy five thousand).
• Question: How many years did he live to compose so many songs? At what age did he start composing? If he composed 4,75,000 songs, how many songs per year did he have to compose?
  • Answer: He would have had to compose an impossibly high number! If he composed for an incredible 60 years, $4,75,000 \div 60 \approx 7,916$ songs a year (which is over 21 songs every single day).
• Question: Calculate the product: $2100 \times 70,000$
  • Answer: 14,70,00,000. (Indian: Fourteen crore seventy lakhs. American: One hundred forty seven million).

Page 17

• Question: Calculate the product: $6400 \times 62,500$
  • Answer: 40,00,00,000. (Indian: Forty crore. American: Four hundred million).
• Question: Calculate the quotient: $13,95,000 \div 150$
  • Answer: 9,300.

Page 18

• Question: Calculate the quotient: $10,50,00,000 \div 700$
  • Answer: 1,50,000. (Indian: One lakh fifty thousand. American: One hundred fifty thousand).
• Question: Calculate the quotient: $52,00,00,00,000 \div 130$
  • Answer: 40,00,00,000. (Indian: Forty crore. American: Four hundred million).

Page 19

• Question: Could the entire population of Mumbai fit into 1 lakh buses?
  • Answer: Assuming 50 people per bus, 1 lakh buses hold 50 lakh people. Mumbai’s population is over 1.24 crore, so no, they could not fit.
• Question: The RMS Titanic ship carried about 2500 passengers. Can the population of Mumbai fit into 5000 such ships?
  • Answer: $5000 \times 2500 = 1,25,00,000$ (1.25 crore). Yes! 1.25 crore is slightly greater than 1.24 crore.
• Question: If I could travel 100 kilometers every day, could I reach the Moon in 10 years?
  • Answer: $100 \times 365 \times 10 = 3,65,000$ km. Since the moon is 3,84,400 km away, no, you would fall short by almost 20,000 km.
• Question: Find out if you can reach the Sun in a lifetime, if you travel 1000 kilometers every day.
  • Answer: The sun is roughly 14,70,00,000 km away. $14,70,00,000 \div (1000 \times 365) \approx 402$ years. No, it would take multiple lifetimes.
• Question: (a) If a single sheet of paper weighs 5 grams, could you lift one lakh sheets of paper together at the same time?
  • Answer: $1,00,000 \times 5\text{g} = 5,00,000\text{g} = 500\text{kg}$. No, a human cannot safely lift 500 kg.
• Question: (b) If 250 babies are born every minute across the world, will a million babies be born in a day?
  • Answer: $250 \times 60 \text{ min} \times 24 \text{ hr} = 3,60,000$ babies. No, it falls short of a million.
• Question: (c) Can you count 1 million coins in a day? Assume you can count 1 coin every second.
  • Answer: Seconds in a day: $60 \times 60 \times 24 = 86,400$ seconds. No.
• Figure it Out: 1. Using all digits from 0-9 exactly once (the first digit cannot be 0) to create a 10-digit number…
  • (a) Largest multiple of 5: 9876543210
  • (b) Smallest even number: 1023456798

Page 20

• Question: 2. Give a 7-digit number name which has the maximum number of letters.
  • Answer: 77,77,777 (Seventy seven lakh seventy seven thousand seven hundred seventy seven – 66 letters).
• Question: 3. Write a 9-digit number where exchanging any two digits results in a bigger number. How many such numbers exist?
  • Answer: The digits must be strictly ascending. 123456789. Only 1 such number exists.
• Question: 4. Strike out 10 digits from the number 12345123451234512345 so that the remaining number is as large as possible.
  • Answer: You want to leave the highest possible numbers starting from the left. Strike ‘1234’ (leaves 5). Strike ‘1234’ (leaves 5). Strike ’12’ (leaves 3). Leftover: 5534512345.
• Question: 5. How far do you have to count to find two consecutive numbers which do not share an English letter in common?
  • Answer: You must count to 5 and 6 (“five” and “six” share no letters).
• Question: 6. (Sequence 1234567891011…)
  • (a) What would the 1000th digit be? -> Single digits take 9 places. Double digits take 180 places. Total so far = 189. We need $1000 – 189 = 811$ more digits. $811 \div 3 = 270$ full 3-digit numbers with 1 remainder. So, $100 + 269 = 369$. The next number is 370. The first digit of 370 is 3.
• Question: 7. A calculator has only ‘+10,000’ and ‘+100’ buttons. Write an expression describing the number of button clicks…
  • (a) 20,800: $(2 \times 10,000) + (8 \times 100)$
  • (b) 92,100: $(9 \times 10,000) + (21 \times 100)$
  • (c) 1,20,500: $(12 \times 10,000) + (5 \times 100)$
  • (d) 65,30,000: $(653 \times 10,000)$
  • (e) 70,25,700: $(702 \times 10,000) + (57 \times 100)$
• Question: 8. How many lakhs make a billion?
  • Answer: $1,00,00,00,000 \div 1,00,000 =$ 10,000 lakhs.
• Question: 9. Place a number card in each box below (Cards 1-9 used once).
  • (a) Largest sum: 97531 + 8642 * (b) Smallest difference: 51234 – 49876

Page 21

• Question: 10. You are given some number cards… (4000, 13000, 300, 70000, 150000, 20, 5). Get as close as you can to the numbers below using any operation.
  • (b) 2,00,000: $150000 + 70000 – 13000 – 4000 – 3000 = 203000$
  • (c) 5,80,000: $(70000 \times 5) + 150000 = 500000$
  • (d) 12,45,000: $(150000 \times 5) + (70000 \times 4000 \div 300) …$
• Question: 11. Find out how many coins should be stacked to match the height of the Statue of Unity. Assume each coin is 1 mm thick.
  • Answer: 180 meters = 180,000 mm. 1,80,000 coins.
• Question: 12. How many days would such a trip take to cross the Pacific Ocean approximately?
  • Answer: 12,000 km $\div$ 1,000 km/day = 12 days.
• Question: 13. Find out the approximate distance it covered every day. Find out the approximate distance it covered every hour.
  • Answer: 13,560 km $\div$ 11 days = 1,232 km/day. $1,232 \div 24 \approx$ 51 km/hour.
• Question: 14. How many times bigger are these heights compared to Somu’s building?
  • Answer: Using 45m for the building:
    • Eagles ($6000 \div 45$): $\sim 133$ times
    • Everest ($8850 \div 45$): $\sim 196$ times
    • Planes ($12800 \div 45$): $\sim 284$ times

Page 23

• Question: 1. Make or write the number 42,019. It would require exactly 23 sticks. 2. Starting with 42,019, add or write two more sticks, and make a bigger number. One example is 42,078. What other numbers… can you make in this way?
  • Answer: You can place a ‘1’ anywhere (it uses 2 sticks). E.g., 142,019 or 421,019.
• Question: 3. 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?
  • Answer: Between the 4 and the 2. 412019 (assuming we insert a stick character). But mathematically inserting 1 anywhere, 421019 is greater.
• Question: 5. Starting with 63,890, rearrange exactly four sticks and make a bigger number.
  • Answer: Moving a stick from the 6 to make a 5, building an 8 out of the 3… 88,078 is given as an example, but you could also do 93,890 by re-arranging the sticks in the first digit.
• Question: 1. Make any number using exactly 24 sticks. 2. What is the biggest number that can be made using 24 sticks? 3. What is the smallest?
  • Answer: * Biggest: Use the digit ‘1’ (it uses the fewest sticks – 2). So, $24 \div 2 = 12$ ones. 111,111,111,111.
    • Smallest: Use the digit ‘8’ (it uses the most sticks – 7) to keep the number short. Three ‘8’s = 21 sticks. You have 3 sticks left, which makes a ‘7’. Arranged small: 7888.