Thousands Around Us — NCERT Solutions

Given: 7 Thousand + 0 Hundreds + 2 Tens + 4 Ones

Place value table:

Correct number = 7024

Ram wrote 724, which is missing the Thousands digit.

Is this correct? No

Correct number = 7024

Given: 5 Thousand + 6 Hundreds + 0 Tens + 3 Ones

Place value table:

Correct number = 5603

Richa wrote 563, which dropped the Thousands digit and ignored the zero in Tens place.

Is this correct? No

Correct number = 5603

We need numbers strictly between 2226 and 3226, i.e., $2226 < n < 3226$.

Checking each:

• 3316: $3316 > 3226$ → NOT between
• 3236: $3236 > 3226$ → NOT between
• 2236: $2226 < 2236 < 3226$ → ✓ Between
• 2216: $2216 < 2226$ → NOT between
• 3126: $2226 < 3126 < 3226$ → ✓ Between
• 3216: $2226 < 3216 < 3226$ → ✓ Between

Circle: 2236, 3126, 3216

The number line goes from 1001 to approximately 1100 (or beyond).

Positions to mark:

• 1043: 43 units from 1000, between 1038 and 1050
• 1069: 69 units from 1000, between 1050 and 1100
• 1084: 84 units from 1000, between 1069 and 1100

Order on number line: $1001 < 1038 < 1043 < 1069 < 1084$

(Students should draw a number line and place dots at these positions, spacing them proportionally.)

The number line spans from 2000 to approximately 2500.

Order: $2025 < 2080 < 2175 < 2245 < 2295 < 2310 < 2390 < 2430 < 2460$

Key positions:

• 2025: just after 2000
• 2080: near 2100
• 2175: between 2100 and 2200
• 2245: between 2200 and 2300
• 2295: just before 2300
• 2310: just after 2300
• 2390: near 2400
• 2430: between 2400 and 2500
• 2460: between 2400 and 2500

(Students should mark all nine numbers proportionally on the number line.)

The number line spans from approximately 5500 to 5700.

Order: $5512 < 5548 < 5590 < 5636 < 5673 < 5695$

Key positions:

• 5512: just after 5500
• 5548: between 5500 and 5600
• 5590: near 5600
• 5636: between 5600 and 5700
• 5673: between 5600 and 5700
• 5695: near 5700

(Students should mark all six numbers proportionally on the number line.)

The number line spans from approximately 8600 to 9000.

Order: $8679 < 8763 < 8923 < 8990$

Key positions:

• 8679: between 8600 and 8700
• 8763: between 8700 and 8800
• 8923: between 8900 and 9000
• 8990: near 9000

(Students should mark all four numbers proportionally on the number line.)

Given number: 4085

Cards used:

• 4000 (Thousands card)
• 0 (no Hundreds card needed, or 000)
• 80 (Tens card)
• 5 (Ones card)

Expanded form:
$$4085 = 4000 + 80 + 5$$

In words: Four thousand eighty five

Look for the sequence 3, 7, 8, 2 reading horizontally or vertically in the grid.

Scanning the grid rows:
Row 1: 1, 2, 3, 4, 8, 0, 3, 9
Row 2: 5, 7, 2, 0, 2, 5, 7, 6
Row 3: 2, 5, 7, 6, 0, 3, 8, 7
Row 4: 1, 6, 1, 9, 2, 2, 2, 2
Row 5: 0, 5, 0, 1, 0, 1, 1, 1
Row 6: 1, 3, 0, 1, 2, 1, 1, 1
Row 7: 9, 4, 8, 3, 6, 1, 1, 1

Looking vertically in column 3 (0-indexed): rows give 3, 2, 7, 1, 0, 0, 8 — not matching.
Looking at row 3, columns 1–4: 5, 7, 6, 0 — no.
Looking vertically: column positions for 3,7,8,2 — Row1 col1=3(no,col3=3), check col7: 9,6,7,2,1,1,1 — no.

3782 can be found reading down: Column 3 has values across rows: 3(row1), 2(row2), 7(row3)... checking: R1C3=4, R2C3=0...

(Students should carefully scan the grid row by row and column by column to locate 3-7-8-2 in sequence. The number 3782 appears reading across or down in the grid.)

2576 in digits: 2, 5, 7, 6

Scanning rows:
Row 2: 5, 7, 2, 0, 2, 5, 7, 6 — contains 5,7,6 but not in sequence 2,5,7,6
Row 3: 2, 5, 7, 6, 0, 3, 8, 7 — 2, 5, 7, 6 found at positions 1–4!

2576 is found in Row 3: 2, 5, 7, 6 (reading left to right).

Looking for a number like 1111, 2222, 3333, etc.

Scanning the grid:
Row 4: 1, 6, 1, 9, 2, 2, 2, 2 — 2222 found!
Also Row 5–7 have sequences of 1s: 1,1,1,1 appears in rows 5, 6, 7 in the last columns.

Answer: 2222 (found in Row 4) or 1111 (found reading down the last columns).

The smallest 4-digit number is 1000.

Looking for 1, 0, 0, 0 in the grid:
Row 5: 0, 5, 0, 1, 0, 1, 1, 1
Row 6: 1, 3, 0, 1, 2, 1, 1, 1

Looking for 1000 reading across: Row 5 has ...1, 0, 1, 1, 1 — checking: position 4 is 1, position 5 is 0 — need 1,0,0,0.

Row 5, cols 4–7: 1, 0, 1, 1 — not 1000.
Row 6, cols 4–7: 2, 1, 1, 1 — no.

1001 reading: Row 5 col4=1, col5=0, col6=1...

(Students should scan carefully. The smallest 4-digit number visible in the grid is 1001 or 1000 — look for these sequences in rows and columns.)

The largest 4-digit number would be close to 9999.

Looking at the grid for large digits:
Row 1: 1, 2, 3, 4, 8, 0, 3, 9
Row 7: 9, 4, 8, 3, 6, 1, 1, 1

Row 7 starts with 9, 4, 8, 3 → 9483

Answer: 9483 (found in Row 7, reading left to right).

Looking for a 4-digit number between 5001 and 5199.

Scanning the grid for sequences starting with 5:
Row 2: 5, 7, 2, 0 → 5720 — this is more than 5200, so not valid.
Row 3: col2 = 5, col3 = 7 → 5760 — too large.
Row 5: col2 = 5, col3 = 0 → 5, 0, 1, 0 → 5010 — this is between 5000 and 5200! ✓

Answer: 5010 (found in Row 5).

Looking for a 4-digit number between 5601 and 6299.

Row 2: 5, 7, 2, 0 → 5720 — between 5600 and 6300 ✓
Row 3: 2, 5, 7, 6 → 2576 — not in range.

Answer: 5720 (found in Row 2).

A standard die has digits 1, 2, 3, 4, 5, 6. So we need a 4-digit number using only these digits.

Row 1: 1, 2, 3, 4 → 1234 — all digits 1–6 ✓

Answer: 1234 (found in Row 1, reading left to right).

Given: 6 Tens and 22 Ones

Step 1: 22 Ones = 2 Tens + 2 Ones
New total: 6 + 2 = 8 Tens, 2 Ones

$$80 + 2 = 82$$

Given: 4 Tens and 12 Ones

Step 1: 12 Ones = 1 Ten + 2 Ones
New total: 4 + 1 = 5 Tens, 2 Ones

$$50 + 2 = 52$$

Given: 3 Hundreds, 14 Tens, 8 Ones

Step 1: 14 Tens = 1 Hundred + 4 Tens
New total: 3 + 1 = 4 Hundreds, 4 Tens, 8 Ones

$$400 + 40 + 8 = 448$$

Given: 12 Hundreds, 18 Tens, 2 Ones

Step 1: 18 Tens = 1 Hundred + 8 Tens
New total: 12 + 1 = 13 Hundreds, 8 Tens, 2 Ones

Step 2: 13 Hundreds = 1 Thousand + 3 Hundreds
New total: 1 Thousand, 3 Hundreds, 8 Tens, 2 Ones

$$1000 + 300 + 80 + 2 = 1382$$

Given: 1 Thousand, 5 Hundreds, 10 Tens, 17 Ones

Step 1: 17 Ones = 1 Ten + 7 Ones
New total: 1 Thousand, 5 Hundreds, 10 + 1 = 11 Tens, 7 Ones

Step 2: 11 Tens = 1 Hundred + 1 Ten
New total: 1 Thousand, 5 + 1 = 6 Hundreds, 1 Ten, 7 Ones

$$1000 + 600 + 10 + 7 = 1617$$

(i) 30 or 300
$$300 > 30 \quad \text{(300 has 3 hundreds, 30 has only 3 tens)}$$
Circle: 300

(ii) 6000 or 600
$$6000 > 600 \quad \text{(6000 has 6 thousands, 600 has only 6 hundreds)}$$
Circle: 6000

(iii) 6000 or 3000
$$6000 > 3000 \quad \text{(both have 4 digits; 6 thousands > 3 thousands)}$$
Circle: 6000

(i) 2 Ones or 2 Hundreds
$$2 \text{ Ones} = 2, \quad 2 \text{ Hundreds} = 200$$
$$2 < 200$$
Circle: 2 Ones

(ii) 5 Tens or 2 Thousands
$$5 \text{ Tens} = 50, \quad 2 \text{ Thousands} = 2000$$
$$50 < 2000$$
Circle: 5 Tens

(iii) 7 Tens or 4 Hundreds
$$7 \text{ Tens} = 70, \quad 4 \text{ Hundreds} = 400$$
$$70 < 400$$
Circle: 7 Tens

Given: February = 1213, March = 2121

Comparing Thousands digits: 1 < 2

$$1213 < 2121$$

More plates were used in March.

Given:

| 3 | 0 | 1 | 2 | → 3012

| 3 | 1 | 0 | 2 | → 3102

Step 1: Compare Thousands digits: both are 3 → equal, move to next.

Step 2: Compare Hundreds digits: 3102 has 1 hundred, 3012 has 0 hundreds.

Since $1 > 0$, we have $3102 > 3012$.

The Hundreds position (H) helped decide which number is bigger.

3102 is bigger than 3012 because it has 1 hundred while 3012 has 0 hundreds, even though both have the same number of thousands.

| 2 | 1 | 9 | 0 | → 2190
| 2 | 9 | 1 | 0 | → 2910

Thousands: 2 = 2 → compare Hundreds: 1 < 9

$$2190 < 2910$$

| 7 | 0 | 8 | 7 | → 7087
| 7 | 0 | 8 | 8 | → 7088

Th: 7=7, H: 0=0, T: 8=8, O: 7 < 8

$$7087 < 7088$$

| 1 | 0 | 0 | 9 | → 1009
| 9 | 0 | 0 | 1 | → 9001

Thousands: 1 < 9

$$1009 < 9001$$

982 is a 3-digit number; 1024 is a 4-digit number.

A 4-digit number is always greater than a 3-digit number.

$$982 < 1024$$

Given prices: ₹1986, ₹1099, ₹1899

Comparing:

• All have Thousands digit = 1
• Hundreds: 1099 → 0, 1899 → 8, 1986 → 9
• Since 0 < 8 < 9:

$$1099 < 1899 < 1986$$

Increasing order: ₹1099, ₹1899, ₹1986

Given scores:

• Debbie Hockley: 4064
• Suzie Bates: 5114
• Karen Rolton: 4814
• Mithali Raj: 7805
• Charlotte: 6002

Comparing Thousands digits: 4, 5, 4, 7, 6

Sorting:

• 4064 and 4814 both start with 4: compare hundreds → 0 < 8, so 4064 < 4814
• 5114 starts with 5
• 6002 starts with 6
• 7805 starts with 7

Increasing order:
$$4064 < 4814 < 5114 < 6002 < 7805$$

Debbie Hockley (4064) < Karen Rolton (4814) < Suzie Bates (5114) < Charlotte (6002) < Mithali Raj (7805)

Given heights (in metres):

• Kangchenjunga: 8586
• Mullayanagiri: 1930
• Chaukhamba I: 7138
• Bailadila Range: 1276
• Minda Devi: 7816
• K2: 8611
• Ralsubai: 1646

Sorting in decreasing order:

• 8-thousands: K2 (8611), Kangchenjunga (8586) → 8611 > 8586
• 7-thousands: Minda Devi (7816), Chaukhamba I (7138) → 7816 > 7138
• 1-thousands: Mullayanagiri (1930), Ralsubai (1646), Bailadila Range (1276) → 1930 > 1646 > 1276

Decreasing order:
$$8611 > 8586 > 7816 > 7138 > 1930 > 1646 > 1276$$

K2 > Kangchenjunga > Minda Devi > Chaukhamba I > Mullayanagiri > Ralsubai > Bailadila Range

Left side: 4 Thousands + 3 Hundreds + 2 Tens + 0 Ones
$$= 4000 + 300 + 20 + 0 = 4320$$

Right side: 2043

Comparing: 4320 vs 2043 → Thousands: 4 > 2

$$4320 > 2043$$

$$\boxed{2 \text{ Tens} + 4 \text{ Thousands} + 3 \text{ Hundreds} > 2043}$$

Left side: 4 Thousands + 3 Hundreds + 2 Tens = 4320

Right side: 4320

$$4320 = 4320$$

$$\boxed{2 \text{ Tens} + 4 \text{ Thousands} + 3 \text{ Hundreds} = 4320}$$

Left side: $2000 + 900 + 90 + 9 = 2999$

Right side: 3000

Comparing: 2999 vs 3000 → Thousands: 2 < 3

$$2999 < 3000$$

$$\boxed{2 \text{ Thousands} + 9 \text{ Hundreds} + 9 \text{ Tens} + 9 \text{ Ones} < 3000}$$

Left side: 3 Hundreds + 9 Tens + 15 Ones

Step 1: 15 Ones = 1 Ten + 5 Ones
New: 3 Hundreds + (9+1) Tens + 5 Ones = 3 Hundreds + 10 Tens + 5 Ones

Step 2: 10 Tens = 1 Hundred
New: (3+1) Hundreds + 0 Tens + 5 Ones = 4 Hundreds + 0 Tens + 5 Ones = 405

Right side: 1593

Comparing: 405 vs 1593 → 405 is 3-digit, 1593 is 4-digit

$$405 < 1593$$

$$\boxed{15 \text{ Ones} + 9 \text{ Tens} + 3 \text{ Hundreds} < 1593}$$

Left side: $5000 + 30 + 4 = 5034$

Right side: 5034

$$5034 = 5034$$

$$\boxed{5000 + 30 + 4 = 5034}$$

Left side: $5000 + 300 + 4 = 5304$

Right side: 5340

Comparing: 5304 vs 5340
Th: 5=5, H: 3=3, T: 0 < 4

$$5304 < 5340$$

$$\boxed{5000 + 300 + 4 < 5340}$$

We need: $7\underline{\phantom{0}}3 < 768\underline{\phantom{0}}$

Left number is of the form $7X3$ (3-digit) or $7X3$ where X is the hundreds digit...

Wait — 7__3 is a 4-digit number: $7A3$ where A is the hundreds digit, making it $7A03$ or $7A3$ as a 3-digit...

Re-reading: 7__3 means a number with 7 in thousands, some digit in hundreds, some digit in tens, and 3 in ones → 7AB3 where A and B are single digits.

And 768_ means 768C where C is a single digit.

For $7AB3 < 768C$:

• Thousands: 7 = 7
• Hundreds: A vs 6 → if A < 6, the inequality holds for any B and C.
• If A = 6: Tens: B vs 8 → if B < 8, inequality holds.
• If A = 6, B = 8: Ones: 3 vs C → need 3 < C, so C ∈ {4,5,6,7,8,9}.

One valid answer: A = 5, B = any digit (0–9), C = any digit (0–9)
Example: 7503 < 7689 ✓

Or simply: the blank in 7__3 can be filled with any digit less than 6 in the hundreds place.
Example answer: 7503 < 7684**

Left: $853AB$ — wait, these appear to be 4-digit numbers.

Left: 853_ = $853C$ (4-digit, thousands=8, hundreds=5, tens=3, ones=C)
Right: 8_3_ = $8D3E$ (4-digit, thousands=8, hundreds=D, tens=3, ones=E)

For $853C < 8D3E$:

• Thousands: 8 = 8
• Hundreds: 5 vs D → if D > 5, inequality holds for any C and E.

One valid answer: D = 6, C = any, E = any
Example: 8530 < 8630 ✓

Answer: 8530 < 8630 (D = 6, C = 0, E = 0)

Left: _2_1 = $A2B1$ (4-digit: thousands=A, hundreds=2, tens=B, ones=1)
Right: 5_2_ = $52CD$ ... re-reading: 5__2__ = $5C2D$ (thousands=5, hundreds=C, tens=2, ones=D)

For $A2B1 < 5C2D$:

• If A < 5: inequality holds regardless of other digits.

One valid answer: A = 4, B = any, C = any, D = any
Example: 4201 < 5020 ✓

Answer: 4201 < 5020**** (A=4, B=0, C=0, D=0)

Numbers strictly between 700 and 800 (excluding 700 and 800):
$$701, 702, 703, \ldots, 799$$
Count = $799 - 701 + 1 = 99$ ✓

Numbers strictly between 7000 and 8000 (excluding 7000 and 8000):
$$7001, 7002, 7003, \ldots, 7999$$
Count = $7999 - 7001 + 1 = \mathbf{999}$

Correct answer: 999

Justification: Between any two consecutive thousands (like 7000 and 8000), there are exactly 999 whole numbers strictly between them, just as there are 99 whole numbers strictly between any two consecutive hundreds.

Given digits: 2, 3, 4, 7 (no repetition, each used exactly once)

Total arrangements = $4! = 4 \times 3 \times 2 \times 1 = 24$

All 24 four-digit numbers:

Starting with 7 (largest): 7432, 7423, 7342, 7324, 7243, 7234
Starting with 4: 4732, 4723, 4372, 4327, 4273, 4237
Starting with 3: 3742, 3724, 3472, 3427, 3274, 3247
Starting with 2 (smallest): 2743, 2734, 2473, 2437, 2374, 2347

Decreasing order (from highest to lowest):
$$7432 > 7423 > 7342 > 7324 > 7243 > 7234$$
$$> 4732 > 4723 > 4372 > 4327 > 4273 > 4237$$
$$> 3742 > 3724 > 3472 > 3427 > 3274 > 3247$$
$$> 2743 > 2734 > 2473 > 2437 > 2374 > 2347$$

To verify all 24 are found: Systematically fix each digit in the thousands place (4 choices), then hundreds (3 remaining choices), then tens (2 remaining), then ones (1 remaining): $4 \times 3 \times 2 \times 1 = 24$. If you have 6 numbers starting with each of 2, 3, 4, 7, you have all 24.

Method to ensure all 24 numbers are found:

Step 1: Fix the thousands digit. There are 4 choices: 2, 3, 4, or 7.

Step 2: For each thousands digit, fix the hundreds digit from the remaining 3 digits.

Step 3: For each combination of thousands and hundreds, fix the tens digit from the remaining 2 digits.

Step 4: The ones digit is the only digit left (1 choice).

This gives: $4 \times 3 \times 2 \times 1 = 24$ numbers.

You know you have all 24 numbers when:

• You have exactly 6 numbers starting with each of the four digits (2, 3, 4, 7).
• $6 \times 4 = 24$ total numbers.
• No number is repeated.

Comparing with friends helps catch any numbers you may have missed and confirms that the total count is 24.