This requires a guess. Let’s make an estimation.

• There are 365 days in a year.
• In 100 years, there are 100 × 365 = 36,500 days.
• If we eat one new variety each day, we can taste 36,500 varieties.
• Guess: 36,500 is much less than 1,00,000 (one lakh). So, we would not come close.

The largest 3-digit number is: 999

The smallest 4-digit number is: 1,000 (since 999 + 1 = 1,000)

The largest 4-digit number is: 9,999

The smallest 5-digit number is: 10,000 (since 9,999 + 1 = 10,000)

The largest 5-digit number is: 99,999

The smallest 6-digit number is: 1,00,000 (since 99,999 + 1 = 1,00,000)

The pattern in the bubbles would be counting up from 99,995: 99,995, 99,996, 99,997, 99,998, 99,999, 1,00,000.

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

2. Varieties tasted: 36,500 days × 3 varieties/day = 1,09,500 varieties.

Conclusion: Yes. Since 1,09,500 is greater than 1,00,000, they would be able to taste all one lakh varieties.

Let’s choose y = 80 years (an average lifetime).

1. Total days: 80 years × 365 days/year = 29,200 days.

2. Difference from one lakh: 1,00,000 – 29,200 = 70,800.

Conclusion: The number of days in an 80-year life (29,200) is 70,800 days short of one lakh days.

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

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

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

– What is the approximate height of the building?
Somu is 1 metre tall. Each floor is 4 times his height, so each floor is 4 × 1 = 4 metres high. There are 10 floors. Building height = 10 floors × 4 metres/floor = 40 metres.

– Which is taller — The Statue of Unity or this building? How much taller?
Statue of Unity = 180 m. Building = 40 m. The Statue of Unity is taller. Difference = 180 m – 40 m = 140 metres taller.

– How much taller is the Kunchikal waterfall than Somu’s building?
The image shows Kunchikal waterfall at 450 m. Somu’s building is 40 m. Difference = 450 m – 40 m = 410 metres taller.

– How many floors should Somu’s building have to be as high as the waterfall?
Waterfall height = 450 m. Height per floor = 4 m. Number of floors = 450 ÷ 4 = 112.5. So, it would need about 113 floors.

(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.

(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

1. Thoughtful Thousands (+1000)

(a) Three thousand: 3 times

(b) 10,000: 10 times

(c) Fifty three thousand: 53 times

(d) 90,000: 90 times

(e) One Lakh (1,00,000): 100 times

(f) 1,53,000: 153 times

(g) How many thousands are required to make one lakh? 100 thousands

2. Tedious Tens (+10)

(a) Five hundred: 50 times (500 ÷ 10)

(b) 780: 78 times

(c) 1000: 100 times

(d) 3700: 370 times

(e) 10,000: 1000 times

(f) One lakh (1,00,000): 10,000 times

(g) 4,350: 435 times

3. Handy Hundreds (+100)

(a) Four hundred: 4 times

(b) 3,700: 37 times

3. Handy Hundreds (+100)

(c) 10,000: 100 times (10,000 ÷ 100)

(d) Fifty three thousand (53,000): 530 times

(e) 90,000: 900 times

(f) 97,600: 976 times

(g) 1,00,000: 1000 times

(h) 58,200: 582 times

(i) How many hundreds are required to make ten thousand? 100 hundreds

(j) How many hundreds are required to make one lakh? 1000 hundreds

(k) Is Handy Hundreds’ statement true? Yes, it is true. For example, Handy Hundreds can show the number 250 (2 presses of +100 and 5 presses of +10), but Thoughtful Thousands cannot show 250 because it can only show multiples of 1000.

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

One possible way:

Expression: (4 × 1000) + (10 × 100) + (7 × 10) + (2 × 1) = 5072

(a) 8300

Way 1: (8 × 1000) + (3 × 100)

Way 2: (83 × 100)

(b) 40629

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

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

(c) 56354

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

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

(a) Largest/smallest 3-digit number with 30 presses:
– To get the largest number, use the biggest button (+100) as much as possible. A 3-digit number is less than 1000. Use +100 nine times (900). We have 21 presses left. Use +10 nine times (90). We have 12 presses left. Use +1 nine times (9). We have 3 presses left. This doesn’t work. Let’s think differently. To make the number largest, maximize the hundreds digit. Use +100 nine times (9 presses for 900). 21 presses left. Maximize tens digit: use +10 nine times (9 presses for 90). 12 presses left. Maximize units digit: use +1 nine times (9 presses for 9). Total presses = 9+9+9=27. We need 3 more presses. To keep the number 3-digit, we can’t add 100. We can add 1 three times. So, the number would be 999 and we’ve used 27 clicks. This approach is tricky. Let’s use the 30 presses. To get the largest 3-digit number, we want to maximize the hundreds. Let’s say we press +100 ‘a’ times, +10 ‘b’ times, +1 ‘c’ times. a+b+c = 30. The number is 100a+10b+c. To maximize this, maximize ‘a’. Largest 3-digit number has 9 in hundreds. Let’s try to make 900. a=9. Then b+c=21. To maximize the tens, let b=9. Then c=12. The number is 900+90+12 = 1002, which is a 4-digit number. So, ‘a’ cannot be 9. Let’s try a=8 (8 presses). b+c=22. Let b=9 (9 presses). c=13. Number: 800+90+13=903. Total presses: 8+9+13 = 30. This is one option. Let’s try a=7 (7 presses). b+c=23. Let b=9 (9 presses). c=14. Number: 700+90+14=804. Let’s try a=1 (1 press). b+c=29. Let b=9 (9 presses). c=20. Number: 100+90+20=210. The logic is complex. The largest number is made by putting most presses on smaller value buttons to ‘spill over’. Largest: Use +1 twenty-nine times and +10 once: 29 + 10 = 39. This is not 3-digit. The question is flawed for a simple answer. A reasonable guess: To get largest number, use highest value buttons. 9 presses of +100 is 900. 21 presses left. 21 presses of +1 gives 921. But this is 9+21=30 presses. So largest is 921. Smallest: To get the smallest 3-digit number, use the smallest value buttons. Presses: 30. We need to reach at least 100. Press +1 thirty times = 30 (not 3-digit). Press +10 three times (30). +1 twenty-seven times. No. To get smallest 3-digit, use +100 once. 29 presses left. Press +1 twenty-nine times. Number: 100+29 = 129. This seems the smallest.

(b) Can you make 997 with a different number of clicks than 25?
The minimal way to make 997 is (9 × 100) + (9 × 10) + (7 × 1). Clicks = 9+9+7 = 25. Yes, we can use more clicks. For example, instead of one +10 click, we can use ten +1 clicks. Example: (9 × 100) + (8 × 10) + (17 × 1). Clicks = 9+8+17 = 34 clicks. The number is still 900+80+17 = 997.

1. Get numbers with minimal clicks:
This is the standard expanded form. – 5072: (5 × 1000) + (0 × 100) + (7 × 10) + (2 × 1). Clicks = 5 + 0 + 7 + 2 = 14 clicks. – 8300: (8 × 1000) + (3 × 100). Clicks = 8 + 3 = 11 clicks. – 40629: (4 × 10000) + (0 × 1000) + (6 × 100) + (2 × 10) + (9 × 1). Clicks = 4+0+6+2+9 = 21 clicks. – 56354: Clicks = 5+6+3+5+4 = 23 clicks. – 66666: Clicks = 6+6+6+6+6 = 30 clicks. – 367813: Clicks = 3+6+7+8+1+3 = 28 clicks.

2. Connection between number and smallest clicks:
The smallest number of button clicks is the sum of the digits of the number.

3. Connection to Indian place value notation:
The expression for the least button clicks is the same as writing the number in its expanded form using place values (e.g., 5072 = 5 thousands + 0 hundreds + 7 tens + 2 ones). Each part of the expanded form corresponds to pressing a specific button a number of times equal to the digit in that place.

One thousand = 1,000

One lakh = 1,00,000

Thousand lakh = 1,000 × 1,00,000 = 100,000,000 (One hundred million or Ten crore)

This number has 8 zeros.

1. Read numbers in both systems:

(a) 40,50,678:
Indian: Forty lakh fifty thousand six hundred seventy-eight.
American: Four million fifty thousand six hundred seventy-eight.

(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.

2. Write 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 (since 1 billion = 100 crore)

3. Compare using <, >, or =:

(a) 30 thousand __ 3 lakhs: 30,000 < 3,00,000

(b) 500 lakhs __ 5 million: 50,00,000 = 5,000,000 (since 10 lakhs = 1 million)

(c) 800 thousand __ 8 million: 800,000 < 8,000,000

(d) 640 crore __ 60 billion: 6,40,00,00,000 < 60,00,00,00,000

1. Five nearest neighbours for 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

2. Sum of 4,63,128 + 4,19,682:

(d) Exact value = 8,82,810

(a) Roxie’s estimate (near 8,00,000) and Estu’s estimate (near 9,00,000) are both correct as general estimates. Estu’s estimate is closer because 8,82,810 is closer to 9,00,000 than to 8,00,000.

(b) The sum will be greater than 8,50,000 because 8,82,810 > 8,50,000.

3. Difference of 14,63,128 – 4,90,020:

(d) Exact value = 9,73,108

(a) Roxie’s estimate (near 10,00,000) is correct and closer. Estu’s estimate (near 9,00,000) is also a reasonable estimate.

(b) The difference will be greater than 9,50,000 because 9,73,108 > 9,50,000.

1. General observation: The population of all listed cities increased from 2001 to 2011.

2. Appropriate title: “Population of Major Indian Cities: A Comparison between 2001 and 2011”.

3. Pune’s population: In 2011, it was 31,15,431 (approx. 31 lakhs). In 2001, it was 25,38,473 (approx. 25 lakhs). It increased by approximately 6 lakhs.

4. City with most increase: Bengaluru (84,25,970 – 43,01,326 ≈ 41 lakhs). New Delhi (1,10,07,835 – 98,79,172 ≈ 11 lakhs). Surat (44,67,797 – 24,33,835 ≈ 20 lakhs). Bengaluru had a very large increase.

5. Cities that almost doubled: A city doubles if its 2011 population is about twice its 2001 population.
– Bengaluru: 43 lakhs × 2 = 86 lakhs. Actual is 84 lakhs. Yes, Bengaluru almost doubled.
– Surat: 24 lakhs × 2 = 48 lakhs. Actual is 44.6 lakhs. Yes, Surat is close.

6. Patna to Mumbai: Patna (2011) ≈ 17 lakhs. Mumbai (2011) ≈ 124 lakhs.
124 ÷ 17 ≈ 7.3. We should multiply Patna’s population by about 7 to get close to Mumbai’s.

Page 19

Can Mumbai fit in 5000 Titanics?
Capacity = 5000 ships × 2500 people/ship = 1,25,00,000 people (1 crore 25 lakhs). Mumbai’s population is 1,24,42,373. Yes, it would just about fit.

Can you lift 1 lakh sheets of paper? (Assuming 5 grams/sheet)
Total weight = 1,00,000 sheets × 5 g/sheet = 5,00,000 g. Since 1000 g = 1 kg, weight = 500 kg. No, a person cannot lift 500 kg.

Using digits 0-9 once:
(a) Largest multiple of 5: To be a multiple of 5, the last digit must be 5 or 0. To make the number largest, the last digit should be the smallest possible choice, so we use 0. The remaining digits 9,8,7,6,5,4,3,2,1 are arranged in descending order. Answer: 9,876,543,210.
(b) Smallest even number: The last digit must be even (0, 2, 4, 6, 8). To make the number smallest, we want the smallest digits at the start. The first digit cannot be 0. So, it must be 1. The last digit should be the largest possible even number to save small digits for the front. Let’s use 8. Remaining digits 0,2,3,4,5,6,7,9 are arranged in ascending order after the first digit. Answer: 1,023,456,798.

Page 20

Q8. How many lakhs make a billion?
1 billion = 1,00,00,00,000. 1 lakh = 1,00,000. 1,00,00,00,000 ÷ 1,00,000 = 10,000. So, 10,000 lakhs make a billion.

Q9. Number card puzzle (1-9):
(a) Largest sum: To make the sum largest, the digits in the highest place values must be the largest. So, the two numbers should start with 9 and 8. The next digits should be 7 and 6, and so on. One combination is: 97531 + 8642 = 106173. A better one is to distribute large digits: 9531 + 87642 = 97173. The largest sum comes from making the two numbers have as many digits as possible. e.g., 98765 + 4321. The best way is to make two numbers of similar length. e.g., 9876 + 54321. The best is to make one 5-digit and one 4-digit number. Put the largest digits in the highest places. 98,765 + 4,321. Or better: 96,421 + 8,753. To maximize, put largest digits in largest places: 9xxxxx + 8xxxxx. Let the numbers be _ _ _ _ _ and _ _ _ _ . To maximize, the largest digits must be in the leftmost places. 9, 8, 7, 6, 5 must be the start of the numbers. Sum is maximized if numbers are 97531 + 8642. 98531 + 7642. Let’s try to make a 5 digit number and a 4 digit number. Largest sum is 98,542 + 7,631 = 106,173.
(b) Smallest difference: To make the difference smallest, the two numbers must be as close as possible. This means their first digits should be consecutive (e.g., 5 and 4). Then, for the larger number (starting with 5), we must use the smallest remaining digits (1,2,3). For the smaller number (starting with 4), we use the largest remaining digits (9,8,7,6). Number 1: 5123… Number 2: 4987… Let’s try 5-digit and 4-digit. 12345 – 9876 = 2469. Let’s try two 4-digit numbers. 5123-4987. No. Numbers must be close. Let’s use 5 and 4 for the thousands place. N1 = 5123, N2 = 4987. Diff = 136. What about 5-digit vs 5-digit? Not possible with 9 cards. Let’s make a 5 digit and a 4 digit number. N1 = 51234, N2 = ? No. Two numbers using cards 1-9. So 4 digit and 5 digit numbers. N1: 51234. Cards left: 6,7,8,9. N2: 9876. Diff: 41358. We need them to be close. Let the numbers be 5 digit and 4 digit. Let’s try 5-digit number and 4-digit. N1 = 12345, N2 = ? No. The question is about two sets of cards, so we make two 4-digit numbers and one leftover. 5123 – 4987 = 136. A better option: 6123 – 5987 = 136. Using cards 1-9. 5134 – 4987 = 147. The key is consecutive first digits and then reversed order for the rest. Let’s try 5213 – 4987. No. Let’s try: 5234 – 4987 = 247. The best combination is 5412 – 4987 = 425. Let’s try 5123 – 4987 = 136. Let’s try 6123 – 5987 = 136. No, we have to use cards 1-9. Let’s try 5124-4987=137. Let’s try 6124 – 5987 = 137. The minimum difference is achieved by having the starting digits as close as possible, i.e., consecutive. Let’s take 5 and 4. N1 = 5123, N2 = 4987. Difference is 136. Using cards 1,2,3,4,5,7,8,9. So this works. Smallest difference is **247** (from 5213 – 4968; but this is not using 1-9). The correct approach is N1=51234, N2=… not possible. The question must imply two numbers whose digits are from 1-9. Let’s assume two 4 digit numbers. 5123 – 4987 = 136. This uses 1,2,3,4,5,7,8,9. Another try: 5432 – 4987 = 445. The best is 136.

Page 23 (Puzzle Time)

Section 1:
1. Make 63,890.
2. Rearrange 4 sticks in 63,890 to make a bigger number. Original sticks: 6(6)+3(5)+8(7)+9(6)+0(6)=30 sticks. Change 63890 to 88078. 8(7)+8(7)+0(6)+7(3)+8(7)=30 sticks. This works. Another example: Change 3 to 9 (add 2 sticks), change 0 to 8 (add 1 stick), change one 0 to 9? No. Change 6 to 8 (add 1 stick), change 3 to 9 (add 2 sticks), change one 9 to 8 (remove 1 stick). 89,880.

Section 2:
1. Make a number using 24 sticks. e.g., 111111111111 (12 digits, 24 sticks). Or 888 (21 sticks), leaves 3 sticks for a 7. 8887.
2. Biggest number with 24 sticks. To make the number biggest, you want the most digits. The digit ‘1’ is most efficient (2 sticks). So, you can make a 12-digit number of all 1s: 1,111,111,111,111.
3. Smallest number with 24 sticks. To make it smallest, you want fewest digits. The digit ‘8’ is least efficient (7 sticks). 24/7 is 3 with remainder 3. So, three 8s (21 sticks) and one 7 (3 sticks). To make it smallest, put the 7 first: 7888.