1. Large Numbers Around Us

1. Population of Chintamani in 2011 was about 75,000. How much less than one lakh is 75,000?

1,00,000 − 75,000 = 25,000. So, 75,000 is 25,000 less than one lakh.

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

1,06,000 − 1,00,000 = 6,000. So, it is 6,000 more than one lakh.

3. By how much did the population increase from 2011 to 2024?

1,06,000 − 75,000 = 31,000.

(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

(k) “There are some numbers which Tedious Tens and Thoughtful Thousands can’t show but I can.” Is this true?

No. Every number made with +100 (like 300 or 4,500) is also a multiple of 10.

So, Tedious Tens can show all of them. (Thoughtful Thousands cannot show 300, but Tedious Tens can.)

(a) 8300

(8 × 1000) + (3 × 100)

(5 × 1000) + (33 × 100)

(b) 40,629

(40 × 1000) + (6 × 100) + (2 × 10) + (9 × 1)

(4 × 10000) + (62 × 10) + (9 × 1)

(c) 56,354

(56 × 1000) + (3 × 100) + (5 × 10) + (4 × 1)

(5 × 10000) + (63 × 100) + (54 × 1)

(d) 66,666

(6 × 10000) + (6 × 1000) + (6 × 100) + (6 × 10) + (6 × 1)

(66 × 1000) + (666 × 1)

(e) 3,67,813

(3 × 100000) + (6 × 10000) + (7 × 1000) + (8 × 100) + (1 × 10) + (3 × 1)

(36 × 10000) + (78 × 1000) + (13 × 1)

(a) With exactly 30 button presses, what is the largest and the smallest 3-digit number?

Largest: (9 × 100) + (8 × 10) + (13 × 1) = 993. Presses = 9 + 8 + 13 = 30.

Smallest: (8 × 10) + (22 × 1) = 102. Presses = 8 + 22 = 30.

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

Yes. (8 × 100) + (19 × 10) + (7 × 1) = 997. Clicks = 8 + 19 + 7 = 34.

Also, (99 × 10) + (7 × 1) = 997 uses 106 clicks.

1. Find the smallest number of clicks for each number of the previous exercise.

(a) 8300 = (8 × 1000) + (3 × 100) → 11 clicks

(b) 40,629 = (4 × 10000) + (6 × 100) + (2 × 10) + (9 × 1) → 21 clicks

(c) 56,354 = (5 × 10000) + (6 × 1000) + (3 × 100) + (5 × 10) + (4 × 1) → 23 clicks

(d) 66,666 = (6 × 10000) + (6 × 1000) + (6 × 100) + (6 × 10) + (6 × 1) → 30 clicks

(e) 3,67,813 = (3 × 100000) + (6 × 10000) + (7 × 1000) + (8 × 100) + (1 × 10) + (3 × 1) → 28 clicks

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

Yes. The smallest number of clicks is the sum of the digits of the number.

3. Why do the expressions for least clicks give the Indian place value notation?

To use fewest clicks, we press the biggest button as many times as possible.

Each button is pressed at most 9 times. So, each count is just the digit in that place.

1. Write the number names in both the Indian and American systems.

2. Write in Indian place value notation.

(a) One crore one lakh one thousand ten → 1,01,01,010

(b) One billion one million one thousand one → 1,00,10,01,001

(c) Ten crore twenty lakh thirty thousand forty → 10,20,30,040

(d) Nine billion eighty million seven hundred thousand six hundred → 9,08,07,00,600

3. Compare and write ‘<’, ‘>’ or ‘=’.

(a) 30 thousand < 3 lakhs (3 lakhs = 300 thousand)

(b) 500 lakhs > 5 million (500 lakhs = 50 million)

(c) 800 thousand < 8 million (8 million = 8000 thousand)

(d) 640 crore < 60 billion (640 crore = 6.4 billion)

(a) Are these estimates correct? Whose estimate is closer?

Both are correct. The sum is 8,82,810. It is closer to 9,00,000, so Estu’s estimate is closer.

(b) Greater than or less than 8,50,000?

Greater. 4,63,128 is more than 4,50,000 and 4,19,682 is more than 4,00,000.

(c) Greater than or less than 8,83,128?

Less. 4,63,128 + 4,20,000 = 8,83,128, and 4,19,682 is less than 4,20,000.

(d) Exact value

4,63,128 + 4,19,682 = 8,82,810

(a) Are these estimates correct? Whose estimate is closer?

Both are correct. The difference is 9,73,108. It is closer to 10,00,000, so Roxie’s estimate is closer.

(b) Greater than or less than 9,50,000?

Greater. 14,63,128 − 5,00,000 = 9,63,128. We subtract less than 5,00,000, so the answer is more than 9,63,128.

(c) Greater than or less than 9,63,128?

Greater, for the same reason: 4,90,020 is less than 5,00,000.

(d) Exact value

14,63,128 − 4,90,020 = 9,73,108

1. What is your general observation about this data?

The population of almost every city grew from 2001 to 2011. Only Kolkata’s population went down a little.

2. What is an appropriate title for the table?

Population of 20 large Indian cities in 2001 and 2011.

3. Population of Pune in 2011? By how much has it increased compared to 2001?

In 2011 it was 31,15,431, about 31 lakh. It increased by about 6 lakh (31 lakh − 25 lakh).

4. Which city’s population increased the most?

Bengaluru. It increased by about 41 lakh (84 lakh − 43 lakh).

5. Are there cities whose population has almost doubled?

Yes. Bengaluru (43 lakh → 84 lakh), Hyderabad (36 lakh → 68 lakh), Surat (24 lakh → 45 lakh) and Vadodara (17 lakh → 36 lakh).

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

Patna ≈ 17 lakh and Mumbai ≈ 124 lakh. 17 × 7 = 119. So, multiply by about 7.

1. Find quick ways to calculate these products.

(a) 2 × 1768 × 50 = 1768 × (2 × 50) = 1768 × 100 = 1,76,800

(b) 72 × 125 = 72 × 10008 = 9 × 1000 = 9000

(c) 125 × 40 × 8 × 25 = (125 × 8) × (40 × 25) = 1000 × 1000 = 10,00,000

2. Calculate these products quickly.

(a) 25 × 12 = 1004 × 12 = 100 × 3 = 300

(b) 25 × 240 = 1004 × 240 = 100 × 60 = 6000

(c) 250 × 120 = 10004 × 120 = 1000 × 30 = 30,000

(d) 2500 × 12 = 100004 × 12 = 10,000 × 3 = 30,000

(e) 12,000 × 10,000 = 12,00,00,000 (another answer: 1200 × 1,00,000)

1250 × 380

= 4,75,000 kīrtanas. Indian: Four lakh seventy five thousand. American: 475 thousand.

If Purandaradāsa composed for 50 years, he made 4,75,000 ÷ 50 = 9,500 songs a year (about 26 a day).

2100 × 70,000

= 14,70,00,000 km. Indian: Fourteen crore seventy lakh. American: 147 million.

6400 × 62,500

= 40,00,00,000 litres. Indian: Forty crore. American: 400 million.

13,95,000 ÷ 150

= 9300 km. Nine thousand three hundred (same in both systems).

10,50,00,000 ÷ 700

= 1,50,000 kg. Indian: One lakh fifty thousand. American: 150 thousand.

52,00,00,00,000 ÷ 130

= 40,00,00,000 tonnes. Indian: Forty crore. American: 400 million.

(a) One sheet weighs 5 g. Could you lift one lakh sheets together?

1,00,000 × 5 g = 5,00,000 g = 500 kg. No, a person cannot lift 500 kg.

(b) 250 babies are born every minute. Will a million babies be born in a day?

250 × 60 × 24 = 3,60,000 babies a day. This is less than 10 lakh (1 million). So, No.

(c) Can you count 1 million coins in a day, counting 1 coin every second?

Seconds in a day = 60 × 60 × 24 = 86,400. So, we can count only 86,400 coins. No.

1. Using all digits 0 – 9 exactly once, write the (a) largest multiple of 5, (b) smallest even number.

(a) 98,76,54,32,10 (it ends in 0, so it is a multiple of 5).

(b) 10,23,45,67,98 (it ends in 8, so it is even).

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

77,77,777 — seventy seven lakh seventy seven thousand seven hundred seventy seven. It has 60 letters.

‘Seventy’ is the longest tens word and ‘seven’ is a 5-letter word, so this gives the most letters. (73,73,373 also has 60 letters.)

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

12,34,56,789. Its digits are in increasing order. Exchanging any two digits puts a bigger digit in a higher place, so the number becomes bigger.

Only one such number exists. Digits must all be different and increasing, and 0 cannot be the first digit.

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

Remaining number: 5534512345.

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

We never find such a pair. Below 100, every pair shares a letter (we can check each one).

Bigger numbers share words like ‘hundred’ or ‘thousand’, or a letter like ‘n’.

6. Writing 1, 2, 3, … one after the other:

(a) Digits used: 1 to 9 → 9 digits; 10 to 99 → 180 digits; total 189.

1000 − 189 = 811 = 270 × 3 + 1. So, 270 three-digit numbers (100 to 369) are written fully.

The 1000th digit is the first digit of 370, that is 3.

(b) The millionth digit is in the number 1,85,185.

(c) The digit ‘5’ is written for the 5000th time in the number 13,495.

7. A calculator has only ‘+10,000’ and ‘+100’ buttons. Write expressions.

(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?

1 billion = 100 crore = 100 × 100 lakh = 10,000 lakh.

9. Place number cards (two sets of 1 – 9) in a 7-digit and a 5-digit row to get (a) the largest sum, (b) the smallest difference.

(a) Put bigger digits in higher places: 99,87,654 + 87,654 = 1,00,75,308.

(b) Make the 7-digit number smallest and the 5-digit number largest: 11,22,334 − 99,887 = 10,22,447.

10. Using the cards 4000, 13000, 300, 70000, 150000, 20, 5, get as close as you can.

(a) 1,10,000: 13000 × 20 − 150000 = 1,10,000 (exact)

(b) 2,00,000: 150000 + 70000 − 4000 × 5 = 2,00,000 (exact)

(c) 5,80,000: 70000 × 5 + 150000 + 4000 × 20 = 5,80,000 (exact)

(d) 12,45,000: (13000 + 70000) × (20 − 5) = 83,000 × 15 = 12,45,000 (exact)

(e) 20,90,800: (70000 + 300) × 20 + (150000 − 13000) × 5 = 14,06,000 + 6,85,000 = 20,91,000 (only 200 more)

11. How many 1 mm thick coins should be stacked to match the Statue of Unity?

180 m = 180 × 1000 mm = 1,80,000 mm. So, 1,80,000 coins.

12. An albatross covers 900 – 1000 km a day. How many days for a 12,000 km trip?

12,000 ÷ 1000 = 12 days and 12,000 ÷ 900 ≈ 13 days. So, about 12 to 13 days.

13. A godwit flew 13,560 km in about 11 days. Find the distance covered every day and every hour.

Every day: 13,560 ÷ 11 ≈ 1,233 km.

Every hour: 1,233 ÷ 24 ≈ 51 km.

14. How many times bigger are these heights compared to Somu’s building (40 m)?

Bald eagles: 4500 ÷ 40 ≈ 112 and 6000 ÷ 40 = 150. About 112 to 150 times.

Mount Everest: 8850 ÷ 40 ≈ 221 times.

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

2. Add two more sticks and make a bigger number.

92,019 (4 → 9), 48,019 (2 → 8), 42,879 (0 → 8, 1 → 7), 42,818 (0 → 8, 9 → 8), 42,049 (1 → 4) and 42,078.

3. Where should the digit ‘1’ be placed to get the biggest number?

After the 2: 4,21,019.

4. What other numbers can he make by placing the digit ‘1’?

1,42,019; 4,12,019; 4,20,119; 4,20,191.

1. Make any number using exactly 24 sticks.

Example: 8887 uses 7 + 7 + 7 + 3 = 24 sticks.

2. Biggest number

‘1’ needs only 2 sticks, so 12 ones use 24 sticks: 1,11,11,11,11,111 (12 digits).

3. Smallest number

We need at least 4 digits (each digit uses at most 7 sticks). The smallest is 2008 (5 + 6 + 6 + 7 = 24).