We the Travellers—I

Given: 456 → 567 → 678 → □ → □ → □

Pattern: Each number increases by 111.
$$456 + 111 = 567,\quad 567 + 111 = 678$$
$$678 + 111 = 789,\quad 789 + 111 = 900,\quad 900 + 111 = 1{,}011$$

Answer: 456 → 567 → 678 → 789 → 900 → 1,011

Given: 1,050 → □ → 3,150 → 4,200 → □ → □ → □

Pattern: Each number increases by 1,050.
$$3{,}150 - 1{,}050 = 2{,}100 \Rightarrow \text{second term} = 2{,}100$$
$$1{,}050 + 1{,}050 = 2{,}100 \checkmark$$
$$4{,}200 + 1{,}050 = 5{,}250,\quad 5{,}250 + 1{,}050 = 6{,}300,\quad 6{,}300 + 1{,}050 = 7{,}350$$

Answer: 1,050 → 2,100 → 3,150 → 4,200 → 5,250 → 6,300 → 7,350

Given: 5,501 → 6,401 → 7,301 → □ → □ → □

Pattern: Each number increases by 900.
$$6{,}401 - 5{,}501 = 900,\quad 7{,}301 - 6{,}401 = 900$$
$$7{,}301 + 900 = 8{,}201,\quad 8{,}201 + 900 = 9{,}101,\quad 9{,}101 + 900 = 10{,}001$$

Answer: 5,501 → 6,401 → 7,301 → 8,201 → 9,101 → 10,001

Given: 10,100 → 10,200 → 10,300 → □ → □ → □ and □ ← 10,900 ← □

Pattern: Each number increases by 100 going right; decreases by 100 going left.
$$10{,}300 + 100 = 10{,}400,\quad 10{,}400 + 100 = 10{,}500,\quad 10{,}500 + 100 = 10{,}600$$
For the reverse chain ending at 10,900:
$$10{,}900 - 100 = 10{,}800 \Rightarrow \text{box before } 10{,}900 = 10{,}800$$
$$10{,}800 - 100 = 10{,}700 \Rightarrow \text{first box} = 10{,}700$$

Answer (forward): 10,100 → 10,200 → 10,300 → 10,400 → 10,500 → 10,600
Answer (reverse): 10,700 ← 10,800 ← 10,900

Given: 10,105 → 10,125 → □ → □ → □ and □ ← □ ← □

Pattern: Each number increases by 20 going right.
$$10{,}125 - 10{,}105 = 20$$
$$10{,}125 + 20 = 10{,}145,\quad 10{,}145 + 20 = 10{,}165,\quad 10{,}165 + 20 = 10{,}185$$
Reverse (continuing the sequence backward from 10,105):
$$10{,}105 - 20 = 10{,}085,\quad 10{,}085 - 20 = 10{,}065,\quad 10{,}065 - 20 = 10{,}045$$

Answer (forward): 10,105 → 10,125 → 10,145 → 10,165 → 10,185
Answer (reverse): 10,045 ← 10,065 ← 10,085

Given: 10,992 → 10,993 → □ → □ → □ and □ ← □ ← □

Pattern: Each number increases by 1 going right.
$$10{,}993 + 1 = 10{,}994,\quad 10{,}994 + 1 = 10{,}995,\quad 10{,}995 + 1 = 10{,}996$$
Reverse (going left from 10,992):
$$10{,}992 - 1 = 10{,}991,\quad 10{,}991 - 1 = 10{,}990,\quad 10{,}990 - 1 = 10{,}989$$

Answer (forward): 10,992 → 10,993 → 10,994 → 10,995 → 10,996
Answer (reverse): 10,989 ← 10,990 ← 10,991

Given: 10,794 → 10,796 → 10,798 → □ → □ → □ and □ ← □ ← □

Pattern: Each number increases by 2 going right.
$$10{,}798 + 2 = 10{,}800,\quad 10{,}800 + 2 = 10{,}802,\quad 10{,}802 + 2 = 10{,}804$$
Reverse (going left from 10,794):
$$10{,}794 - 2 = 10{,}792,\quad 10{,}792 - 2 = 10{,}790,\quad 10{,}790 - 2 = 10{,}788$$

Answer (forward): 10,794 → 10,796 → 10,798 → 10,800 → 10,802 → 10,804
Answer (reverse): 10,788 ← 10,790 ← 10,792

Given: 73,005 → 72,004 → □ → □ → □ and □ ← □ ← □

Pattern: Each number decreases by 1,001 going right.
$$73{,}005 - 72{,}004 = 1{,}001$$
$$72{,}004 - 1{,}001 = 71{,}003,\quad 71{,}003 - 1{,}001 = 70{,}002,\quad 70{,}002 - 1{,}001 = 69{,}001$$
Reverse (going left, i.e., increasing by 1,001 from 73,005):
$$73{,}005 + 1{,}001 = 74{,}006,\quad 74{,}006 + 1{,}001 = 75{,}007,\quad 75{,}007 + 1{,}001 = 76{,}008$$

Answer (forward): 73,005 → 72,004 → 71,003 → 70,002 → 69,001
Answer (reverse): 76,008 ← 75,007 ← 74,006

Given: 82,350 → 83,350 → □ → □ → □ and □ ← □ ← □

Pattern: Each number increases by 1,000 going right.
$$83{,}350 - 82{,}350 = 1{,}000$$
$$83{,}350 + 1{,}000 = 84{,}350,\quad 84{,}350 + 1{,}000 = 85{,}350,\quad 85{,}350 + 1{,}000 = 86{,}350$$
Reverse (going left from 82,350):
$$82{,}350 - 1{,}000 = 81{,}350,\quad 81{,}350 - 1{,}000 = 80{,}350,\quad 80{,}350 - 1{,}000 = 79{,}350$$

Answer (forward): 82,350 → 83,350 → 84,350 → 85,350 → 86,350
Answer (reverse): 79,350 ← 80,350 ← 81,350

We write the number name for each number and the number for each name.

| Number | Number Name |
|---|---|
| 8,045 | Eight thousand forty-five |
| 7,209 | Seven thousand two hundred nine |
| 10,599 | Ten thousand five hundred ninety-nine |
| 10,743 | Ten thousand seven hundred forty-three |
| 20,869 | Twenty thousand eight hundred sixty-nine |
| 13,579 | Thirteen thousand five hundred seventy-nine |
| 10,010 | Ten thousand ten |
| 56,491 | Fifty-six thousand four hundred ninety-one |
| 45,045 | Forty-five thousand forty-five |
| 39,593 | Thirty-nine thousand five hundred ninety-three |
| 50,005 | Fifty thousand five |
| 26,050 | Twenty-six thousand fifty |
| 81,200 | Eighty-one thousand two hundred |
| 90,009 | Ninety thousand nine |
| 23,230 | Twenty-three thousand two hundred thirty |
| 36,001 | Thirty-six thousand one |

Note: The exact numbers in the figure are not visible in the OCR. The method to arrange numbers in increasing order is:

Step 1: Count the digits. A number with fewer digits is smaller.
Step 2: If the digit count is the same, compare the leftmost (highest place value) digit first.
Step 3: If those are equal, move to the next digit, and so on.

Arrange from the smallest to the largest value on the number line.

*(Students should apply this method to the numbers shown in their textbook figure and write them from left to right on the number line in increasing order.)*

Given numbers: 9,990 and 49,014

Place Value Chart:

| TTh | Th | H | T | O |
|---|---|---|---|---|
| — | 9 | 9 | 9 | 0 |
| 4 | 9 | 0 | 1 | 4 |

Step 1: Count the digits.
- 9,990 has 4 digits (no Ten-Thousands digit).
- 49,014 has 5 digits (Ten-Thousands digit = 4).

Step 2: A 5-digit number is always greater than a 4-digit number.
$$49{,}014 > 9{,}990$$

Conclusion: The student is incorrect. The student compared only the leading digits (9 and 4) without considering the number of digits. Since 49,014 has 5 digits and 9,990 has only 4 digits, 49,014 is much greater than 9,990. We must always compare the number of digits first.

Given number: 1,478

We need a number larger than 5,500.

The digits are: 1, 4, 7, 8.

To get a number larger than 5,500, the thousands digit must be at least 6, or it must be 5 with the hundreds digit ≥ 5.

Interchange 1 and 8: we get 8,471.
$$8{,}471 > 5{,}500 \checkmark$$

Alternatively, interchange 1 and 7: we get 7,418.
$$7{,}418 > 5{,}500 \checkmark$$

Answer: For example, interchange digits 1 and 8 to get 8,471, which is greater than 5,500.

Given number: 10,593. Digits: 1, 0, 5, 9, 3.

We need a 5-digit number between 11,000 and 15,000, so the ten-thousands digit stays 1 and the thousands digit should be 1–4.

Interchange 0 and 1 (thousands and ten-thousands positions): gives 01,593 — not valid (leading zero).

Interchange 0 and 3: gives 13,590.
$$11{,}000 < 13{,}590 < 15{,}000 \checkmark$$

Answer: Interchange digits 0 and 3 to get 13,590, which lies between 11,000 and 15,000.

Given number: 10,593. Digits: 1, 0, 5, 9, 3.

We need the ten-thousands digit to be at least 4 (for a number > 35,000, it should be ≥ 4; for > 35,000 with digit 3, hundreds must be large).

Interchange 1 and 9: gives 90,513.
$$90{,}513 > 35{,}000 \checkmark$$

Alternatively, interchange 1 and 5: gives 50,193.
$$50{,}193 > 35{,}000 \checkmark$$

Answer: Interchange digits 1 and 9 to get 90,513, which is more than 35,000.

Given number: 48,247. Digits: 4, 8, 2, 4, 7.

To make the number as small as possible, we want the ten-thousands digit to be as small as possible.

The smallest digit available is 2 (in the hundreds place).

Interchange 4 (ten-thousands) and 2 (hundreds): gives 28,447.

Check: Can we do better? Interchange 4 (ten-thousands) and the other 4 (ones): gives 48,244 — not smaller in ten-thousands.
Interchange 4 (ten-thousands) and 2: gives 28,447. This is the smallest possible.

Answer: Interchange the ten-thousands digit 4 and the hundreds digit 2 to get 28,447.

Given number: 48,247. Digits: 4, 8, 2, 4, 7.

To make the number as large as possible, we want the ten-thousands digit to be as large as possible.

The largest digit is 8 (already in the thousands place).

Interchange 4 (ten-thousands) and 8 (thousands): gives 84,247.

Check other options: interchange 4 (ten-thousands) and 7 (ones) → 78,244; interchange 4 (ten-thousands) and 8 → 84,247. The largest is 84,247.

Answer: Interchange the ten-thousands digit 4 and the thousands digit 8 to get 84,247.

Given: Rabbit is at 2,346.

Neighbouring hundreds of 2,346 are 2,300 and 2,400.

Distance to 2,300: $2{,}346 - 2{,}300 = 46$ jumps.
Distance to 2,400: $2{,}400 - 2{,}346 = 54$ jumps.

Since 46 < 54, the nearest hundred is 2,300.

Answer: 2,300 is the nearest hundred of 2,346. It will need 46 jumps to reach 2,300.

Given: Rabbit is at 2,346.

Neighbouring thousands of 2,346 are 2,000 and 3,000.

Distance to 2,000: $2{,}346 - 2{,}000 = 346$ jumps.
Distance to 3,000: $3{,}000 - 2{,}346 = 654$ jumps.

Since 346 < 654, the nearest thousand is 2,000.

Answer: 2,000 is the nearest thousand of 2,346. It will need 346 jumps to reach 2,000.

Method:
- Nearest Ten: Look at the ones digit. If ones < 5, round down; if ones ≥ 5, round up.
- Nearest Hundred: Look at the tens digit. If tens < 5, round down; if tens ≥ 5, round up.
- Nearest Thousand: Look at the hundreds digit. If hundreds < 5, round down; if hundreds ≥ 5, round up.

3,176:
- Ones digit = 6 ≥ 5 → Nearest Ten = 3,180
- Tens digit = 7 ≥ 5 → Nearest Hundred = 3,200
- Hundreds digit = 1 < 5 → Nearest Thousand = 3,000

4,017:
- Ones digit = 7 ≥ 5 → Nearest Ten = 4,020
- Tens digit = 1 < 5 → Nearest Hundred = 4,000
- Hundreds digit = 0 < 5 → Nearest Thousand = 4,000

5,789:
- Ones digit = 9 ≥ 5 → Nearest Ten = 5,790
- Tens digit = 8 ≥ 5 → Nearest Hundred = 5,800
- Hundreds digit = 7 ≥ 5 → Nearest Thousand = 6,000

8,203:
- Ones digit = 3 < 5 → Nearest Ten = 8,200
- Tens digit = 0 < 5 → Nearest Hundred = 8,200
- Hundreds digit = 2 < 5 → Nearest Thousand = 8,000

| Number | Nearest Tens | Nearest Hundreds | Nearest Thousands |
|---|---|---|---|
| 3,176 | 3,180 | 3,200 | 3,000 |
| 4,017 | 4,020 | 4,000 | 4,000 |
| 5,789 | 5,790 | 5,800 | 6,000 |
| 8,203 | 8,200 | 8,200 | 8,000 |

We need a number whose nearest hundred and nearest thousand are the same.

7,126:
- Nearest hundred: tens digit = 2 < 5 → 7,100
- Nearest thousand: hundreds digit = 1 < 5 → 7,000
- 7,100 ≠ 7,000 ✗

7,835:
- Nearest hundred: tens digit = 3 < 5 → 7,800
- Nearest thousand: hundreds digit = 8 ≥ 5 → 8,000
- 7,800 ≠ 8,000 ✗

7,030:
- Nearest hundred: tens digit = 3 < 5 → 7,000
- Nearest thousand: hundreds digit = 0 < 5 → 7,000
- 7,000 = 7,000 ✓

6,999:
- Nearest hundred: tens digit = 9 ≥ 5 → 7,000
- Nearest thousand: hundreds digit = 9 ≥ 5 → 7,000
- 7,000 = 7,000 ✓

Answer: The numbers are 7,030 and 6,999.

Example: 23 and 27 both have nearest ten = 20 (since 23 rounds to 20 and 27 rounds to 30 — let us pick better ones).

Actually: 21 and 24 → nearest ten of 21 is 20, nearest ten of 24 is 20. ✓

Answer: 21 and 24 (both round to 20). Another example: 31 and 33 (both round to 30).

Example: 320 and 340 → nearest hundred of 320 is 300, nearest hundred of 340 is 300. ✓

Answer: 320 and 340 (both round to 300). Another example: 510 and 530 (both round to 500).

Example: 3,200 and 3,400 → nearest thousand of 3,200 is 3,000, nearest thousand of 3,400 is 3,000. ✓

Answer: 3,200 and 3,400 (both round to 3,000). Another example: 5,100 and 5,300 (both round to 5,000).