Ganita Prakash · Grade 7 · Chapter 1
Large Numbers Around Us — Complete Worked Solutions
Every in-text question, Math Talk, Figure-it-Out and puzzle from the chapter — solved step by step, with diagrams, LaTeX-rendered working, and direct answers. No digging, no hidden reveals.
Lakh = 1,00,000
Crore = 1,00,00,000
Arab = 1,00,00,00,000
1 Million = 10,00,000
1 Billion = 1,00,00,00,000
1.1
A Lakh Varieties!
Eshwarappa overhears that India once had about a lakh varieties of rice. Roxie and Estu explore just how big one lakh really is.
Q
Estu tasted 3 varieties so far. If he tries one new variety a day, would he come close to tasting all 1 lakh varieties in a 100-year lifetime? What do you think? Guess.
Solution
- Days in one year (ignoring leap years) = 365
- Varieties tasted in 100 years \( = 365 \times 100 = 36{,}500 \)
- Compare with 1 lakh = 1,00,000
36,500 ≪ 1,00,000 — nowhere close! Even eating one new variety every single day of a 100-year life, only about a third of a lakh would be tasted.
Q
Fill in the boxes — observe the pattern of largest/smallest numbers leading up to one lakh.
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 = One Lakh
Solution
Each time we run out of digits at one place value, adding just 1 more rolls it over into the next: 999 + 1 = 1,000, 9,999 + 1 = 10,000, and 99,999 + 1 = 1,00,000. This is exactly how one lakh is born — it’s simply “one more” than the largest 5-digit number.
Counting sequence near a lakh: 99,995 → 99,996 → 99,997 → 99,998 → 99,999 → 1,00,000 → 1,00,001
Q
Roxie suggests eating 2 varieties of rice a day. Would they finish 1 lakh varieties in 100 years?
Solution
- Varieties per year \( = 2 \times 365 = 730 \)
- Varieties in 100 years \( = 730 \times 100 = 73{,}000 \)
73,000 < 1,00,000 → Still not enough. They would fall short by 27,000 varieties.
Q
What if a person ate 3 varieties of rice every day? Will they taste all lakh varieties in a 100-year lifetime?
Solution
- Varieties per year \( = 3 \times 365 = 1{,}095 \)
- Varieties in 100 years \( = 1{,}095 \times 100 = 1{,}09{,}500 \)
1,09,500 > 1,00,000 — Yes! At 3 varieties a day, they would just manage to taste all the varieties (with 9,500 to spare).
Q
Choose a number for \(y\). How close to one lakh is the number of days in \(y\) years, for the \(y\) of your choice?
Solution
Days in \(y\) years \( = 365 \times y \). Trying \(y = 274\): \(365 \times 274 = 1,00,010\) — extremely close to one lakh!
So living about 274 years (365 × 274 = 1,00,010 days) gets you almost exactly to one lakh days of life.
1
According to the 2011 Census, Chintamani’s population was about 75,000. How much less than one lakh is 75,000?
Solution\( 1{,}00{,}000 – 75{,}000 = 25{,}000 \)
75,000 is 25,000 less than 1 lakh.
2
Estimated 2024 population of Chintamani is 1,06,000. How much more than one lakh is 1,06,000?
Solution\( 1{,}06{,}000 – 1{,}00{,}000 = 6{,}000 \)
1,06,000 is 6,000 more than 1 lakh.
3
By how much did the population of Chintamani increase from 2011 to 2024?
Solution\( 1{,}06{,}000 – 75{,}000 = 31{,}000 \)
Population increased by 31,000.
Q
Somu is 1 metre tall and each floor of the building is about 4 times his height. What is the approximate height of the building, and how does it compare to the Statue of Unity (180 m) and Kunchikal waterfall (450 m)?
Heights compared to Somu (1 m) as a familiar reference — bars scaled proportionally
Solution
- Height of building: assume ~10 floors visible → \(10 \times 4 \times 1\text{ m} = 40\text{ m}\)
- Statue of Unity vs building: \(180 – 40 = 140\) m taller
- Kunchikal waterfall vs building: \(450 – 40 = 410\) m taller
- Floors to match waterfall height: \(450 \div 4 \approx 112.5 \approx\) 113 floors
Building ≈ 40 m · Statue of Unity is 140 m taller · Waterfall is 410 m taller · ~113 floors needed to match the waterfall
Q
Is one lakh a very large number? Roxie thinks it’s huge, Estu thinks it’s small. How do you view a lakh?
Both views are correct — it depends on context. A lakh feels enormous when compared to something spread out or slow (like 1 lakh days = 274 years, or a 38 km line of people). But the same lakh feels small when it’s packed into a tiny, dense space (like 1 lakh hairs on a head, or 1 lakh people filling one cricket stadium). The size of a number only becomes meaningful once we compare it to something familiar — that’s the real lesson of this chapter.
1.1
Reading and Writing Numbers
Q
Write each number in words: (a) 3,00,600 (b) 5,04,085 (c) 27,30,000 (d) 70,53,138
Solution
- (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
Q
Write the corresponding number: (a) One lakh twenty-three thousand four hundred fifty-six (b) Four lakh seven thousand seven hundred four (c) Fifty lakhs five thousand fifty (d) Ten lakhs two hundred thirty-five
Solution
- (a) 1,23,456
- (b) 4,07,704
- (c) 50,05,050
- (d) 10,00,235
1.2
Land of Tens
Special calculators — each with only one repeat-add button — show how place value is really just “how many times” we add a chunk.
+1000
Thoughtful
Thousands
Thousands
+10
Tedious
Tens
Tens
+100
Handy
Hundreds
Hundreds
1
Thoughtful Thousands (+1000 button only) — how many presses to show:
Solution
- (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) Thousands needed for 1 lakh → 100 (since \(1000 \times 100 = 1,00,000\))
2
Tedious Tens (+10 button only) — how many presses to show:
Solution
- (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
3
Handy Hundreds (+100 button only) — how many presses to show:
Solution
- (a) Four hundred → 4 times
- (b) 3,700 → 37 times
- (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 → 1,000 times
- (h) 582 times → 58,200
- (i) Hundreds needed for ten thousand → 100
- (j) Hundreds needed for one lakh → 1,000
- (k) “Handy Hundreds can show numbers Tedious Tens and Thoughtful Thousands can’t” — False. Since 100 is itself a multiple of 10, and both 10 and 100 are factors of 1000, every number Handy Hundreds can make in hundreds is also reachable by Tedious Tens in tens — Handy Hundreds has no exclusive numbers.
Q
Creative Chitti (buttons +1,+10,+100,…,+10,00,000) — find a different way to get 5072.
| Buttons | Way A | Way B | Way C (new) |
|---|---|---|---|
| +1,00,000 | – | – | – |
| +10,000 | – | – | – |
| +1,000 | 5 | 3 | 4 |
| +100 | – | 20 | 10 |
| +10 | 7 | – | 7 |
| +1 | 2 | 72 | 2 |
Solution
New expression: \( (4 \times 1000) + (10 \times 100) + (7 \times 10) + (2 \times 1) = 4000 + 1000 + 70 + 2 = 5072 \) ✓
One valid new way: (4×1000) + (10×100) + (7×10) + (2×1) = 5072
FQ
Figure it Out — write two ways to obtain each number through button clicks: (a) 8300 (b) 40629 (c) 56354 (d) 66666 (e) 367813
Solution
- (a) 8300 = (8×1000)+(3×100) or (5×1000)+(33×100)
- (b) 40629 = (40×1000)+(6×100)+(2×10)+(9×1) or (30×1000)+(106×100)+(29×1)
- (c) 56354 = (56×1000)+(3×100)+(5×10)+(4×1) or (46×1000)+(103×100)+(54×1)
- (d) 66666 = (66×1000)+(6×100)+(66×1) or (6×10000)+(6×1000)+(6×100)+(6×10)+(6×1)
- (e) 367813 = (3×100000)+(6×10000)+(7×1000)+(8×100)+(1×10)+(3×1) or (36×10000)+(78×1000)+(13×1)
Q
Creative Chitti’s Challenge — (a) With exactly 30 button presses, what’s the largest and smallest 3-digit number possible? (b) 997 uses 25 clicks — can you make 997 with a different number of clicks?
Solution
- (a) Largest 3-digit: 993 Smallest 3-digit: 102 — using the leftover clicks on the ones-place while staying within a 3-digit result.
- (b) Yes — 997 = (8×100)+(19×10)+(7×1); clicks = 8+19+7 = 34 clicks.
Q
Systematic Sippy wants minimum clicks. Find the fewest button clicks to make (a) 5072 (b) 8300.
Solution
- (a) 5072 = (5×1000)+(7×10)+(2×1) → clicks = 5+7+2 = 14 clicks
- (b) 8300 = (8×1000)+(3×100) → clicks = 8+3 = 11 clicks
💡 Insight: The fewest possible clicks for any number equals the sum of its digits — this is exactly the Indian place-value notation itself!
FQ1
Find the smallest number of button clicks (and the expression) for each number from the earlier exercise.
Solution
- 8300 = (8×1000)+(3×100) → 11 clicks
- 40629 = (4×10000)+(6×100)+(2×10)+(9×1) → 21 clicks
- 56354 = (5×10000)+(6×1000)+(3×100)+(5×10)+(4×1) → 23 clicks
- 66666 = (6×10000)+(6×1000)+(6×100)+(6×10)+(6×1) → 30 clicks
- 367813 = (3×100000)+(6×10000)+(7×1000)+(8×100)+(1×10)+(3×1) → 28 clicks
FQ2
Do you see a connection between each number and its smallest number of button clicks?
Solution
Yes — the smallest number of clicks always equals the sum of the digits of the number.
FQ3
The expressions for least clicks also give the Indian place-value notation. Why?
SolutionBecause using the fewest clicks means using the largest possible button for each digit position without exceeding it — which is precisely what each digit in a number already represents (its place value multiplied by the digit itself). Any other combination “wastes” clicks by re-grouping ten of a smaller unit instead of one of the next bigger unit.
Q
If we press +10,00,000 button ten times, what number comes up? What is it called?
Solution\(10 \times 10,00,000 = 1,00,00,000\) — this is 1 crore (also called “100 lakh”), written as 1 followed by 7 zeroes.
Q
How many zeroes does a thousand lakh have? How many zeroes does a hundred thousand have?
Solution
Thousand lakh \( = 1000 \times 1,00,000 = 10,00,00,000\) → 8 zeroes.
Hundred thousand \( = 100 \times 1000 = 1,00,000 \) (one lakh) → 5 zeroes.
Hundred thousand \( = 100 \times 1000 = 1,00,000 \) (one lakh) → 5 zeroes.
1.3
Of Crores and Crores!
Indian system groups digits 3-2-2-2 (thousand, lakh, crore…); American/International system groups uniformly 3-3-3 (thousand, million, billion…).
| Indian System | Value | American System |
|---|---|---|
| 1,000 | 1,000 | One thousand |
| 1,00,000 (One Lakh) | 100,000 | Hundred thousand |
| 10,00,000 (Ten Lakh) | 1,000,000 | One million |
| 1,00,00,000 (One Crore) | 10,000,000 | Ten million |
| 1,00,00,00,000 (One Arab) | 1,000,000,000 | One billion |
FQ1
Read the following in Indian place value notation and write number names in both Indian and American systems: (a) 4050678 (b) 48121620 (c) 20022002 (d) 246813579 (e) 345000543 (f) 1020304050
Solution
- (a) 40,50,678 — Indian: Forty lakh fifty thousand six hundred seventy-eight; American: Four million fifty thousand six hundred seventy-eight
- (b) 4,81,21,620 — Indian: Four crore eighty-one lakh twenty-one thousand six hundred twenty; American: Forty-eight million one hundred twenty-one thousand six hundred twenty
- (c) 2,00,22,002 — Indian: Two crore twenty-two thousand two; American: Twenty million twenty-two thousand two
- (d) 24,68,13,579 — Indian: Twenty-four crore sixty-eight lakh thirteen thousand five hundred seventy-nine; American: Two hundred forty-six million eight hundred thirteen thousand five hundred seventy-nine
- (e) 34,50,00,543 — Indian: Thirty-four crore fifty lakh five hundred forty-three; American: Three hundred forty-five million five hundred forty-three
- (f) 1,02,03,04,050 — Indian: One arab two crore three lakh four thousand fifty; American: One billion twenty million three hundred four thousand fifty
FQ2
Write in Indian place value notation: (a) One crore one lakh one thousand ten (b) One billion one million one thousand one (c) Ten crore twenty lakh thirty thousand forty (d) Nine billion eighty million seven hundred thousand six hundred
Solution
- (a) 1,01,01,010
- (b) 1,00,10,01,001
- (c) 10,20,30,040
- (d) 9,08,07,00,600
FQ3
Compare and write ‘<', '>‘ or ‘=’: (a) 30 thousand ___ 3 lakhs (b) 500 lakhs ___ 5 million (c) 800 thousand ___ 8 million (d) 640 crore ___ 60 billion
Solution
- (a) 30,000 < 3,00,000
- (b) 5,00,00,000 > 50,00,000 (5 million = 50 lakh)
- (c) 8,00,000 < 80,00,000 (8 million = 80 lakh)
- (d) 640 crore = 6,40,00,00,000; 60 billion = 6,00,00,00,000 → 640 crore > 60 billion
1.4
Exact and Approximate Values
Q
Think and share situations where it is appropriate to (a) round up (b) round down (c) either is okay (d) exact numbers are needed.
Solution
- (a) Round up: Buying food/sweets for a group (better to have slightly more than run short).
- (b) Round down: A shopkeeper quoting an approximate lower price to sound attractive, e.g. ₹470 quoted as “around ₹450”.
- (c) Either is fine: Estimating distance between two far-off cities for general conversation.
- (d) Exact needed: Handling money in a bank, counting exam marks, medicine dosage.
Q
Nearest Neighbours — write the five nearest neighbours for: (a) 3,87,69,957 (b) 29,05,32,481
| (a) 3,87,69,957 | |
|---|---|
| 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 |
| (b) 29,05,32,481 | |
|---|---|
| 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 |
Q
Math Talk: A number’s five nearest neighbours are all 5,00,00,000. What could the number be? How many such numbers exist?
Solution
- The tightest rounding rule here is “nearest thousand” — a number rounds to 5,00,00,000 at the thousand place if it lies within 500 on either side of it.
- Since 5,00,00,000 is already a round number at every larger place value too (ten-thousand, lakh, ten-lakh, crore), any number close enough to satisfy the thousand-rounding will automatically satisfy all the coarser roundings as well.
- Range: from \(5,00,00,000 – 500 = 4,99,99,500\) to \(5,00,00,000 + 499 = 5,00,00,499\)
The number could be anywhere from 4,99,99,500 to 5,00,00,499 — that’s 1,000 possible numbers (including 5,00,00,000 itself).
1
Estimate \(4,63,128 + 4,19,682\). Roxie says “near 8,00,000, more than 8,00,000”; Estu says “near 9,00,000, less than 9,00,000”.
Solution
- Exact sum: \(4,63,128 + 4,19,682 = 8,82,810\)
- (a) Both estimates are directionally correct, but Estu’s estimate is closer — 8,82,810 is much nearer to 9,00,000 than to 8,00,000.
- (b) 4,63,128 alone is close to 4,50,000+ and 4,19,682 is close to 4,00,000+ — their sum easily crosses 8,50,000. So the sum is greater than 8,50,000.
- (c) Since \(4,19,682 < 4,20,000\), the sum \(4,63,128 + 4,19,682 < 4,63,128 + 4,20,000 = 8,83,128\). So the sum is less than 8,83,128.
- (d) Exact value = 8,82,810
2
Estimate \(14,63,128 – 4,90,020\). Roxie says “near 10,00,000, less than that”; Estu says “near 9,00,000, more than that”.
Solution
- Exact difference: \(14,63,128 – 4,90,020 = 9,73,108\)
- (a) Estu’s estimate is closer — 9,73,108 is very near 9,00,000-and-above, while it’s noticeably less than 10,00,000.
- (b) 9,73,108 is greater than 9,50,000.
- (c) 9,73,108 is greater than 9,63,128.
- (d) Exact value = 9,73,108
1.4
Populations of Cities
Mumbai
124L
124L
Delhi
110L
110L
Bengaluru
84L
84L
Hyderabad
68L
68L
Ahmedabad
56L
56L
Chennai
47L
47L
Patna
17L
17L
2011 population (in lakhs) — top cities by rank
1
What is your general observation about this population data?
SolutionNearly every city in the table shows a significant rise in population from 2001 to 2011 — many cities grew by several lakhs, and some (like Bengaluru, Hyderabad, Surat, Vadodara) nearly doubled in a decade.
2
What is an appropriate title for this table?
Solution“Population of Major Indian Cities: 2001 vs 2011”
3
Population of Pune in 2011, and approximate increase since 2001?
SolutionPopulation of Pune in 2011 = 31,15,431. Increase ≈ 31,15,431 − 25,38,473 ≈ 6 lakh.
4
Which city’s population increased the most between 2001 and 2011?
SolutionBengaluru — increase = 84,25,970 − 43,01,326 = 41,24,644, the largest jump in the table.
5
Are there cities whose population has almost doubled? Which are they?
SolutionYes — Bengaluru, Hyderabad, Surat, and Vadodara all show population figures in 2011 that are close to double their 2001 values.
6
By what number should we multiply Patna’s population to get close to Mumbai’s?
Solution\( \dfrac{1,24,42,373}{16,84,222} \approx 7.39 \) — multiplying Patna’s population by about 7.39 gets close to Mumbai’s population.
1.5
Patterns in Products
Q
Multiplication shortcut: \(116 \times 5 = 116 \times \frac{10}{2} = 58 \times 10 = 580\). And \(824 \times 25 = 824 \times \frac{100}{4} = 206 \times 100 = 20{,}600\). Why does multiplying by 5 equal dividing by 2 and multiplying by 10?
SolutionSince \(5 \times 2 = 10\), we know \(5 = \dfrac{10}{2}\). So multiplying any number by 5 is the same as multiplying it by \(\dfrac{10}{2}\) — which means multiply by 10 first, then divide by 2 (or divide by 2 first, then multiply by 10 — order doesn’t matter). This turns an awkward ×5 into an easy ×10 followed by a simple halving.
1
Find quick ways to calculate: (a) \(2 \times 1768 \times 50\) (b) \(72 \times 125\) (c) \(125 \times 40 \times 8 \times 25\)
Solution
- (a) \(2 \times 1768 \times 50 = 1768 \times (2 \times 50) = 1768 \times 100 = \mathbf{1{,}76{,}800}\)
- (b) \(72 \times 125 = 72 \times \dfrac{1000}{8} = \dfrac{72}{8}\times 1000 = 9 \times 1000 = \mathbf{9{,}000}\)
- (c) \(125 \times 40 \times 8 \times 25 = (125 \times 8)\times(40\times 25) = 1000 \times 1000 = \mathbf{10{,}00{,}000}\)
2
Calculate quickly: (a) 25×12 (b) 25×240 (c) 250×120 (d) 2500×12 (e) ___ × ___ = 12,00,00,000
Solution
- (a) \(25 \times 12 = \frac{100}{4}\times 12 = 100\times 3 = \mathbf{300}\)
- (b) \(25\times 240 = \frac{100}{4}\times240 = 100\times60 = \mathbf{6{,}000}\)
- (c) \(250\times120 = \frac{1000}{4}\times120=1000\times30=\mathbf{30{,}000}\)
- (d) \(2500\times12=\frac{10000}{4}\times12=10000\times3=\mathbf{30{,}000}\)
- (e) \(1200 \times 1{,}00{,}000 = \mathbf{12{,}00{,}00{,}000}\)
Q
How Long is the Product? Evaluate and extend the pattern in each box.
Box 1
11 × 11 = 121
111 × 111 = 12,321
1111 × 1111 = 12,34,321
Pattern: digits rise then fall symmetrically (1-2-3-…-3-2-1).
111 × 111 = 12,321
1111 × 1111 = 12,34,321
Pattern: digits rise then fall symmetrically (1-2-3-…-3-2-1).
Box 2
66 × 61 = 4,026
666 × 661 = 4,40,226
6666 × 6661 = 4,44,02,226
Pattern: extra “0” and “2” digits appear in the middle as numbers get longer.
666 × 661 = 4,40,226
6666 × 6661 = 4,44,02,226
Pattern: extra “0” and “2” digits appear in the middle as numbers get longer.
Box 3
3 × 5 = 15
33 × 35 = 1,155
333 × 335 = 1,11,555
Pattern: growing blocks of 1’s followed by equal blocks of 5’s.
33 × 35 = 1,155
333 × 335 = 1,11,555
Pattern: growing blocks of 1’s followed by equal blocks of 5’s.
Box 4
101 × 101 = 10,201
102 × 102 = 10,404
103 × 103 = 10,609
Pattern: these are \((100+n)^2 = 10000+200n+n^2\).
102 × 102 = 10,404
103 × 103 = 10,609
Pattern: these are \((100+n)^2 = 10000+200n+n^2\).
Q
Is there a connection between the digits of the numbers multiplied and the digits in the product? Is Roxie correct that two 2-digit numbers always give a 3- or 4-digit product? Can two 3-digit numbers give a 4-digit product? Can a 4-digit × 2-digit give a 5-digit product?
Solution
- Yes, Roxie is correct. Smallest 2-digit × 2-digit = \(10\times10=100\) (3 digits); largest = \(99\times99=9801\) (4 digits) — so it’s always 3 or 4 digits.
- We don’t need to try every combination — just check the smallest possible product and largest possible product for the digit-lengths involved; everything in between automatically falls in that digit range.
- 3-digit × 3-digit → 4 digits? No. Smallest = \(100\times100=10{,}000\) (5 digits) already, so the product is always 5 or 6 digits, never 4.
- 4-digit × 2-digit → 5 digits? Yes. Smallest = \(1000\times10=10{,}000\) (5 digits); largest = \(9999\times99=9,89,901\) (6 digits) — so the product can indeed be 5 digits.
Q
Complete the digit-count pattern table.
| Multiplying | Result is |
|---|---|
| 1-digit × 1-digit | 1-digit or 2-digit |
| 2-digit × 1-digit | 2-digit or 3-digit |
| 2-digit × 2-digit | 3-digit or 4-digit |
| 3-digit × 3-digit | 5-digit or 6-digit |
| 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 |
SolutionRule: For an \(m\)-digit number × an \(n\)-digit number, the product has either \((m+n-1)\) or \((m+n)\) digits.
1.5
Fascinating Facts about Large Numbers
Q
\(1250 \times 380 = \) ? (number of kirtanas composed by Purandaradāsa)
Solution\(1250 \times 380 = 4{,}75{,}000\)
4,75,000 kirtanas — Four lakh seventy-five thousand
If he composed for, say, 60 active years: \(4{,}75{,}000 \div 60 \approx 7{,}917\) songs/year ≈ 22 songs a day — an extraordinary (almost legendary) creative pace, which is why historians treat the number as a traditional estimate rather than a literal record.
Q
\(2100 \times 70{,}000 = \) ? (approx. distance from Earth to Sun, in km)
Solution\(2100 \times 70{,}000 = 14{,}70{,}00{,}000 \text{ km} = 147 \text{ million km}\)
≈ 14.7 crore km (close to the true average Earth–Sun distance of ~15 crore km / 1 Astronomical Unit)
How is this distance measured? Scientists use radar ranging (bouncing radio waves off planets and timing the echo) combined with Kepler’s laws of planetary motion to calculate the Astronomical Unit (AU) extremely precisely.
Q
\(6400 \times 62{,}500 = \) ? (litres of water the Amazon discharges into the Atlantic every second)
Solution\(6400 \times 62{,}500 = 40{,}00{,}00{,}000\) litres/second
40 crore litres every second — no wonder drinkable freshwater is found 160 km out at sea!
Q
\(13{,}95{,}000 \div 150 = \) ? (distance in km of the longest single-train journey, Moscow–Vladivostok)
Solution\(13{,}95{,}000 \div 150 = 9{,}300\) km
9,300 km — more than twice India’s longest train route (Dibrugarh–Kanyakumari, 4,219 km)
Q
\(10{,}50{,}00{,}000 \div 700 = \) ? (kg an adult blue whale can weigh)
Solution\(10{,}50{,}00{,}000 \div 700 = 1{,}50{,}000\) kg
1,50,000 kg — that’s roughly 55 times the weight of the largest land animal, Argentinosaurus (90,000 kg)!
Q
\(52{,}00{,}00{,}00{,}000 \div 130 = \) ? (tonnes of global plastic waste generated in 2021)
Solution\(52{,}00{,}00{,}00{,}000 \div 130 = 40{,}00{,}00{,}000\) tonnes
40 crore (400 million) tonnes of plastic waste in a single year — a genuinely alarming large number.
1.6
Did You Ever Wonder…?
Q
Could the entire population of Mumbai fit into 1 lakh buses (each carrying 50 people)?
SolutionCapacity = \(1{,}00{,}000 \times 50 = 50{,}00{,}000\) (50 lakh) people.
Mumbai’s population (2011) = 1,24,42,373 (more than 1 crore 24 lakh).
No — 50 lakh is far less than Mumbai’s population. The buses would fall short by about 74 lakh people.
Q
The RMS Titanic carried about 2,500 passengers. Can Mumbai’s population fit into 5,000 such ships?
SolutionCapacity = \(5000 \times 2500 = 1,25,00,000\) (1 crore 25 lakh).
Yes — just barely! 1,25,00,000 is slightly more than Mumbai’s population of 1,24,42,373.
Q
Roxie travels 100 km every day. Could she reach the Moon (3,84,400 km away) in 10 years?
Solution
- Distance in 1 year \( = 100 \times 365 = 36{,}500\) km
- Distance in 10 years \( = 36{,}500 \times 10 = 3{,}65{,}000\) km
3,65,000 km < 3,84,400 km — Not quite! She’d fall about 19,400 km short. She’d need roughly 3,844 days (≈10.5 years) to actually reach the Moon.
Q
Can you reach the Sun (≈14,70,00,000 km) in a lifetime, travelling 1000 km every day?
SolutionDays needed \( = \dfrac{14{,}70{,}00{,}000}{1000} = 1{,}47{,}000\) days \(= \dfrac{1{,}47{,}000}{365} \approx 403\) years.
No — it would take about 403 years, far beyond any human lifetime.
(a)
If a sheet of paper weighs 5 g, could you lift one lakh sheets together?
SolutionWeight \( = 1{,}00{,}000 \times 5\text{ g} = 5{,}00{,}000\text{ g} = 500\text{ kg}\)
No — 500 kg is far beyond normal human lifting capacity.
(b)
If 250 babies are born every minute worldwide, will a million be born in a day?
SolutionMinutes in a day = 1,440. Babies/day \(= 250 \times 1440 = 3,60,000\)
No — 3,60,000 is far short of 1 million (10 lakh).
(c)
Can you count 1 million coins in a day, at 1 coin per second?
SolutionSeconds in a day \(= 24 \times 60 \times 60 = 86{,}400\)
No — 86,400 coins/day is nowhere near 1 million (10,00,000).
Exercise
Figure It Out — 14 Questions
The chapter’s concluding exercise (pages 19–21) — brain-teasers on digits, place value, estimation, and real-world large numbers.
1
Using digits 0–9 exactly once (first digit ≠ 0) to form a 10-digit number, write: (a) largest multiple of 5 (b) smallest even number
Solution
- (a) To be a multiple of 5, last digit must be 0 or 5. To maximize, keep digits descending and put 5 last: 9876543210 is the largest multiple of 5.
- (b) To be smallest & even, first digit should be smallest non-zero (1), remaining digits ascending, and last digit even: 1023456798
2
10,30,285 in words has 42 letters. Give a 7-digit number name with the maximum number of letters.
SolutionRepeating the “heaviest” word (seventy-seven, seven) works well: 77,77,777 = “Seventy seven lakh seventy seven thousand seven hundred seventy seven” — 60 letters, the maximum among 7-digit numbers.
3
Write a 9-digit number where exchanging any two digits results in a bigger number. How many such numbers exist?
SolutionThis requires digits to be in strictly descending order (any swap must increase the number). Example: 987654312 → swapping the last two digits gives 987654321, which is bigger.
Such numbers must have all 9 distinct digits arranged in strictly decreasing order — try other digit selections arranged the same way.
4
Strike out 10 digits from 12345123451234512345 so the remaining number is as large as possible.
SolutionUsing a greedy “keep the largest possible leading digits” strategy on the 20-digit string, the largest remaining 10-digit number is:
5534512345
5
‘Zero’ & ‘one’ share ‘e’,’o’. ‘One’ & ‘two’ share ‘o’. ‘Two’ & ‘three’ share ‘t’. How far must you count to find two consecutive numbers sharing no letter?
Solution
- one & two share o
- two & three share t
- three & four share r
- four & five share f
- five & six share i
- six & seven share s
- seven & eight share e
- eight & nine share e, i, n
- nine & ten share n, e
- …and this pattern continues (numbers containing “-teen”, “-ty” etc. keep sharing letters like t, e, n)
Surprisingly — you never find such a pair! Every two consecutive numbers in English share at least one letter, no matter how far you count. It’s a neat, counter-intuitive fact about the English number system.
6
Writing 1,2,3,4,…,9,10,11,… continuously as digits: (a) What is the 1000th digit, and at which number does it occur? (b) What number contains the millionth digit? (c) When would you write digit ‘5’ for the 5000th time?
Solution
- (a) Digits from 1–9: 9 digits. Digits from 10–99: 90 numbers × 2 = 180 digits. Total through 99: 189 digits.
Remaining to reach 1000th digit: \(1000 – 189 = 811\).
3-digit numbers used: \(811 \div 3 = 270\) full numbers + 1 extra digit.
270th 3-digit number after 99: \(100 + 270 – 1 = 369\). Next number is 370, and its first digit “3” is the 1000th digit. - (b) Careful place-value counting through 1-digit, 2-digit, 3-digit, 4-digit, 5-digit, 6-digit number blocks shows the millionth digit falls among the 6-digit numbers, specifically at the number 1,85,185.
- (c) Counting how often digit ‘5’ appears across the number sequence (units, tens, hundreds places etc.) shows the 5000th occurrence of digit ‘5’ happens by the number 13,995.
7
A calculator has only ‘+10,000’ and ‘+100’ buttons. Write click-expressions for: (a) 20,800 (b) 92,100 (c) 1,20,500 (d) 65,30,000 (e) 70,25,700
Solution
- (a) 20,800 = (2×10,000)+(8×100) → 10 clicks
- (b) 92,100 = (9×10,000)+(21×100) → 30 clicks
- (c) 1,20,500 = (12×10,000)+(5×100) → 17 clicks
- (d) 65,30,000 = (653×10,000) → 653 clicks
- (e) 70,25,700 = (702×10,000)+(57×100) → 759 clicks
8
How many lakhs make a billion?
Solution\(1 \text{ billion} = 1,00,00,00,000\), and \(1 \text{ lakh} = 1,00,000\).
\( \dfrac{1,00,00,00,000}{1,00,000} = 10{,}000 \)
10,000 lakh make a billion.
9
Using cards 1–9 twice, place a digit in each box to get (a) the largest possible sum (b) the smallest possible difference of the two resulting numbers.
Solution
- (a) Largest sum: put the biggest digits in the highest place values of both numbers (9,9,8,8,7,7,6 on top and 6,5,5,4,4 on bottom, aligned by place value): Sum = 1,00,54,320.
- (b) Smallest difference: keep both numbers as close as possible by pairing similar-value digits (1,1,2,2,3,3,4 vs 9,9,8,8,7): Difference = 10,22,447.
10
Using cards 4000, 13000, 300, 70000, 150000, 20, 5 (each used once per number), get as close as possible to: (a) 1,10,000 (b) 2,00,000 (c) 5,80,000 (d) 12,45,000 (e) 20,90,800
Solution
- (a) \(4000 \times (20+5) + 13000 = 1{,}13{,}000\) (closest to 1,10,000)
- (b) \(1{,}50{,}000 + 70{,}000 – (4000 \times 5) = 2{,}00{,}000\) ✓ exact!
- (c) \((1{,}50{,}000 \times 4) – (4000 \times 5) = 5{,}80{,}000\) ✓ exact!
- (d) \((70{,}000 \times 20) – 1{,}50{,}000 – 4{,}000 – (300\times5) = 12{,}44{,}500\) (closest to 12,45,000)
- (e) \((1{,}50{,}000\times14) + 4{,}000 – 13{,}000 = 20{,}91{,}000\) (closest to 20,90,800)
11
How many 1-mm-thick coins must be stacked to match the height of the Statue of Unity (180 m)?
SolutionHeight of statue in mm \(= 180 \times 100 \times 10 = 1{,}80{,}000\) mm
Number of coins \(= 1{,}80{,}000 \div 1 = \)
1,80,000 coins
12
Grey-headed albatrosses fly 900–1000 km/day. A recorded trip covers 12,000 km. How many days would this take?
Solution
At 900 km/day: \(12{,}000 \div 900 \approx 13.3\) days
At 1000 km/day: \(12{,}000 \div 1000 = 12\) days
At 1000 km/day: \(12{,}000 \div 1000 = 12\) days
Approximately 12 to 14 days to cross the ocean.
13
A bar-tailed godwit flew 13,560 km from Alaska to Australia non-stop over ~11 days. Find the approximate distance covered per day, and per hour.
Solution
Per day: \(13{,}560 \div 11 \approx 1{,}233\) km
Per hour: \(1{,}233 \div 24 \approx 51\) km
Per hour: \(1{,}233 \div 24 \approx 51\) km
≈ 1,233 km/day and ≈ 51 km/hour — an astonishing non-stop flight.
14
Bald eagles fly 4500–6000 m high; Mount Everest is 8850 m; aeroplanes fly 10,000–12,800 m. How many times bigger are these compared to Somu’s 40 m building?
Solution
- Bald eagles: \(4500\div40=112.5\) to \(6000\div40=150\) → about 112 to 150 times taller
- Mount Everest: \(8850\div40=221.25\) → about 221 times taller
- Aeroplanes: \(10000\div40=250\) to \(12800\div40=320\) → about 250 to 320 times taller
Puzzle Time
Toothpick Digits
Digits can be built from seven-segment-style sticks — just like a calculator screen. To make digit 7, three sticks are needed.
0 1 2 3 4 5 6 7 8 9 — built from sticks (seven-segment style)
Q
Write or make the number 5108. How many sticks are required?
SolutionSticks needed per digit: 5→5, 1→2, 0→6, 8→7.
Total \(= 5+2+6+7 = \)
20 sticks
Q
Make 42,019 (needs exactly 23 sticks). Adding 2 more sticks (like 42,019 → 42,078), what other bigger numbers can you make?
SolutionBy changing a digit to one that uses 2 more sticks and is visually reachable by adding segments (e.g., 1→7 adds 1 stick, 0→8 adds 1 stick, 9 stays as is), several bigger numbers are possible: 42,079, 42,098, 48,019, 42,048 — try more combinations that add exactly 2 sticks total across the digits.
Q
Preetham wants to insert digit ‘1’ among 4, 2, 0, 1, 9 to get the biggest possible number. Where should he place it?
SolutionTo maximize the number, place the new ‘1’ where it keeps the larger digits as far left as possible, inserted just before the smaller trailing digits.
Placing it as 4, 2, 1, 0, 1, 9 → 4,21,019 gives the largest possible number from this insertion.
Q
Make 63,890. Rearranging exactly 4 sticks (like 63,890 → 88,078), what other bigger numbers can you make?
SolutionBy reshaping digits using exactly 4 sticks worth of change (e.g., 6→8 needs 1 stick, 3→8 needs 2 sticks), other bigger numbers include: 88,890, 68,898, 89,890 — explore more combinations that use exactly 4 rearranged sticks.
Q
(1) Make any number using exactly 24 sticks. (2) Biggest number possible with 24 sticks? (3) Smallest number possible with 24 sticks?
Solution
- Stick-cost per digit: 0→6, 1→2, 2→5, 3→5, 4→4, 5→5, 6→6, 7→3, 8→7, 9→6
- Biggest number: use the cheapest digit (1, cost 2 sticks) as many times as possible: \(24 \div 2 = 12\) ones → 111111111111 (twelve 1’s) — the most digits gives the numerically largest result.
- Smallest number: we want the fewest digits, so use the costliest digit per stick (8, cost 7): \(24 \div 7 = 3\) full 8’s (21 sticks) with 3 sticks left over for one more digit (a 7, cost 3). Arranging smallest-first: 7,888 uses only 4 digits and all 24 sticks exactly (3+7+7+7=24).
EduGrown · Ganita Prakash Grade 7 · Chapter 1 — Large Numbers Around Us · Full worked solutions