Large Numbers Around Us

Given: 1 year ≈ 365 days. We need to find y such that y × 365 is close to 1,00,000.

Let us choose y = 274.

Number of days = 274 × 365 = 1,00,010.

This is just 10 more than one lakh, so it is very close to one lakh.

Alternatively, choose y = 273:
Number of days = 273 × 365 = 99,645.
This is 1,00,000 − 99,645 = 355 less than one lakh.

So for y = 274, the number of days (1,00,010) is closest to one lakh — just 10 more than one lakh.

Given: Population = 75,000; One lakh = 1,00,000.

Concept: Subtraction.

$$1,00,000 - 75,000 = 25,000$$

∴ 75,000 is 25,000 less than one lakh.

Given: Population in 2024 = 1,06,000; One lakh = 1,00,000.

Concept: Subtraction.

$$1,06,000 - 1,00,000 = 6,000$$

∴ 1,06,000 is 6,000 more than one lakh.

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

Concept: Subtraction to find increase.

$$1,06,000 - 75,000 = 31,000$$

∴ The population increased by 31,000 from 2011 to 2024.

We need to find how many hundreds make 10,000.

$$10,000 ÷ 100 = 100$$

∴ 100 hundreds are required to make 10,000.

Fifty three thousand = 53,000.

$$53,000 ÷ 100 = 530$$

∴ 530 hundreds are required to make 53,000.

$$90,000 ÷ 100 = 900$$

∴ 900 hundreds are required to make 90,000.

$$97,600 ÷ 100 = 976$$

∴ 976 hundreds are required to make 97,600.

$$1,00,000 ÷ 100 = 1,000$$

∴ 1,000 hundreds are required to make one lakh.

Number = 582 × 100 = 58,200.

∴ The number is 58,200.

Ten thousand = 10,000.

$$10,000 ÷ 100 = 100$$

∴ 100 hundreds are required to make ten thousand.

One lakh = 1,00,000.

$$1,00,000 ÷ 100 = 1,000$$

∴ 1,000 hundreds are required to make one lakh.

Let us think carefully:

- Tedious Tens can show any multiple of 10 (10, 20, 30, …).
- Thoughtful Thousands can show any multiple of 1000 (1000, 2000, …).
- Handy Hundreds can show any multiple of 100 (100, 200, 300, …).

Now, every multiple of 1000 is also a multiple of 100 (e.g., 1000 = 10 × 100), and every multiple of 100 is also a multiple of 10 (e.g., 100 = 10 × 10).

So Tedious Tens can show ALL multiples of 100 (since 100 is itself a multiple of 10). Therefore, there is NO number that Handy Hundreds can show but Tedious Tens cannot.

However, Handy Hundreds CAN show numbers like 100, 200, 300, 500, 700 that Thoughtful Thousands cannot (since these are not multiples of 1000).

∴ The statement is partially true: Handy Hundreds can show numbers that Thoughtful Thousands cannot (e.g., 100, 200, 500), but Tedious Tens can show everything Handy Hundreds can. So the statement is true with respect to Thoughtful Thousands, but false with respect to Tedious Tens.

Method 1: Press +10 thirty-two times and +1 once.
$$32 \times 10 + 1 \times 1 = 320 + 1 = 321 ✓$$

Method 2: Press +100 two times, +10 twelve times, +1 once.
$$2 \times 100 + 12 \times 10 + 1 \times 1 = 200 + 120 + 1 = 321 ✓$$

Both methods correctly give 321.

We need to express 5072 in a different way using the available buttons.

Method 3: Press +1000 five times, +10 seven times, +1 two times.
$$5 \times 1000 + 7 \times 10 + 2 \times 1 = 5000 + 70 + 2 = 5072 ✓$$

Expression: $(5 \times 1000) + (7 \times 10) + (2 \times 1) = 5072$

Another Method: Press +100 forty-seven times, +10 twenty-two times, +1 two times.
$$47 \times 100 + 22 \times 10 + 2 \times 1 = 4700 + 220 + 2 = 4922 \neq 5072$$

Let us try: +1000 four times, +100 ten times, +1 two times:
$$4 \times 1000 + 10 \times 100 + 7 \times 10 + 2 \times 1 = 4000 + 1000 + 70 + 2 = 5072 ✓$$

Expression: $(4 \times 1000) + (10 \times 100) + (7 \times 10) + (2 \times 1) = 5072$

Way 1 (using thousands and hundreds):
$$8 \times 1000 + 3 \times 100 = 8000 + 300 = 8300$$
Expression: $(8 \times 1000) + (3 \times 100) = 8300$

Way 2 (using hundreds only):
$$83 \times 100 = 8300$$
Expression: $(83 \times 100) = 8300$

Way 3 (mixing differently):
$$5 \times 1000 + 33 \times 100 = 5000 + 3300 = 8300$$
Expression: $(5 \times 1000) + (33 \times 100) = 8300$

Way 1:
$$4 \times 10000 + 6 \times 100 + 2 \times 10 + 9 \times 1 = 40000 + 600 + 20 + 9 = 40629$$
Expression: $(4 \times 10000) + (6 \times 100) + (2 \times 10) + (9 \times 1) = 40629$

Way 2:
$$3 \times 10000 + 10 \times 1000 + 6 \times 100 + 2 \times 10 + 9 \times 1 = 30000 + 10000 + 600 + 20 + 9 = 40629$$
Expression: $(3 \times 10000) + (10 \times 1000) + (6 \times 100) + (2 \times 10) + (9 \times 1) = 40629$

Way 1:
$$5 \times 10000 + 6 \times 1000 + 3 \times 100 + 5 \times 10 + 4 \times 1 = 50000 + 6000 + 300 + 50 + 4 = 56354$$
Expression: $(5 \times 10000) + (6 \times 1000) + (3 \times 100) + (5 \times 10) + (4 \times 1) = 56354$

Way 2:
$$4 \times 10000 + 16 \times 1000 + 3 \times 100 + 5 \times 10 + 4 \times 1 = 40000 + 16000 + 300 + 50 + 4 = 56354$$
Expression: $(4 \times 10000) + (16 \times 1000) + (3 \times 100) + (5 \times 10) + (4 \times 1) = 56354$

Way 1:
$$6 \times 10000 + 6 \times 1000 + 6 \times 100 + 6 \times 10 + 6 \times 1 = 60000 + 6000 + 600 + 60 + 6 = 66666$$
Expression: $(6 \times 10000) + (6 \times 1000) + (6 \times 100) + (6 \times 10) + (6 \times 1) = 66666$

Way 2:
$$5 \times 10000 + 16 \times 1000 + 6 \times 100 + 6 \times 10 + 6 \times 1 = 50000 + 16000 + 600 + 60 + 6 = 66666$$
Expression: $(5 \times 10000) + (16 \times 1000) + (6 \times 100) + (6 \times 10) + (6 \times 1) = 66666$

Way 1:
$$3 \times 100000 + 6 \times 10000 + 7 \times 1000 + 8 \times 100 + 1 \times 10 + 3 \times 1$$
$$= 300000 + 60000 + 7000 + 800 + 10 + 3 = 367813$$
Expression: $(3 \times 100000) + (6 \times 10000) + (7 \times 1000) + (8 \times 100) + (1 \times 10) + (3 \times 1) = 367813$

Way 2:
$$2 \times 100000 + 16 \times 10000 + 7 \times 1000 + 8 \times 100 + 1 \times 10 + 3 \times 1$$
$$= 200000 + 160000 + 7000 + 800 + 10 + 3 = 367813$$
Expression: $(2 \times 100000) + (16 \times 10000) + (7 \times 1000) + (8 \times 100) + (1 \times 10) + (3 \times 1) = 367813$

Available buttons: +1, +10, +100, +1000, +10000, +100000, +1000000.
We need exactly 30 button presses and the result must be a 3-digit number (100–999).

Largest 3-digit number with 30 presses:
To maximise the number while keeping it 3-digit (≤ 999) and using exactly 30 presses:
- Press +100 nine times: 9 × 100 = 900 (9 presses used, 21 remaining)
- Press +10 nine times: 9 × 10 = 90 (18 more presses, 3 remaining)
- Press +1 three times: 3 × 1 = 3
- Total = 900 + 90 + 3 = 993, using 9 + 9 + 3 = 21 presses. Need 9 more.

Let us try: Press +100 nine times (9 presses) = 900, press +10 nine times (9 presses) = 90, press +1 twelve times (12 presses) = 12. Total presses = 30. Number = 900 + 90 + 12 = 1002. This exceeds 3 digits.

Adjust: Press +100 nine times = 900 (9 presses), press +10 two times = 20 (2 presses), press +1 nineteen times = 19 (19 presses). Total = 30 presses. Number = 939. But can we do better?

Press +100 nine times = 900 (9 presses), press +10 nine times = 90 (9 presses), press +1 nine times = 9 (9 presses), press +10 three times = 30 (3 presses). Total = 30 presses. Number = 900 + 90 + 9 + 30 = 1029. Too big.

Best approach: We want the largest 3-digit number ≤ 999 using exactly 30 presses.
Press +100 nine times = 900 (9 presses), press +10 nine times = 90 (9 presses), press +1 nine times = 9 (9 presses). Total = 27 presses, number = 999. Need 3 more presses without changing the number — but every press adds at least 1. So we cannot use exactly 30 presses to get 999.

With 30 presses, each press adds at least 1, so minimum value = 30. We want a 3-digit number.
If we press +100 once (1 press) and +1 twenty-nine times (29 presses): 100 + 29 = 129. (30 presses)
If we press +100 nine times (9 presses) and +1 twenty-one times (21 presses): 900 + 21 = 921. (30 presses)
If we press +100 nine times (9 presses), +10 two times (2 presses), +1 nineteen times (19 presses): 900 + 20 + 19 = 939. (30 presses)
If we press +100 nine times (9 presses), +10 nine times (9 presses), +1 twelve times (12 presses): 900 + 90 + 12 = 1002. Too big.
If we press +100 nine times (9 presses), +10 eight times (8 presses), +1 thirteen times (13 presses): 900 + 80 + 13 = 993. (30 presses) ✓
If we press +100 nine times (9 presses), +10 eight times (8 presses), +1 thirteen times (13 presses): 993. Can we do better?
If we press +100 nine times (9 presses), +10 nine times (9 presses), +1 eleven times (11 presses): 900 + 90 + 11 = 1001. Too big.
So the largest 3-digit number = 993 (press +100 nine times, +10 eight times, +1 thirteen times).

Smallest 3-digit number with 30 presses:
We want the smallest number ≥ 100 using exactly 30 presses.
Press +100 once (1 press) and +1 twenty-nine times (29 presses): 100 + 29 = 129. (30 presses)
Can we get smaller? Press +100 once and +10 once and +1 twenty-eight times: 100 + 10 + 28 = 138. Bigger.
Press +100 once and +1 twenty-nine times = 129 is the smallest since we must press +100 at least once (to reach 3 digits) and each remaining press adds at least 1.

So the smallest 3-digit number = 129 (press +100 once, +1 twenty-nine times).

First, verify 25 clicks:
Standard way: 9 × 100 + 9 × 10 + 7 × 1 = 9 + 9 + 7 = 25 clicks. ✓

Different number of clicks:

With fewer clicks (using larger buttons):
Press +1000 once (1 click) — but 1000 > 997, so we'd need to subtract, which isn't possible.

So we cannot use +1000 directly. Let us try:
- Press +100 nine times (9 clicks) = 900
- Press +10 nine times (9 clicks) = 90
- Press +1 seven times (7 clicks) = 7
- Total = 25 clicks (same as before)

Alternative with more clicks:
- Press +100 eight times (8 clicks) = 800
- Press +10 nineteen times (19 clicks) = 190
- Press +1 seven times (7 clicks) = 7
- Total = 34 clicks, number = 997 ✓

Alternative with fewer clicks:
- Press +100 nine times = 900 (9 clicks)
- Press +10 eight times = 80 (8 clicks)
- Press +1 seventeen times = 17 (17 clicks)
- Total = 34 clicks, number = 997 ✓

For fewer than 25: We need to use bigger denominations. Since +1000 overshoots, we cannot reduce below 25 clicks with the given buttons for 997.

∴ Yes, 997 can be made with 34 clicks (or other numbers greater than 25). The minimum is 25 clicks.

The minimum number of clicks equals the sum of all digits of the number (since each digit tells us how many times to press that place-value button).

(a) 8300:
Digits: 8, 3, 0, 0
Minimum clicks = 8 + 3 = 11
Expression: $(8 \times 1000) + (3 \times 100) = 8300$
Press +1000 eight times and +100 three times.

(b) 40629:
Digits: 4, 0, 6, 2, 9
Minimum clicks = 4 + 0 + 6 + 2 + 9 = 21
Expression: $(4 \times 10000) + (6 \times 100) + (2 \times 10) + (9 \times 1) = 40629$

(c) 56354:
Digits: 5, 6, 3, 5, 4
Minimum clicks = 5 + 6 + 3 + 5 + 4 = 23
Expression: $(5 \times 10000) + (6 \times 1000) + (3 \times 100) + (5 \times 10) + (4 \times 1) = 56354$

(d) 66666:
Digits: 6, 6, 6, 6, 6
Minimum clicks = 6 + 6 + 6 + 6 + 6 = 30
Expression: $(6 \times 10000) + (6 \times 1000) + (6 \times 100) + (6 \times 10) + (6 \times 1) = 66666$

(e) 367813:
Digits: 3, 6, 7, 8, 1, 3
Minimum clicks = 3 + 6 + 7 + 8 + 1 + 3 = 28
Expression: $(3 \times 100000) + (6 \times 10000) + (7 \times 1000) + (8 \times 100) + (1 \times 10) + (3 \times 1) = 367813$

Yes! The minimum number of button clicks = sum of all the digits of the number.

For example:
- 8300 → 8 + 3 + 0 + 0 = 11 clicks
- 56354 → 5 + 6 + 3 + 5 + 4 = 23 clicks

This is because each digit tells us exactly how many times to press the corresponding place-value button, and pressing it that many times is the most efficient way.

In the Indian place value system, every number is written as a sum of its digits multiplied by their respective place values:
$$N = d_n \times 10^n + d_{n-1} \times 10^{n-1} + \cdots + d_1 \times 10 + d_0 \times 1$$

For example: $56354 = 5 \times 10000 + 6 \times 1000 + 3 \times 100 + 5 \times 10 + 4 \times 1$

This is exactly what Systematic Sippy does with minimum clicks — it presses each place-value button exactly as many times as the digit in that place. So the expression for minimum clicks directly mirrors the place value expansion of the number. This is why the two are the same.

Hundred thousand = 1,00,000.

Counting the zeros: 1-0-0-0-0-0 → 5 zeros.

(a) 4050678
Indian notation: 40,50,678
Indian name: Forty lakh fifty thousand six hundred seventy-eight
American notation: 4,050,678
American name: Four million fifty thousand six hundred seventy-eight

(b) 48121620
Indian notation: 4,81,21,620
Indian name: Four crore eighty-one lakh twenty-one thousand six hundred twenty
American notation: 48,121,620
American name: Forty-eight million one hundred twenty-one thousand six hundred twenty

(c) 20022002
Indian notation: 2,00,22,002
Indian name: Two crore twenty-two thousand two
American notation: 20,022,002
American name: Twenty million twenty-two thousand two

(d) 246813579
Indian notation: 24,68,13,579
Indian name: Twenty-four crore sixty-eight lakh thirteen thousand five hundred seventy-nine
American notation: 246,813,579
American name: Two hundred forty-six million eight hundred thirteen thousand five hundred seventy-nine

(e) 345000543
Indian notation: 34,50,00,543
Indian name: Thirty-four crore fifty lakh five hundred forty-three
American notation: 345,000,543
American name: Three hundred forty-five million five hundred forty-three

(f) 1020304050
Indian notation: 1,02,03,04,050
Indian name: One arab two crore three lakh four thousand fifty
American notation: 1,020,304,050
American name: One billion twenty million three hundred four thousand fifty

(a) One crore one lakh one thousand ten:
= 1,00,00,000 + 1,00,000 + 1,000 + 10
= 1,01,01,010

(b) One billion one million one thousand one:
1 billion = 1,00,00,00,000
1 million = 10,00,000
1 thousand = 1,000
1 = 1
= 1,00,00,00,000 + 10,00,000 + 1,000 + 1
= 1,00,10,01,001

(c) Ten crore twenty lakh thirty thousand forty:
= 10,00,00,000 + 20,00,000 + 30,000 + 40
= 10,20,30,040

(d) Nine billion eighty million seven hundred thousand six hundred:
9 billion = 9,00,00,00,000
80 million = 8,00,00,000
7 hundred thousand = 7,00,000
600 = 600
= 9,00,00,00,000 + 8,00,00,000 + 7,00,000 + 600
= 9,08,07,00,600

(a) 30 thousand vs 3 lakhs:
30 thousand = 30,000
3 lakhs = 3,00,000
30,000 < 3,00,000
$$30 \text{ thousand} < 3 \text{ lakhs}$$

(b) 500 lakhs vs 5 million:
500 lakhs = 500 × 1,00,000 = 5,00,00,000 = 5 crore
5 million = 50 lakh = 50,00,000
5,00,00,000 > 50,00,000
$$500 \text{ lakhs} > 5 \text{ million}$$

(c) 800 thousand vs 8 million:
800 thousand = 8,00,000
8 million = 80,00,000
8,00,000 < 80,00,000
$$800 \text{ thousand} < 8 \text{ million}$$

(d) 640 crore vs 60 billion:
640 crore = 640 × 1,00,00,000 = 6,40,00,00,000
60 billion = 60 × 1,00,00,00,000 = 6,00,00,00,000
6,40,00,00,000 > 6,00,00,00,000
$$640 \text{ crore} > 60 \text{ billion}$$

We know that $25 = \dfrac{100}{4}$.

So, multiplying any number $n$ by 25:
$$n \times 25 = n \times \frac{100}{4} = \frac{n}{4} \times 100$$

This means: divide $n$ by 4, then multiply by 100.

Similarly, multiplying by 5 = multiplying by $\dfrac{10}{2}$ = dividing by 2 and multiplying by 10.

This works because $5 = \dfrac{10}{2}$, so $n \times 5 = n \times \dfrac{10}{2} = \dfrac{n}{2} \times 10$.

We know that $2 \times 50 = 100$.

Rearranging:
$$2 \times 1768 \times 50 = 1768 \times (2 \times 50) = 1768 \times 100 = \mathbf{1,76,800}$$

Using the hint: $125 = \dfrac{1000}{8}$

$$72 \times 125 = 72 \times \frac{1000}{8} = \frac{72}{8} \times 1000 = 9 \times 1000 = \mathbf{9000}$$

Rearrange to group convenient pairs:
$$125 \times 8 = 1000 \quad \text{and} \quad 40 \times 25 = 1000$$

So:
$$125 \times 40 \times 8 \times 25 = (125 \times 8) \times (40 \times 25) = 1000 \times 1000 = \mathbf{10,00,000}$$

We need to express 12,00,00,000 as a product of two numbers.

One way: $12,00,00,000 = 1200 \times 100000$

Another way: $12,00,00,000 = 12000 \times 10000$

Or: $12,00,00,000 = 40000 \times 3000$

So for example: $\mathbf{12000 \times 10000 = 12,00,00,000}$

$$11 \times 11 = 121$$
$$111 \times 111 = 12321$$
$$1111 \times 1111 = 1234321$$

Pattern observed: The product of $n$ ones multiplied by itself gives a palindrome: 1, 2, 3, …, n, …, 3, 2, 1.

Extension:
$$11111 \times 11111 = 123454321$$
$$111111 \times 111111 = 12345654321$$

$$66 \times 61 = 4026$$
$$666 \times 661 = 4,40,226 = 440226$$
$$6666 \times 6661 = 44,402,226 = 4,44,02,226$$

Pattern observed: The product has the form 4, 44, 444, … followed by 0, 02, 002, … and ending in 26, 226, 2226, …

Extension:
$$66666 \times 66661 = 4,44,40,02,226$$

$$3 \times 5 = 15$$
$$33 \times 35 = 1155$$
$$333 \times 335 = 111555$$

Pattern observed: The product of $n$ threes × $n$ fives gives $n$ ones followed by $n$ fives.

Extension:
$$3333 \times 3335 = 11115555$$
$$33333 \times 33335 = 1111155555$$

$$101 \times 101 = 10201$$
$$102 \times 102 = 10404$$
$$103 \times 103 = 10609$$

Pattern observed: $(100 + n)^2 = 10000 + 200n + n^2$. The middle two digits increase by 2 each time (for small n), and the last two digits are $n^2$.

Extension:
$$104 \times 104 = 10816$$
$$105 \times 105 = 11025$$