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Class 7 Ganita Prakash · Chapter 1
Large Numbers Around Us

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✔ Understood 0/125

🗝️ Key ideas of this chapter

1 lakh = 1,00,000 (5 zeros) · 1 crore = 1,00,00,000 (7 zeros) · 1 arab = 100 crore (9 zeros).
Indian commas: 3, then 2, 2, 2 … from the right. American (international) commas: 3, 3, 3 … from the right.
10 lakh = 1 million · 100 crore = 1 billion · 1 billion = 10,000 lakh.
Rounding: look at the digit to the right — 5 or more → round up, less than 5 → round down.
a-digit × b-digit numbers give a product with (a + b − 1) or (a + b) digits.
Quick products: × 5 = ÷ 2 then × 10 · × 25 = ÷ 4 then × 100 · × 125 = ÷ 8 then × 1000.

📖 Textbook Page 1

📖 Page 1Guess

Estu tastes one new variety of rice every day. Can he taste all 1 lakh varieties in a lifetime of 100 years?

1Days in 1 year = 365 (ignore leap years).

2Days in 100 years = 365 × 100 = 36,500 days.

3One variety a day → 36,500 varieties in 100 years.

41,00,000 − 36,500 = 63,500 varieties are still left.

✅ Answer: No. In 100 years he tastes only about 36,500 varieties — far less than 1 lakh.

💡 Tip: To live 1 lakh days, a person would need about 274 years (1,00,000 ÷ 365 ≈ 274).

📖 Textbook Page 2

📖 Page 2Fill the boxes

Observe the pattern and fill in the boxes: largest 3-digit number, smallest 4-digit number, … smallest 6-digit number.

1Largest 3-digit number = 999. Add 1 → 999 + 1 = 1,000 (smallest 4-digit number).

2Largest 4-digit number = 9,999. Add 1 → 10,000 (smallest 5-digit number).

3Largest 5-digit number = 99,999. Add 1 → 1,00,000 (smallest 6-digit number).

Gold boxes are the answers. Adding 1 to the largest number with some digits gives the smallest number with one more digit.

✅ Answer: 999 → 1,000; 9,999 → 10,000; 99,999 → 1,00,000 (one lakh).

💡 Tip: Largest n-digit number + 1 = smallest (n + 1)-digit number.

📖 Page 2Fill the boxes

Fill the number chain that starts 99,995, 99,996, ___, 99,998, …

1Each box is 1 more than the box before it.

299,996 + 1 = 99,997

399,998 + 1 = 99,999

499,999 + 1 = 1,00,000 (one lakh)

5Then 1,00,001, 1,00,002, 1,00,003, 1,00,004.

Counting on by 1: 99,999 + 1 = 1,00,000 (one lakh). Gold boxes are the ones to fill.

✅ Answer: 99,997; 99,999; 1,00,000; 1,00,001; 1,00,002; 1,00,003; 1,00,004

📖 Page 2Think

Roxie: “What if we ate 2 varieties of rice every day? Would we be able to eat 1 lakh varieties in 100 years?”

1Days in 100 years = 365 × 100 = 36,500.

2Varieties eaten = 2 × 36,500 = 73,000.

373,000 is less than 1,00,000 (short by 27,000).

✅ Answer: No. With 2 varieties a day, only 73,000 varieties can be eaten in 100 years.

📖 Page 2Find out

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?

1Varieties in 100 years = 3 × 36,500 = 1,09,500.

21,09,500 is more than 1,00,000.

3Days needed = 1,00,000 ÷ 3 ≈ 33,334 days.

4Years needed = 33,334 ÷ 365 ≈ 91 years and 4 months.

✅ Answer: Yes. 3 varieties a day gives 1,09,500 in 100 years; 1 lakh is reached in about 91–92 years.

📖 Textbook Page 3

📖 Page 3Think

Choose a number for y. How close to one lakh is the number of days in y years (365 × y)?

1Try y = 100: 365 × 100 = 36,500 days. This is 63,500 less than 1 lakh.

2Try y = 250: 365 × 250 = 91,250 days. This is 8,750 less than 1 lakh.

3Try y = 274: 365 × 274 = 1,00,010 days. This is just 10 more than 1 lakh!

4Try y = 273: 365 × 273 = 99,645 days. This is 355 less than 1 lakh.

✅ Answer: y = 274 years gives 1,00,010 days — the closest to one lakh.

💡 Tip: So 1 lakh days ≈ 274 years. No human lives that long!

📖 Page 3Figure it Out · Q1

In 2011 the population of Chintamani was about 75,000. How much less than one lakh is 75,000?

11 lakh = 1,00,000

21,00,000 − 75,000 = 25,000

✅ Answer: 25,000 less than one lakh.

📖 Page 3Figure it Out · Q2

The estimated population of Chintamani in 2024 is 1,06,000. How much more than one lakh is 1,06,000?

11,06,000 − 1,00,000 = 6,000

✅ Answer: 6,000 more than one lakh.

📖 Page 3Figure it Out · Q3

By how much did the population of Chintamani increase from 2011 to 2024?

1Population in 2024 = 1,06,000; population in 2011 = 75,000.

2Increase = 1,06,000 − 75,000 = 31,000

✅ Answer: It increased by about 31,000.

⚠️ Common mistake: Writing 1,06,000 − 75,000 = 1,31,000 by adding instead of subtracting. ‘Increase’ means find the difference.

📖 Page 3Feel of large numbers

Somu is 1 metre tall. Each floor of his building is about 4 times his height. What is the approximate height of the building?

Somu’s building has 10 floors; each floor is about 4 times Somu’s height.

1Height of one floor = 4 × 1 m = 4 m.

2The building has about 10 floors.

3Height = 10 × 4 m = 40 m.

Heights compared: the building is 40 m (10 floors × 4 m), the statue 180 m and the waterfall 450 m.

✅ Answer: About 40 metres.

📖 Page 3Feel of large numbers

Which is taller — the Statue of Unity (180 m) or Somu’s building (40 m)? How much taller?

1180 m is more than 40 m, so the statue is taller.

2Difference = 180 − 40 = 140 m.

3Also, 180 ÷ 40 = 4.5 → the statue is about 4 times as tall.

✅ Answer: The Statue of Unity is taller, by 140 m.

📖 Page 3Feel of large numbers

How much taller is the Kunchikal waterfall (450 m) than Somu’s building (40 m)?

1Difference = 450 − 40 = 410 m.

✅ Answer: 410 m taller.

📖 Page 3Feel of large numbers

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

1Each floor is 4 m high.

2Number of floors = 450 ÷ 4 = 112.5

3We cannot have half a floor, so we need about 113 floors.

✅ Answer: About 113 floors (112½ floors exactly).

⚠️ Common mistake: Dividing by 40 instead of 4. Use the height of ONE floor (4 m).

📖 Textbook Page 4

📖 Page 4Think

Is one lakh a big number or a small number?

1Big, when we count one by one: 1 lakh days ≈ 274 years; 1 lakh people in a line would stretch about 38 km.

2Small, when things are tiny or crowded: a cricket stadium can seat more than 1 lakh people; a person has about 1 lakh hairs on the head; some fish lay 1 lakh eggs at once.

3So whether a number feels big or small depends on what we are counting.

✅ Answer: Both views are right — one lakh is big for days or distances, but small for hairs, eggs or stadium seats.

💡 Tip: Always compare a large number with something familiar to ‘feel’ its size.

📖 Page 4Write in words (a)

Write 3,00,600 in words.

1Groups from the left: 3 lakh, 00 thousand, 600 (hundreds, tens, ones).

2Read each group and add its place name (lakh, thousand). Skip a group if it is 0.

✅ Answer: Three lakh six hundred

⚠️ Common mistake: Reading the zero groups aloud (like ‘zero thousand’). A group of zeros is simply skipped.

📖 Page 4Write in words (b)

Write 5,04,085 in words.

1Groups from the left: 5 lakh, 04 thousand, 085 (hundreds, tens, ones).

2Read each group and add its place name (lakh, thousand). Skip a group if it is 0.

✅ Answer: Five lakh four thousand eighty-five

📖 Page 4Write in words (c)

Write 27,30,000 in words.

1Groups from the left: 27 lakh, 30 thousand, 000 (hundreds, tens, ones).

2Read each group and add its place name (lakh, thousand). Skip a group if it is 0.

✅ Answer: Twenty-seven lakh thirty thousand

📖 Page 4Write in words (d)

Write 70,53,138 in words.

1Groups from the left: 70 lakh, 53 thousand, 138 (hundreds, tens, ones).

2Read each group and add its place name (lakh, thousand). Skip a group if it is 0.

✅ Answer: Seventy lakh fifty-three thousand one hundred thirty-eight

📖 Textbook Page 5

📖 Page 5Write the number (a)

Write the number in the Indian place value system: One lakh twenty-three thousand four hundred and fifty-six.

1Write each group: 1 lakh | 23 thousand | 456

2Thousand group always has 2 digits and the last group 3 digits — fill gaps with 0.

3Put commas: 1,23,456

✅ Answer: 1,23,456

📖 Page 5Write the number (b)

Write the number in the Indian place value system: Four lakh seven thousand seven hundred and four.

1Write each group: 4 lakh | 07 thousand | 704

2Thousand group always has 2 digits and the last group 3 digits — fill gaps with 0.

3Put commas: 4,07,704

✅ Answer: 4,07,704

⚠️ Common mistake: Writing 4,7,704 or 47,704. Every group after the first needs its full number of digits (use 0s).

📖 Page 5Write the number (c)

Write the number in the Indian place value system: Fifty lakhs five thousand and fifty.

1Write each group: 50 lakh | 05 thousand | 050

2Thousand group always has 2 digits and the last group 3 digits — fill gaps with 0.

3Put commas: 50,05,050

✅ Answer: 50,05,050

📖 Page 5Write the number (d)

Write the number in the Indian place value system: Ten lakhs two hundred and thirty-five.

1Write each group: 10 lakh | 00 thousand | 235

2Thousand group always has 2 digits and the last group 3 digits — fill gaps with 0.

3Put commas: 10,00,235

✅ Answer: 10,00,235

📖 Page 5Land of Tens · Q1

Thoughtful Thousands has only a +1000 button. How many presses show (a) 3,000 (b) 10,000 (c) 53,000 (d) 90,000 (e) 1 lakh? (f) What number shows after 153 presses? (g) How many thousands make one lakh?

1Each press adds 1,000, so presses = number ÷ 1,000.

2The table shows (a) to (e).

3(f) 153 presses → 153 × 1,000 = 1,53,000.

4(g) 1,00,000 ÷ 1,000 = 100.

Number	Value	Presses
(a) Three thousand	3,000	3,000 ÷ 1,000 = 3
(b) 10,000	10,000	10,000 ÷ 1,000 = 10
(c) Fifty-three thousand	53,000	53,000 ÷ 1,000 = 53
(d) 90,000	90,000	90,000 ÷ 1,000 = 90
(e) One lakh	1,00,000	1,00,000 ÷ 1,000 = 100

✅ Answer: (a) 3 (b) 10 (c) 53 (d) 90 (e) 100 (f) 1,53,000 (g) 100 thousands make 1 lakh.

📖 Page 5Land of Tens · Q2

Tedious Tens has only a +10 button. How many presses show (a) 500 (b) 780 (c) 1000 (d) 3700 (e) 10,000 (f) 1 lakh? (g) Which number shows after 435 presses?

1Each press adds 10, so presses = number ÷ 10.

2The table shows (a) to (f).

3(g) 435 presses → 435 × 10 = 4,350.

Number	Value	Presses
(a) Five hundred	500	500 ÷ 10 = 50
(b) 780	780	780 ÷ 10 = 78
(c) 1000	1,000	1,000 ÷ 10 = 100
(d) 3700	3,700	3,700 ÷ 10 = 370
(e) 10,000	10,000	10,000 ÷ 10 = 1,000
(f) One lakh	1,00,000	1,00,000 ÷ 10 = 10,000

✅ Answer: (a) 50 (b) 78 (c) 100 (d) 370 (e) 1,000 (f) 10,000 (g) 4,350

💡 Tip: Dividing by 10 removes one zero from the end.

📖 Textbook Page 6

📖 Page 6Land of Tens · Q3

Handy Hundreds has only a +100 button. How many presses show (a) 400 (b) 3,700 (c) 10,000 (d) 53,000 (e) 90,000 (f) 97,600 (g) 1,00,000? (h) Which number shows after 582 presses? (i) How many hundreds make ten thousand? (j) How many hundreds make one lakh?

1Each press adds 100, so presses = number ÷ 100.

2The table shows (a) to (g).

3(h) 582 × 100 = 58,200.

4(i) 10,000 ÷ 100 = 100.

5(j) 1,00,000 ÷ 100 = 1,000.

Number	Value	Presses
(a) Four hundred	400	400 ÷ 100 = 4
(b) 3,700	3,700	3,700 ÷ 100 = 37
(c) 10,000	10,000	10,000 ÷ 100 = 100
(d) Fifty-three thousand	53,000	53,000 ÷ 100 = 530
(e) 90,000	90,000	90,000 ÷ 100 = 900
(f) 97,600	97,600	97,600 ÷ 100 = 976
(g) 1,00,000	1,00,000	1,00,000 ÷ 100 = 1,000

✅ Answer: (a) 4 (b) 37 (c) 100 (d) 530 (e) 900 (f) 976 (g) 1,000 (h) 58,200 (i) 100 (j) 1,000

📖 Page 6Land of Tens · Q3(k)

Handy Hundreds says, “There are some numbers which Tedious Tens and Thoughtful Thousands can’t show but I can.” Is this true?

1Handy Hundreds can show only multiples of 100 (100, 200, 300, …).

2Thoughtful Thousands cannot show 300 or 3,700 — so Hundreds does beat Thousands.

3But every multiple of 100 is also a multiple of 10 (for example 300 = 30 × 10). So Tedious Tens can show every number that Hundreds shows.

4So no number can be shown by Hundreds but by neither of the other two.

✅ Answer: Not true. Tedious Tens can show every number Handy Hundreds shows (it just needs more presses).

💡 Tip: Tens can show 10, 20, 30 … ; Hundreds cannot show 50 or 780. So Tens can show MORE numbers.

📖 Page 6Creative Chitti

To get 321, Chitti presses +10 thirty-two times and +1 once. Will it get 321? Check the other way too: +100 two times, +10 twelve times and +1 once.

1(32 × 10) + (1 × 1) = 320 + 1 = 321 ✔

2(2 × 100) + (12 × 10) + (1 × 1) = 200 + 120 + 1 = 321 ✔

✅ Answer: Yes, both ways give 321.

📖 Page 6Creative Chitti

Find a different way to get 5072 and write an expression for it.

1Use 4 thousands = 4,000. We still need 5,072 − 4,000 = 1,072.

2Use 10 hundreds = 1,000. Still need 72.

3Use 7 tens = 70 and 2 ones = 2.

4Check: 4,000 + 1,000 + 70 + 2 = 5,072 ✔

Buttons	way 1	way 2	way 3
+10,00,000
+1,00,000
+10,000
+1,000	4		5
+100	10	50
+10	7		5
+1	2	72	22

✅ Answer: (4 × 1000) + (10 × 100) + (7 × 10) + (2 × 1) = 5072 (other answers are possible, e.g. (50 × 100) + (72 × 1))

📖 Page 6Figure it Out · (a)

Write expressions for at least two different ways to get 8,300 with Chitti’s buttons.

1Way 1 (using place value): (8 × 1000) + (3 × 100)

2Way 2 (mixing buttons): (80 × 100) + (30 × 10)

3Check: both expressions equal 8,300 ✔

✅ Answer: (8 × 1000) + (3 × 100) = 8,300;
(80 × 100) + (30 × 10) = 8,300

💡 Tip: There are many correct answers. Always check by adding.

📖 Page 6Figure it Out · (b)

Write expressions for at least two different ways to get 40,629 with Chitti’s buttons.

1Way 1 (using place value): (4 × 10000) + (6 × 100) + (2 × 10) + (9 × 1)

2Way 2 (mixing buttons): (40 × 1000) + (62 × 10) + (9 × 1)

3Check: both expressions equal 40,629 ✔

✅ Answer: (4 × 10000) + (6 × 100) + (2 × 10) + (9 × 1) = 40,629;
(40 × 1000) + (62 × 10) + (9 × 1) = 40,629

📖 Page 6Figure it Out · (c)

Write expressions for at least two different ways to get 56,354 with Chitti’s buttons.

1Way 1 (using place value): (5 × 10000) + (6 × 1000) + (3 × 100) + (5 × 10) + (4 × 1)

2Way 2 (mixing buttons): (56 × 1000) + (35 × 10) + (4 × 1)

3Check: both expressions equal 56,354 ✔

✅ Answer: (5 × 10000) + (6 × 1000) + (3 × 100) + (5 × 10) + (4 × 1) = 56,354;
(56 × 1000) + (35 × 10) + (4 × 1) = 56,354

📖 Textbook Page 7

📖 Page 7Figure it Out · (d)

Write expressions for at least two different ways to get 66,666 with Chitti’s buttons.

1Way 1 (using place value): (6 × 10000) + (6 × 1000) + (6 × 100) + (6 × 10) + (6 × 1)

2Way 2 (mixing buttons): (66 × 1000) + (666 × 1)

3Check: both expressions equal 66,666 ✔

✅ Answer: (6 × 10000) + (6 × 1000) + (6 × 100) + (6 × 10) + (6 × 1) = 66,666;
(66 × 1000) + (666 × 1) = 66,666

📖 Page 7Figure it Out · (e)

Write expressions for at least two different ways to get 3,67,813 with Chitti’s buttons.

1Way 1 (using place value): (3 × 100000) + (6 × 10000) + (7 × 1000) + (8 × 100) + (1 × 10) + (3 × 1)

2Way 2 (mixing buttons): (36 × 10000) + (78 × 100) + (13 × 1)

3Check: both expressions equal 3,67,813 ✔

✅ Answer: (3 × 100000) + (6 × 10000) + (7 × 1000) + (8 × 100) + (1 × 10) + (3 × 1) = 3,67,813;
(36 × 10000) + (78 × 100) + (13 × 1) = 3,67,813

📖 Page 7Chitti’s challenge (a)

With exactly 30 button presses, what is the largest 3-digit number you can make? What is the smallest 3-digit number?

1Largest: use as many +100 as possible but stay below 1,000. 9 hundreds = 900 (9 presses).

221 presses are left. If all were +10 we get 900 + 210 = 1,110 (too big). Use 8 tens and 13 ones: 900 + 80 + 13 = 993.

3Check: 9 + 8 + 13 = 30 presses ✔; with 9 tens we would need 12 ones → 900 + 90 + 12 = 1,002 (4 digits ✘).

4Smallest: use no +100. With t tens and (30 − t) ones the number is 10t + (30 − t) = 30 + 9t.

5We need at least 100: 30 + 9t ≥ 100 → t ≥ 8. With t = 8: 80 + 22 = 102 (8 + 22 = 30 presses ✔).

✅ Answer: Largest = 993 (9 × 100 + 8 × 10 + 13 × 1). Smallest = 102 (8 × 10 + 22 × 1).

📖 Page 7Chitti’s challenge (b)

997 can be made using 25 clicks. Can you make 997 with a different number of clicks?

125 clicks: (9 × 100) + (9 × 10) + (7 × 1) → 9 + 9 + 7 = 25.

2Another way: (8 × 100) + (19 × 10) + (7 × 1) = 800 + 190 + 7 = 997.

3Clicks = 8 + 19 + 7 = 34.

4Breaking one +100 into ten +10s (or one +10 into ten +1s) always adds 9 clicks, so the counts go 25, 34, 43, … up to 997 (all +1s).

✅ Answer: Yes. For example (8 × 100) + (19 × 10) + (7 × 1) = 997 uses 34 clicks. Possible click counts are 25, 34, 43, … (9 more each time) up to 997.

📖 Page 7Systematic Sippy

Get (a) 5072 (b) 8300 using as few button clicks as possible. Is there a way to get 5072 with fewer than 23 clicks?

1The 23-click way: (5 × 1000) + (6 × 10) + (12 × 1) → 5 + 6 + 12 = 23.

2Fewer clicks: use the biggest buttons first and never press a button 10 or more times.

3(a) 5072 = (5 × 1000) + (7 × 10) + (2 × 1) → 5 + 7 + 2 = 14 clicks.

4(b) 8300 = (8 × 1000) + (3 × 100) → 8 + 3 = 11 clicks.

Buttons	5072	8300
+10,00,000
+1,00,000
+10,000
+1,000	5	8
+100		3
+10	7
+1	2

✅ Answer: Yes. 5072 needs only 14 clicks; 8300 needs 11 clicks.

💡 Tip: Pressing a button 10 times = pressing the next bigger button once. So minimum clicks never press a button 10 times.

📖 Page 7Figure it Out · Q1

For the numbers 8300, 40629, 56354, 66666 and 367813, find the smallest number of button clicks and write the expression.

1Use each place-value button as many times as its digit.

2The table shows each expression and the number of clicks.

Number	Expression (fewest clicks)	Clicks
8,300	(8 × 1000) + (3 × 100)	8 + 3 = 11
40,629	(4 × 10000) + (6 × 100) + (2 × 10) + (9 × 1)	4 + 6 + 2 + 9 = 21
56,354	(5 × 10000) + (6 × 1000) + (3 × 100) + (5 × 10) + (4 × 1)	5 + 6 + 3 + 5 + 4 = 23
66,666	(6 × 10000) + (6 × 1000) + (6 × 100) + (6 × 10) + (6 × 1)	6 + 6 + 6 + 6 + 6 = 30
3,67,813	(3 × 100000) + (6 × 10000) + (7 × 1000) + (8 × 100) + (1 × 10) + (3 × 1)	3 + 6 + 7 + 8 + 1 + 3 = 28

✅ Answer: 8300 → 11, 40629 → 21, 56354 → 23, 66666 → 30, 367813 → 28 clicks.

📖 Page 7Figure it Out · Q2

Do you see any connection between each number and its smallest number of button clicks?

18300 → 8 + 3 + 0 + 0 = 11.

240629 → 4 + 0 + 6 + 2 + 9 = 21.

3Each time, the clicks equal the sum of the digits.

✅ Answer: Smallest number of clicks = sum of the digits of the number.

📖 Page 7Figure it Out · Q3 (Math Talk)

Why do the expressions for the least button clicks also give the Indian place value form of the numbers?

1For the fewest clicks no button is pressed 10 or more times (10 presses can be replaced by 1 press of the next button).

2So each button is pressed 0 to 9 times — exactly one digit.

3The button for 1,000 is pressed ‘thousands digit’ times, for 100 ‘hundreds digit’ times, and so on.

4That is exactly how place value writes a number: 5072 = 5 thousands + 0 hundreds + 7 tens + 2 ones.

✅ Answer: Because the least-click way uses each place value button ‘digit’ times (0 to 9), which is the place value form.

📖 Textbook Page 8

📖 Page 8Think

What if we press the +10,00,000 button ten times? What number comes up? How many zeros will it have? What do we call it?

110 × 10,00,000 = 1,00,00,000

2Count the zeros: 7 zeros.

3100 lakh = 1 crore.

✅ Answer: 1,00,00,000 — one crore, with 7 zeros.

📖 Page 8Think

How many zeros does a thousand lakh have?

11 lakh = 1,00,000 (5 zeros).

21,000 lakh = 1,000 × 1,00,000 = 10,00,00,000.

3Zeros: 3 + 5 = 8.

✅ Answer: 8 zeros (1,000 lakh = 10 crore = 10,00,00,000).

💡 Tip: When multiplying numbers that end in zeros, add the numbers of zeros.

📖 Textbook Page 9

📖 Page 9Think

How many zeros does a hundred thousand have?

1100 × 1,000 = 1,00,000

2Zeros: 2 + 3 = 5

✅ Answer: 5 zeros (a hundred thousand = 1 lakh).

📖 Page 9Figure it Out · Q1(a)

Read 4050678. Write its number name in the Indian and the American systems.

1Indian commas (3, then 2, 2, …): 40,50,678

2Indian name: Forty lakh fifty thousand six hundred seventy-eight.

3American commas (3, 3, 3 …): 4,050,678

4American name: Four million fifty thousand six hundred seventy-eight.

✅ Answer: 40,50,678 → Forty lakh fifty thousand six hundred seventy-eight
4,050,678 → Four million fifty thousand six hundred seventy-eight

⚠️ Common mistake: Putting commas from the left. Always start from the right.

📖 Page 9Figure it Out · Q1(b)

Read 48121620. Write its number name in the Indian and the American systems.

1Indian commas (3, then 2, 2, …): 4,81,21,620

2Indian name: Four crore eighty-one lakh twenty-one thousand six hundred twenty.

3American commas (3, 3, 3 …): 48,121,620

4American name: Forty-eight million one hundred twenty-one thousand six hundred twenty.

✅ Answer: 4,81,21,620 → Four crore eighty-one lakh twenty-one thousand six hundred twenty
48,121,620 → Forty-eight million one hundred twenty-one thousand six hundred twenty

📖 Page 9Figure it Out · Q1(c)

Read 20022002. Write its number name in the Indian and the American systems.

1Indian commas (3, then 2, 2, …): 2,00,22,002

2Indian name: Two crore twenty-two thousand two.

3American commas (3, 3, 3 …): 20,022,002

4American name: Twenty million twenty-two thousand two.

✅ Answer: 2,00,22,002 → Two crore twenty-two thousand two
20,022,002 → Twenty million twenty-two thousand two

📖 Page 9Figure it Out · Q1(d)

Read 246813579. Write its number name in the Indian and the American systems.

1Indian commas (3, then 2, 2, …): 24,68,13,579

2Indian name: Twenty-four crore sixty-eight lakh thirteen thousand five hundred seventy-nine.

3American commas (3, 3, 3 …): 246,813,579

4American name: Two hundred forty-six million eight hundred thirteen thousand five hundred seventy-nine.

✅ Answer: 24,68,13,579 → Twenty-four crore sixty-eight lakh thirteen thousand five hundred seventy-nine
246,813,579 → Two hundred forty-six million eight hundred thirteen thousand five hundred seventy-nine

📖 Page 9Figure it Out · Q1(e)

Read 345000543. Write its number name in the Indian and the American systems.

1Indian commas (3, then 2, 2, …): 34,50,00,543

2Indian name: Thirty-four crore fifty lakh five hundred forty-three.

3American commas (3, 3, 3 …): 345,000,543

4American name: Three hundred forty-five million five hundred forty-three.

✅ Answer: 34,50,00,543 → Thirty-four crore fifty lakh five hundred forty-three
345,000,543 → Three hundred forty-five million five hundred forty-three

📖 Page 9Figure it Out · Q1(f)

Read 1020304050. Write its number name in the Indian and the American systems.

1Indian commas (3, then 2, 2, …): 1,02,03,04,050

2Indian name: One arab two crore three lakh four thousand fifty (or one hundred two crore three lakh four thousand fifty).

3American commas (3, 3, 3 …): 1,020,304,050

4American name: One billion twenty million three hundred four thousand fifty.

✅ Answer: 1,02,03,04,050 → One arab two crore three lakh four thousand fifty (or one hundred two crore three lakh four thousand fifty)
1,020,304,050 → One billion twenty million three hundred four thousand fifty

📖 Page 9Figure it Out · Q2(a)

Write in Indian place value notation: One crore one lakh one thousand ten.

11 crore + 1 lakh + 1 thousand + 10

2Now put Indian commas: 1,01,01,010

✅ Answer: 1,01,01,010

📖 Page 9Figure it Out · Q2(b)

Write in Indian place value notation: One billion one million one thousand one.

11,000,000,000 + 1,000,000 + 1,000 + 1 = 1,001,001,001

2Now put Indian commas: 1,00,10,01,001

✅ Answer: 1,00,10,01,001

📖 Page 9Figure it Out · Q2(c)

Write in Indian place value notation: Ten crore twenty lakh thirty thousand forty.

110 crore + 20 lakh + 30 thousand + 40

2Now put Indian commas: 10,20,30,040

✅ Answer: 10,20,30,040

📖 Page 9Figure it Out · Q2(d)

Write in Indian place value notation: Nine billion eighty million seven hundred thousand six hundred.

19,000,000,000 + 80,000,000 + 700,000 + 600 = 9,080,700,600

2Now put Indian commas: 9,08,07,00,600

✅ Answer: 9,08,07,00,600

📖 Page 9Figure it Out · Q3(a)

Compare and write ‘<’, ‘>’ or ‘=’: 30 thousand ___ 3 lakhs

130 thousand = 30,000

23 lakhs = 3,00,000

3Compare the numbers of digits / values.

✅ Answer: 30 thousand < 3 lakhs

📖 Page 9Figure it Out · Q3(b)

Compare and write ‘<’, ‘>’ or ‘=’: 500 lakhs ___ 5 million

1500 lakhs = 500 lakh = 5 crore = 5,00,00,000

25 million = 5 million = 50,00,000

3Compare the numbers of digits / values.

✅ Answer: 500 lakhs > 5 million

💡 Tip: 1 million = 10 lakh; 1 billion = 100 crore.

📖 Page 9Figure it Out · Q3(c)

Compare and write ‘<’, ‘>’ or ‘=’: 800 thousand ___ 8 million

1800 thousand = 8,00,000

28 million = 80,00,000

3Compare the numbers of digits / values.

✅ Answer: 800 thousand < 8 million

📖 Page 9Figure it Out · Q3(d)

Compare and write ‘<’, ‘>’ or ‘=’: 640 crore ___ 60 billion

1640 crore = 640 crore = 6,40,00,00,000 (6.4 billion)

260 billion = 60 billion = 60,00,00,00,000

3Compare the numbers of digits / values.

✅ Answer: 640 crore < 60 billion

📖 Textbook Page 10

📖 Page 10Think

“1 lakh people visited the book fair.” “If I had not gone, they would have written ‘99,999 people visited’.” What do you think of this conversation?

1The number 1 lakh in the headline is a rounded (approximate) number.

2The real number of visitors may be 98,700 or 1,02,350 — not exactly 1,00,000.

3So one person missing does not change the headline. The joke is that the person thinks the number is exact.

✅ Answer: Headlines use approximate (rounded) numbers; one person more or less does not change ‘about 1 lakh’.

📖 Page 10Think and share

Give situations where it is right to (a) round up (b) round down (c) either is okay (d) exact numbers are needed.

1(a) Round up: buying sweets, chairs or food for a group; buying tiles or paint — having a few extra is safer.

2(b) Round down: telling how much money you can spend; how many full boxes you can pack; time left before the school bus comes.

3(c) Either: population of a city, distance between two cities, number of people in a stadium.

4(d) Exact: bank amounts, bills, medicine doses, phone numbers, exam marks, train timings.

✅ Answer: Answers will vary — the examples in the steps are good ones.

📖 Textbook Page 11

📖 Page 11Nearest neighbours (a)

Write the five nearest neighbours of 3,87,69,957.

1For each place, look at the digit just to its right: 5 or more → round up; less than 5 → round down.

2Change all digits after that place to 0.

3The table shows all five.

Neighbour	Value
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

✅ Answer: 3,87,70,000; 3,87,70,000; 3,88,00,000; 3,90,00,000; 4,00,00,000

📖 Page 11Nearest neighbours (b)

Write the five nearest neighbours of 29,05,32,481.

1For each place, look at the digit just to its right: 5 or more → round up; less than 5 → round down.

2Change all digits after that place to 0.

3The table shows all five.

Neighbour	Value
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: 29,05,32,000; 29,05,30,000; 29,05,00,000; 29,10,00,000; 29,00,00,000

⚠️ Common mistake: For the nearest ten lakh of 29,05,32,481, writing 29,00,00,000. The lakhs part 5,32,481 is more than 5,00,000, so round UP to 29,10,00,000.

📖 Page 11Math Talk

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?

1The nearest thousand must be 5,00,00,000.

2So the number is at most 499 more or at least 500 less: from 4,99,99,500 to 5,00,00,499.

3For all these numbers the nearest ten thousand, lakh, ten lakh and crore are also 5,00,00,000.

4Count: from 4,99,99,500 to 5,00,00,499 there are 500 + 500 = 1,000 numbers.

✅ Answer: Any number from 4,99,99,500 to 5,00,00,499 (e.g. 5,00,00,123) — 1,000 such numbers.

📖 Page 11Estimation · Q1

4,63,128 + 4,19,682. Roxie: “near 8,00,000 and more than 8,00,000.” Estu: “near 9,00,000 and less than 9,00,000.” (a) Are they correct? Whose estimate is closer? (b) Is the sum more or less than 8,50,000? (c) More or less than 8,83,128? (d) Find the exact sum.

1(a) 4,63,128 > 4,00,000 and 4,19,682 > 4,00,000, so the sum > 8,00,000 ✔ (Roxie).

24,63,128 < 5,00,000 and 4,19,682 < 4,20,000, so the sum < 9,00,000 ✔ (Estu). Both are correct.

3(b) 4,63,128 > 4,50,000 and 4,19,682 > 4,00,000 → sum > 4,50,000 + 4,00,000 = 8,50,000.

4(c) 4,19,682 < 4,20,000 → sum < 4,63,128 + 4,20,000 = 8,83,128.

5(d) 4,63,128 + 4,19,682 = 8,82,810.

68,82,810 is 17,190 away from 9,00,000 but 82,810 away from 8,00,000 → Estu is closer.

✅ Answer: (a) Both are correct; Estu’s estimate is closer. (b) More than 8,50,000 (c) Less than 8,83,128 (d) 8,82,810

📖 Textbook Page 12

📖 Page 12Estimation · Q2

14,63,128 − 4,90,020. Roxie: “near 10,00,000 and less than 10,00,000.” Estu: “near 9,00,000 and more than 9,00,000.” (a) Are they correct? Whose is closer? (b) More or less than 9,50,000? (c) More or less than 9,63,128? (d) Exact value?

1(a) We take away almost 5 lakh from almost 14.6 lakh → about 9.7 lakh. Both statements are correct.

2(b) 14,63,128 > 14,50,000 and 4,90,020 < 5,00,000 → difference > 14,50,000 − 5,00,000 = 9,50,000.

3(c) 14,63,128 − 5,00,000 = 9,63,128. We take away LESS than 5,00,000, so more is left → difference > 9,63,128.

4(d) 14,63,128 − 4,90,020 = 9,73,108.

59,73,108 is 26,892 from 10,00,000 but 73,108 from 9,00,000 → Roxie is closer.

✅ Answer: (a) Both correct; Roxie is closer. (b) More than 9,50,000 (c) More than 9,63,128 (d) 9,73,108

⚠️ Common mistake: In (c), thinking that subtracting less gives a smaller answer. Taking away less leaves MORE.

📖 Textbook Page 13

📖 Page 13Populations · Q1

What is your general observation about the population table of 20 Indian cities?

1Almost every city grew between 2001 and 2011.

2Mumbai and New Delhi have more than 1 crore people.

3Kolkata is the only city whose population went down (45,72,876 → 44,86,679).

4Cities are ranked by their 2011 population.

✅ Answer: Most cities grew a lot from 2001 to 2011; Mumbai is the biggest, and only Kolkata became smaller.

📖 Page 13Populations · Q2

What is an appropriate title for the table?

1The table shows the populations of 20 big Indian cities in two census years.

✅ Answer: “Populations of the 20 largest Indian cities in 2001 and 2011”

📖 Page 13Populations · Q3

What is the population of Pune in 2011? By about how much has it increased since 2001?

1Pune 2011: 31,15,431. Pune 2001: 25,38,473.

2Round: about 31 lakh − 25 lakh = 6 lakh.

3Exact: 31,15,431 − 25,38,473 = 5,76,958.

✅ Answer: 31,15,431; it increased by about 6 lakh (5,76,958 exactly).

📖 Page 13Populations · Q4

Which city’s population increased the most between 2001 and 2011?

1Approximate increases: Bengaluru 84 − 43 = 41 lakh; Hyderabad 68 − 36 = 32 lakh; Ahmedabad and Surat about 20 lakh each; Mumbai only about 5 lakh.

2Exact for Bengaluru: 84,25,970 − 43,01,326 = 41,24,644.

✅ Answer: Bengaluru (an increase of about 41 lakh).

📖 Page 13Populations · Q5

Are there cities whose population has almost doubled? Which ones?

1Double the 2001 population and compare with 2011.

2Bengaluru: 43 lakh × 2 = 86 lakh ≈ 84 lakh ✔

3Hyderabad: 36 lakh × 2 = 72 lakh ≈ 68 lakh ✔

4Surat: 24 lakh × 2 = 48 lakh ≈ 45 lakh ✔

5Vadodara: 17 lakh × 2 = 34 lakh; 2011 is 35.5 lakh — a little MORE than double ✔

✅ Answer: Yes — Bengaluru, Hyderabad and Surat almost doubled; Vadodara more than doubled.

📖 Page 13Populations · Q6

By what number should we multiply Patna’s population to get a number close to Mumbai’s?

1Patna ≈ 17 lakh; Mumbai ≈ 1,24 lakh (1 crore 24 lakh).

2124 ÷ 17 ≈ 7.3

3Check: 7 × 16,84,222 = 1,17,89,554 — close to 1,24,42,373.

✅ Answer: About 7 (more exactly about 7.4).

📖 Textbook Page 14

📖 Page 14Math Talk

Using the meaning of multiplication and division, explain why multiplying by 5 is the same as dividing by 2 and multiplying by 10.

15 is half of 10: 5 = 10 ÷ 2.

2So n × 5 = n × (10 ÷ 2) = (n ÷ 2) × 10.

3Example: 116 × 5 = (116 ÷ 2) × 10 = 58 × 10 = 580.

4In the same way 25 = 100 ÷ 4, so 824 × 25 = (824 ÷ 4) × 100 = 206 × 100 = 20,600.

✅ Answer: Because 5 = 10 ÷ 2, multiplying by 5 = halving and then multiplying by 10.

📖 Page 14Figure it Out · Q1(a)

Find a quick way: 2 × 1768 × 50

1Re-order: (2 × 50) × 1768

2= 100 × 1768

3= 1,76,800

✅ Answer: 1,76,800

📖 Page 14Figure it Out · Q1(b)

Find a quick way: 72 × 125 [Hint: 125 = 1000 ÷ 8]

172 × 125 = 72 × 1000 ÷ 8

2= (72 ÷ 8) × 1000

3= 9 × 1000 = 9,000

✅ Answer: 9,000

📖 Page 14Figure it Out · Q1(c)

Find a quick way: 125 × 40 × 8 × 25

1Re-group: (125 × 8) × (40 × 25)

2= 1000 × 1000

3= 10,00,000

✅ Answer: 10,00,000 (ten lakh / one million)

💡 Tip: Look for pairs that make 100 or 1000: 2 × 50, 4 × 25, 8 × 125.

📖 Page 14Figure it Out · Q2(a)

Calculate quickly: 25 × 12

125 = 100 ÷ 4

225 × 12 = (12 ÷ 4) × 100 = 3 × 100

3= 300

✅ Answer: 300

📖 Page 14Figure it Out · Q2(b)

Calculate quickly: 25 × 240

125 × 240 = (240 ÷ 4) × 100

2= 60 × 100

3= 6,000

✅ Answer: 6,000

📖 Page 14Figure it Out · Q2(c)

Calculate quickly: 250 × 120

1250 = 1000 ÷ 4

2250 × 120 = (120 ÷ 4) × 1000 = 30 × 1000

3= 30,000

✅ Answer: 30,000

📖 Page 14Figure it Out · Q2(d)

Calculate quickly: 2500 × 12

12500 = 10,000 ÷ 4

22500 × 12 = (12 ÷ 4) × 10,000 = 3 × 10,000

3= 30,000

✅ Answer: 30,000

📖 Page 14Figure it Out · Q2(e)

Fill in: ____ × ____ = 120000000

1120000000 = 12,00,00,000 (12 crore).

2One pair: 12 × 1,00,00,000 (12 × 1 crore).

3Another: 1,200 × 1,00,000 = 12,00,00,000.

4Another: 2,500 × 48,000 = 12,00,00,000 (since 25 × 48 = 1200).

✅ Answer: For example 1,200 × 1,00,000 (many answers).

📖 Page 14How long is the product?

Evaluate and extend the patterns: 11 × 11, 111 × 111, …; 66 × 61, 666 × 661, …; 3 × 5, 33 × 35, …; 101 × 101, 102 × 102, …

1Multiply each one (the tables show the answers and the next step of each pattern).

211s: the digits rise to the number of 1s and come back down: 1 2 1, 1 2 3 2 1, …

366 × 61 pattern: one more 4 at the front and one more 2 near the end each time.

433 × 35 pattern: the 1s and 5s grow by one each time.

5101 × 101 pattern: 102 → 10404 = 1 | 04 | 04, 103 → 1 | 06 | 09: middle part is double, end part is the square.

Product	Answer
11 × 11	121
111 × 111	12321
1111 × 1111	1234321
11111 × 11111	123454321

Product	Answer
66 × 61	4026
666 × 661	440226
6666 × 6661	44402226
66666 × 66661	4444022226

Product	Answer
3 × 5	15
33 × 35	1155
333 × 335	111555
3333 × 3335	11115555

Product	Answer
101 × 101	10201
102 × 102	10404
103 × 103	10609
104 × 104	10816

✅ Answer: 121, 12321, 1234321, (123454321); 4026, 440226, 44402226, (4444022226); 15, 1155, 111555, (11115555); 10201, 10404, 10609, (10816)

📖 Textbook Page 15

📖 Page 15Observe

Look at the number of digits in the numbers multiplied and in their product. Is there a connection?

111 × 11 (2 + 2 digits) = 121 (3 digits); 66 × 61 (2 + 2) = 4026 (4 digits).

2111 × 111 (3 + 3) = 12321 (5 digits); 666 × 661 (3 + 3) = 440226 (6 digits).

3So the product has either the total number of digits, or one less.

✅ Answer: If the numbers have a and b digits, the product has (a + b) or (a + b − 1) digits.

📖 Page 15Roxie’s claim

Roxie says the product of two 2-digit numbers can only be a 3- or 4-digit number. Is she correct? Do we need to try all multiplications?

1Smallest product: 10 × 10 = 100 (3 digits).

2Every 2-digit number is less than 100, so the product is less than 100 × 100 = 10,000 → at most 4 digits.

3So every product lies from 100 to 9,999 → 3 or 4 digits.

4We do not need to try all — checking the smallest and the largest possible products is enough.

✅ Answer: Yes, she is correct. Checking the smallest (10 × 10) and the upper limit (100 × 100) proves it.

📖 Page 15Think

Can multiplying a 3-digit number by another 3-digit number give a 4-digit number?

1The smallest such product is 100 × 100 = 10,000, which has 5 digits.

✅ Answer: No. The product always has 5 or 6 digits.

📖 Page 15Think

Can multiplying a 4-digit number by a 2-digit number give a 5-digit number?

1Smallest: 1,000 × 10 = 10,000 (5 digits) ✔

2Largest: 9,999 × 99 = 9,89,901 (6 digits).

✅ Answer: Yes. For example 1,000 × 10 = 10,000.

📖 Page 15Complete the table

Complete: 5-digit × 5-digit = ___ or ___; 8-digit × 3-digit = ___ or ___; 12-digit × 13-digit = ___ or ___

1Rule: a-digit × b-digit gives (a + b − 1) or (a + b) digits.

25 + 5 = 10 → 9 or 10

38 + 3 = 11 → 10 or 11

412 + 13 = 25 → 24 or 25

Multiplication	Digits in the product
1-digit × 1-digit	1 or 2
2-digit × 1-digit	2 or 3
2-digit × 2-digit	3 or 4
3-digit × 3-digit	5 or 6
5-digit × 5-digit	9 or 10
8-digit × 3-digit	10 or 11
12-digit × 13-digit	24 or 25

✅ Answer: 9 or 10; 10 or 11; 24 or 25

📖 Textbook Page 16

📖 Page 16Fascinating fact

1250 × 380 is the number of kīrtanas composed by Purandaradāsa (according to legends). Find it and read it in both systems.

11250 × 380 = 125 × 38 × 100

2125 × 38 = 125 × 40 − 125 × 2 = 5,000 − 250 = 4,750

34,750 × 100 = 4,75,000

✅ Answer: 4,75,000 — four lakh seventy-five thousand (475 thousand in the American system).

📖 Page 16Think

If Purandaradāsa composed 4,75,000 songs, how many songs per year did he have to compose?

1He lived about 80 years (1484–1564). Suppose he composed songs for about 50 years.

2Songs per year = 4,75,000 ÷ 50 = 9,500.

3Songs per day ≈ 9,500 ÷ 365 ≈ 26.

✅ Answer: About 9,500 songs a year (about 26 a day) if he composed for 50 years — which shows why the number is a legend!

💡 Tip: Your answer depends on the number of years you assume. Show your assumption.

📖 Page 16Fascinating fact

2100 × 70,000 is the approximate distance (km) between the Earth and the Sun. Find it and read it.

121 × 7 = 147

22100 × 70,000 has 2 + 4 = 6 zeros → 147 followed by 6 zeros

3= 14,70,00,000

✅ Answer: 14,70,00,000 km — fourteen crore seventy lakh (147 million) km.

📖 Textbook Page 17

📖 Page 17Fascinating fact

6400 × 62,500 is the average number of litres of water the Amazon river pours into the ocean every second. Find it.

164 × 625 = 40,000

2Add the 2 + 2 = 4 zeros: 40,000 × 10,000 = 40,00,00,000

✅ Answer: 40,00,00,000 litres — forty crore (400 million) litres per second.

📖 Page 17Fascinating fact

13,95,000 ÷ 150 is the length (km) of the longest single train journey (Moscow–Vladivostok). Find it.

1Remove one zero from both: 1,39,500 ÷ 15

21,39,500 ÷ 15 = 9,300

3Check: 9,300 × 150 = 13,95,000 ✔

✅ Answer: 9,300 km.

📖 Textbook Page 18

📖 Page 18Fascinating fact

Adult blue whales can weigh more than 10,50,00,000 ÷ 700 kg. Find the weight.

1Remove two zeros from both: 10,50,000 ÷ 7

2= 1,50,000

3Check: 1,50,000 × 700 = 10,50,00,000 ✔

✅ Answer: 1,50,000 kg (one lakh fifty thousand kg = 150 tonnes).

📖 Page 18Fascinating fact

52,00,00,00,000 ÷ 130 was the weight (tonnes) of plastic waste made in the world in 2021. Find it.

1Remove one zero from both: 5,20,00,00,000 ÷ 13

252 ÷ 13 = 4, so 5,20,00,00,000 ÷ 13 = 40,00,00,000

3Check: 40,00,00,000 × 130 = 52,00,00,00,000 ✔

✅ Answer: 40,00,00,000 tonnes — forty crore (400 million) tonnes.

📖 Textbook Page 19

📖 Page 19Did you ever wonder?

The RMS Titanic carried about 2,500 passengers. Can the population of Mumbai fit into 5,000 such ships?

1People in 5,000 ships = 5,000 × 2,500 = 1,25,00,000 (1 crore 25 lakh).

2Mumbai’s population = 1,24,42,373.

31,25,00,000 is a little more than 1,24,42,373.

✅ Answer: Yes — just about! 5,000 ships hold 1,25,00,000 people, about 57,600 more than Mumbai’s population.

📖 Page 19Did you ever wonder?

Roxie travels 100 km every day. Can she reach the Moon (3,84,400 km away) in 10 years?

1In 1 year: 100 × 365 = 36,500 km.

2In 10 years: 36,500 × 10 = 3,65,000 km.

33,65,000 < 3,84,400 — she is still 19,400 km short.

✅ Answer: No. In 10 years she covers only 3,65,000 km (she needs about 10½ years).

📖 Page 19Did you ever wonder?

If you travel 1,000 km every day, can you reach the Sun in a lifetime? (Earth–Sun ≈ 14,70,00,000 km)

1Days needed = 14,70,00,000 ÷ 1,000 = 1,47,000 days.

2Years = 1,47,000 ÷ 365 ≈ 403 years.

✅ Answer: No. It would take about 403 years.

📖 Page 19Reasonable assumptions (a)

A sheet of paper weighs 5 g. Could you lift one lakh sheets together?

1Weight = 1,00,000 × 5 g = 5,00,000 g.

21,000 g = 1 kg → 5,00,000 g = 500 kg.

✅ Answer: No — 1 lakh sheets weigh 500 kg, far too heavy for a person.

📖 Page 19Reasonable assumptions (b)

If 250 babies are born every minute in the world, will a million babies be born in a day?

1Minutes in a day = 24 × 60 = 1,440.

2Babies = 250 × 1,440 = 3,60,000.

31 million = 10,00,000. 3,60,000 < 10,00,000.

✅ Answer: No — about 3,60,000 babies are born in a day (less than a million).

📖 Page 19Reasonable assumptions (c)

Can you count 1 million coins in a day if you count 1 coin every second?

1Seconds in a day = 24 × 60 × 60 = 86,400.

2So at most 86,400 coins in a day (without sleeping!).

31 million = 10,00,000 → it would take about 12 days non-stop.

✅ Answer: No. You can count only 86,400 coins in a day.

📖 Page 19Figure it Out · Q1

Using all digits 0–9 exactly once (first digit not 0), write the (a) largest multiple of 5 (b) smallest even 10-digit number.

1(a) A multiple of 5 ends in 0 or 5. Put 0 at the end and the other digits in decreasing order: 9876543210.

2(b) Smallest: start with 1, then 0, then increasing: 1023456789 — but it ends in 9 (odd).

3Swap the last two digits so it ends in an even digit: 1023456798.

✅ Answer: (a) 9876543210 (b) 1023456798

📖 Textbook Page 20

📖 Page 20Figure it Out · Q2

“Ten lakhs thirty thousand two hundred eighty five” (10,30,285) has 42 letters. Give a 7-digit number whose name has the maximum number of letters.

1Make every part of the name as long as possible.

2Longest tens names: ‘seventy’ (7 letters). Longest ones names: ‘three’, ‘seven’, ‘eight’ (5 letters). So ‘seventy-seven’ = 12 letters.

3Lakh part: seventy-seven (12) + lakhs (5) = 17. Thousand part: seventy-seven (12) + thousand (8) = 20.

4Hundreds: seven (5) + hundred (7) = 12. Last part: seventy-seven (12).

5Total = 17 + 20 + 12 + 12 = 61 letters.

✅ Answer: 77,77,777 — ‘seventy seven lakhs seventy seven thousand seven hundred seventy seven’ has 61 letters (60 if we write ‘lakh’). 73,73,373 or 78,78,878 are just as long.

📖 Page 20Figure it Out · Q3

Write a 9-digit number where exchanging ANY two digits gives a bigger number. How many such numbers exist?

1If two digits are equal, swapping them changes nothing — so all 9 digits must be different.

2Swapping must always bring a bigger digit to the front → the digits must be in increasing order from left to right.

30 cannot be used: it cannot be first, and if it is later, swapping it with the first digit makes the number smaller (or not 9-digit).

4So we use 1 to 9 in increasing order.

✅ Answer: 123456789 — it is the only such number.

📖 Page 20Figure it Out · Q4

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

120 digits − 10 = 10 digits must remain. We want the biggest possible first digit, then second, and so on.

21st digit: choose from the first 11 digits (we must leave 9 after it). Biggest = 5 (the 5th digit).

32nd digit: from digits 6 to 12 → biggest = 5 (the 10th digit).

43rd digit: from digits 11 to 13 (1, 2, 3) → 3.

54th digit: only digit 14 remains possible → 4. The rest are forced: 5, 1, 2, 3, 4, 5.

✅ Answer: 5534512345

📖 Page 20Figure it Out · Q5

‘zero’–‘one’ share e, o; ‘one’–‘two’ share o; ‘two’–‘three’ share t. How far must you count to find two consecutive numbers whose names share no letter?

1Check pairs: three–four (r), four–five (f), five–six (i), six–seven (s), seven–eight (e), eight–nine (e, i, n), nine–ten (e, n), ten–eleven (e, n) …

2Bigger numbers share whole words: ‘twenty-one, twenty-two’ share ‘twenty’; 99 → 100: ‘ninety-nine’ and ‘one hundred’ share n, e.

3Even at jumps like 9,999 → 10,000 or 99,999 → 1,00,000, the names share letters (‘nine … thousand’ and ‘ten thousand’ / ‘one lakh’ share n, e).

4Checking every number up to 30 lakh, every pair of consecutive number names shares at least one letter.

✅ Answer: You never find one in ordinary counting — every two consecutive number names share at least one letter (checked up to 30 lakh).

💡 Tip: Letters e, n and t appear in almost every number name.

📖 Page 20Figure it Out · Q6(a)

Writing 1, 2, 3, … in a row, what is the 1000th digit? In which number does it occur?

11 to 9: 9 digits.

210 to 99: 90 numbers × 2 = 180 digits. Total so far 189.

31000 − 189 = 811 digits more, from 3-digit numbers.

4811 ÷ 3 = 270, remainder 1 → 270 full numbers (100 to 369) and 1 more digit.

5The next number is 370, and the 1000th digit is its first digit.

✅ Answer: The 1000th digit is 3 — the first digit of 370.

📖 Page 20Figure it Out · Q6(b)

Which number contains the millionth (10,00,000th) digit?

1Digits used: 1–9 → 9; 10–99 → 180; 100–999 → 2,700; 1000–9999 → 36,000; 10000–99999 → 4,50,000.

2Total = 4,88,889 digits.

3Left: 10,00,000 − 4,88,889 = 5,11,111 digits from 6-digit numbers.

45,11,111 ÷ 6 = 85,185 remainder 1 → 85,185 full numbers (1,00,000 to 1,85,184) and 1 more digit.

5So the millionth digit is the first digit of the next number, 1,85,185.

✅ Answer: 1,85,185 (the millionth digit is its first digit, 1).

📖 Page 20Figure it Out · Q6(c)

When would you have written the digit ‘5’ for the 5000th time?

1From 1 to 9,999 each place (ones, tens, hundreds, thousands) shows 5 exactly 1,000 times → 4,000 fives.

2We need 1,000 more fives after 10,000. Count the 5s in the last four digits from 10,000 onward:

310,000 to 12,999: 3 × 300 = 900 fives (each block of 1,000 has 300 fives).

413,000 to 13,494: 49 fives in the ones place + 50 in the tens place = 99 fives.

5Total so far: 4,000 + 900 + 99 = 4,999. The next 5 is written in 13,495 — the 5,000th five.

✅ Answer: At the number 13,495.

💡 Tip: Some answer keys write 13,995 — counting carefully shows 13,495.

📖 Page 20Figure it Out · Q7(a)

A calculator has only ‘+10,000’ and ‘+100’ buttons. Write an expression for 20,800 and find the clicks.

120,800 = 20,000 + 800

2Expression: (2 × 10,000) + (8 × 100)

3Clicks = 2 + 8 = 10

✅ Answer: (2 × 10,000) + (8 × 100); 10 clicks

📖 Page 20Figure it Out · Q7(b)

A calculator has only ‘+10,000’ and ‘+100’ buttons. Write an expression for 92,100 and find the clicks.

192,100 = 90,000 + 2,100

2Expression: (9 × 10,000) + (21 × 100)

3Clicks = 9 + 21 = 30

✅ Answer: (9 × 10,000) + (21 × 100); 30 clicks

📖 Page 20Figure it Out · Q7(c)

A calculator has only ‘+10,000’ and ‘+100’ buttons. Write an expression for 1,20,500 and find the clicks.

11,20,500 = 1,20,000 + 500

2Expression: (12 × 10,000) + (5 × 100)

3Clicks = 12 + 5 = 17

✅ Answer: (12 × 10,000) + (5 × 100); 17 clicks

📖 Page 20Figure it Out · Q7(d)

A calculator has only ‘+10,000’ and ‘+100’ buttons. Write an expression for 65,30,000 and find the clicks.

165,30,000 = 653 × 10,000

2Expression: (653 × 10,000)

3Clicks = 653 + 0 = 653

✅ Answer: (653 × 10,000); 653 clicks

📖 Page 20Figure it Out · Q7(e)

A calculator has only ‘+10,000’ and ‘+100’ buttons. Write an expression for 70,25,700 and find the clicks.

170,25,700 = 70,20,000 + 5,700

2Expression: (702 × 10,000) + (57 × 100)

3Clicks = 702 + 57 = 759

✅ Answer: (702 × 10,000) + (57 × 100); 759 clicks

📖 Page 20Figure it Out · Q8

How many lakhs make a billion?

11 billion = 1,000,000,000 = 1,00,00,00,000 = 100 crore.

21 crore = 100 lakh → 100 crore = 100 × 100 = 10,000 lakh.

✅ Answer: 10,000 lakhs make a billion.

📖 Page 20Figure it Out · Q9

Two sets of cards 1–9 (each digit twice). Fill a 7-digit number (top) and a 5-digit number (bottom) to get (a) the largest possible sum (b) the smallest possible difference.

1(a) Put the biggest cards in the biggest places. The top number’s first two places (ten lakhs and lakhs) are alone: 9 and 9.

2From ten thousands down, each place appears in BOTH numbers: give them 8 & 8, 7 & 7, 6 & 6, 5 & 5, 4 & 4.

399,87,654 + 87,654 = 1,00,75,308.

4(b) Make the 7-digit number as small as possible: 11,22,334. Make the 5-digit number as large as possible: 99,887.

511,22,334 − 99,887 = 10,22,447.

✅ Answer: (a) 99,87,654 + 87,654 = 1,00,75,308 (b) 11,22,334 − 99,887 = 10,22,447

⚠️ Common mistake: Using 9988776 + 65544: this gives only 1,00,54,320. The ten-thousands place is in both numbers, so it should get the 8s.

📖 Textbook Page 21

📖 Page 21Figure it Out · Q10(a)

Cards: 4000, 13000, 300, 70000, 150000, 20, 5 (each used once). Get as close as you can to 1,10,000.

1Try: 13,000 × 20 − 1,50,000

2Work out: = 1,10,000

3This is exactly 1,10,000 ✔

✅ Answer: 13,000 × 20 − 1,50,000 = 1,10,000

💡 Tip: (a) The book’s example gives 1,13,000; 13,000 × 20 − 1,50,000 gets 1,10,000 exactly!

📖 Page 21Figure it Out · Q10(b)

Cards: 4000, 13000, 300, 70000, 150000, 20, 5 (each used once). Get as close as you can to 2,00,000.

1Try: 1,50,000 + 70,000 − 4,000 × 5

2Work out: = 2,00,000

3This is exactly 2,00,000 ✔

✅ Answer: 1,50,000 + 70,000 − 4,000 × 5 = 2,00,000

📖 Page 21Figure it Out · Q10(c)

Cards: 4000, 13000, 300, 70000, 150000, 20, 5 (each used once). Get as close as you can to 5,80,000.

1Try: 70,000 × 5 + 1,50,000 + 4,000 × 20

2Work out: = 5,80,000

3This is exactly 5,80,000 ✔

✅ Answer: 70,000 × 5 + 1,50,000 + 4,000 × 20 = 5,80,000

📖 Page 21Figure it Out · Q10(d)

Cards: 4000, 13000, 300, 70000, 150000, 20, 5 (each used once). Get as close as you can to 12,45,000.

1Try: (13,000 + 70,000) × 300 ÷ 20

2Work out: = 12,45,000

3This is exactly 12,45,000 ✔

✅ Answer: (13,000 + 70,000) × 300 ÷ 20 = 12,45,000

📖 Page 21Figure it Out · Q10(e)

Cards: 4000, 13000, 300, 70000, 150000, 20, 5 (each used once). Get as close as you can to 20,90,800.

1Try: (70,000 + 300) × 20 + (1,50,000 − 13,000) × 5

2Work out: = 20,91,000

3This is only 200 away from 20,90,800.

✅ Answer: (70,000 + 300) × 20 + (1,50,000 − 13,000) × 5 = 20,91,000

📖 Page 21Figure it Out · Q11

How many coins (each 1 mm thick) must be stacked to match the height of the Statue of Unity (180 m)?

11 m = 100 cm = 1,000 mm.

2180 m = 180 × 1,000 = 1,80,000 mm.

3Coins = 1,80,000 ÷ 1 = 1,80,000.

✅ Answer: 1,80,000 coins.

📖 Page 21Figure it Out · Q12

An albatross flies about 900–1,000 km a day. About how many days would a 12,000 km trip take?

1At 1,000 km a day: 12,000 ÷ 1,000 = 12 days.

2At 900 km a day: 12,000 ÷ 900 ≈ 13.3 days.

✅ Answer: About 12 to 14 days.

📖 Page 21Figure it Out · Q13

A bar-tailed godwit flew 13,560 km non-stop in about 11 days. About how far did it fly each day? Each hour?

1Per day: 13,560 ÷ 11 ≈ 1,233 km (estimate: 13,200 ÷ 11 = 1,200).

2Hours in 11 days = 11 × 24 = 264.

3Per hour: 13,560 ÷ 264 ≈ 51 km.

✅ Answer: About 1,233 km a day and about 51 km an hour.

📖 Page 21Figure it Out · Q14

Bald eagles fly 4,500–6,000 m high, Mount Everest is about 8,850 m, aeroplanes fly 10,000–12,800 m. How many times the height of Somu’s 40 m building are these?

1Eagles: 4,500 ÷ 40 = 112.5 and 6,000 ÷ 40 = 150 → about 112 to 150 times.

2Everest: 8,850 ÷ 40 ≈ 221 times.

3Aeroplanes: 10,000 ÷ 40 = 250 and 12,800 ÷ 40 = 320 → 250 to 320 times.

✅ Answer: Eagles ≈ 112–150 times, Everest ≈ 221 times, aeroplanes ≈ 250–320 times.

📖 Textbook Page 23

📖 Page 23Toothpick Digits

Write 5108 with sticks. How many sticks are needed?

Digits made with sticks (number of sticks under each digit).

1Sticks for each digit: 5 → 5, 1 → 2, 0 → 6, 8 → 7.

2Total = 5 + 2 + 6 + 7 = 20.

5108 needs 5 + 2 + 6 + 7 = 20 sticks.

✅ Answer: 20 sticks.

📖 Page 23Toothpick Digits · Set 1

(1) Check that 42,019 needs 23 sticks. (2) Add exactly two sticks to make a bigger number (one example is 42,078). (3) Where should the digit ‘1’ be inserted to get the biggest number? (4) What other numbers can be made by inserting ‘1’?

1(1) 4 + 5 + 6 + 2 + 6 = 23 ✔

2(2) Adding sticks to the digits: 1 → 4 or 1 → 7 (+2 or +1), 0 → 8 (+1), 9 → 8 (+1), 2 → 8 (+2), 4 → 9 (+2).

3Bigger numbers with exactly 2 more sticks: 42,049, 42,078, 42,818, 42,879, 48,019, 92,019. The biggest is 92,019.

4(3) Inserting ‘1’ (2 sticks): 1,42,019; 4,12,019; 4,21,019; 4,20,119; 4,20,191.

5Biggest is 4,21,019 (put 1 after 2).

42,019 needs 4 + 5 + 6 + 2 + 6 = 23 sticks.

✅ Answer: (2) 42,049, 42,078, 42,818, 42,879, 48,019, 92,019 (3) 4,21,019 (4) 1,42,019; 4,12,019; 4,20,119; 4,20,191

📖 Page 23Toothpick Digits · Set 2

Make 63,890. Rearrange exactly four sticks to make bigger numbers (one example is 88,078). What others can you make?

163,890 uses 6 + 5 + 7 + 6 + 6 = 30 sticks. After moving 4 sticks we still have 30 sticks.

2Example 99,985: take away 4 sticks (one from 6, one from 8, two from 0) and put them back (one each on 6, 3, 9 and 0) → 9, 9, 9, 8, 5.

3Other bigger numbers: 99,983, 99,982, 99,969, 99,966.

4There are 704 such bigger numbers in all; the biggest is 99,985.

✅ Answer: For example 99,985, 99,983, 99,982, 99,969, 99,966 (biggest 99,985)

📖 Page 23Toothpick Digits · Set 3

(1) Make any number using exactly 24 sticks. (2) What is the biggest number with 24 sticks? (3) The smallest?

1(1) Example: 8,888 uses 28 — too many; 2,008 uses 5 + 6 + 6 + 7 = 24 ✔.

2(2) Biggest: use as many digits as possible. ‘1’ uses only 2 sticks → 24 ÷ 2 = 12 ones: 111111111111.

3(3) Smallest: use as few digits as possible. 3 digits need at most 7 × 3 = 21 sticks, so we need 4 digits.

4First digit as small as possible: ‘1’ (2) leaves 22 for 3 digits — too many. ‘2’ (5) leaves 19 → 0 (6), 0 (6), 8 (7) = 19 ✔.

✅ Answer: (2) 1,11,11,11,11,111 (twelve 1s) (3) 2,008

🌍 Where do we use large numbers?

Population of towns, cities and countries; votes in elections.
Money: government budgets, bank amounts, prices of houses.
Science: distance to the Sun and stars, number of bacteria, water in rivers.

🚀 Link to higher classes

Class 8: powers of 10 and scientific notation (Power Play), estimation.
Class 9: real numbers and place value of decimals.

⬆ Top 🏠 Class 7 Hub

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