Estimation

Estimation is the process of finding an approximate value that is “close enough” to the actual value to be useful. It simplifies large numbers to make mental math faster.

1. General Rules for Rounding Off

To round off any number, we look at the digit to the immediate right of the place we are rounding to:

• Round Down: If the digit is less than 5 (0, 1, 2, 3, 4), keep the target digit the same and turn all digits to its right into zeros.
• Round Up: If the digit is 5 or more (5, 6, 7, 8, 9), add 1 to the target digit and turn all digits to its right into zeros.

2. Common Rounding Scenario

Rounding Level	Look at this Digit	Example (Number: 8,462)	Result
Nearest Ten	One’s Place	2 is less than 5	8,460
Nearest Hundred	Ten’s Place	6 is more than 5	8,500
Nearest Thousand	Hundred’s Place	4 is less than 5	8,000

3. Estimating Outcomes (Operations)

In the ICSE curriculum, students are often asked to estimate the results of arithmetic operations.

• Estimated Sum & Difference: Round off each number to its greatest place value (or the place specified), then add or subtract.
  • Example: Estimate 5,290 + 17,986 to the nearest thousand.
  • 5,000 + 18,000 = 23,000.
• Estimated Product: Round off each factor to its greatest place value, then multiply.
  • Example: Estimate 63 *182
  • 60 * 200 = 12,000.
• Estimated Quotient: Round off the dividend and divisor to numbers that are “compatible” (easy to divide).
  • Example: Estimate 78 / 18
  • 80/ 20 = 4.

4. Roman Numerals (Often included in this chapter)

ICSE Class 6 usually pairs Estimation with Roman Numerals. Key symbols to include on your site:

• I = 1, V = 5, X = 10, L = 50, C = 100, D = 500, M = 1000.
• Rules: 1. A symbol can be repeated up to 3 times (e.g., XXX = 30).
• 2. A smaller symbol before a larger one means subtraction (IX = 9).
• 3. A smaller symbol after a larger one means addition (XI = 11).

EXERCISE 2(A)

Question 1.
Round off each of the following to the nearest ten:
(i) 62
(ii) 265
(iii) 543
(iv) 8261
(v) 6294
(vi) 3008
(vii) 72326

Solution :

(i) 62

• Ones digit is 2 (less than 5).
• Round down.
• Result: 60

(ii) 265

• Ones digit is 5 (5 or more).
• Round up.
• Result: 270

(iii) 543

• Ones digit is 3 (less than 5).
• Round down.
• Result: 540

(iv) 8261

• Ones digit is 1 (less than 5).
• Round down.
• Result: 8260

(v) 6294

• Ones digit is 4 (less than 5).
• Round down.
• Result: 6290

(vi) 3008

• Ones digit is 8 (5 or more).
• Round up.
• Result: 3010

(vii) 72326

• Ones digit is 6 (5 or more).
• Round up.
• Result: 72330

Question 2.
Round off each of the following to the nearest hundred :
(i) 748
(ii) 784
(iii) 2667
(iv) 5432
(v) 6388
(vi) 59237

Solution :

(i) 748

• Tens digit is 4 (less than 5).
• Round down.
• Result: 700

(ii) 784

• Tens digit is 8 (5 or more).
• Round up.
• Result: 800

(iii) 2667

• Tens digit is 6 (5 or more).
• Round up. (Add 1 to 6 in the hundreds place).
• Result: 2700

(iv) 5432

• Tens digit is 3 (less than 5).
• Round down.
• Result: 5400

(v) 6388

• Tens digit is 8 (5 or more).
• Round up. (Add 1 to 3 in the hundreds place).
• Result: 6400

(vi) 59237

• Tens digit is 3 (less than 5).
• Round down. (The 2 stays the same).
• Result: 59200

Question 3.
Round off each of the following to the nearest thousand:
(i) 6475
(ii) 6732 (iii) 5352
(iv) 32568
(v) 9248
(vi) 83294

Solution :

(i) 6475

• Hundreds digit is 4 (less than 5).
• Action: Round down.
• Result: 6,000

(ii) 6732

• Hundreds digit is 7 (5 or more).
• Action: Round up. Add 1 to the 6 in the thousands place.
• Result: 7,000

(iii) 5352

• Hundreds digit is 3 (less than 5).
• Action: Round down.
• Result: 5,000

(iv) 32568

• Hundreds digit is 5 (5 or more).
• Action: Round up. Add 1 to the 2 in the thousands place.
• Result: 33,000

(v) 9248

• Hundreds digit is 2 (less than 5).
• Action: Round down.
• Result: 9,000

(vi) 83294

• Hundreds digit is 2 (less than 5).
• Action: Round down.
• Result: 83,000

Question 4.
Round off
(i) 578 to the nearest ten.
(ii) 578 to the nearest hundred.
(iii) 4327 to the nearest thousand.
(iv) 32974 to the nearest ten-thousand.
(v) 27487 to the nearest ten-thousand.

Solution :

(i) 578 to the nearest ten

• Target: Tens place (7).
• Deciding Digit: Ones place (8).
• Since 8 is “5 or more,” we round up. Add 1 to the tens place.
• Result: 580

(ii) 578 to the nearest hundred

• Target: Hundreds place (5).
• Deciding Digit: Tens place (7).
• Since 7 is “5 or more,” we round up. Add 1 to the hundreds place.
• Result: 600

(iii) 4327 to the nearest thousand

• Target: Thousands place (4).
• Deciding Digit: Hundreds place (3).
• Since 3 is “less than 5,” we round down. The thousands digit stays the same.
• Result: 4,000

(iv) 32974 to the nearest ten-thousand

• Target: Ten-thousands place (3).
• Deciding Digit: Thousands place (2).
• Since 2 is “less than 5,” we round down. The 3 stays the same, and all digits to the right become zero.
• Result: 30,000

(v) 27487 to the nearest ten-thousand

• Target: Ten-thousands place (2).
• Deciding Digit: Thousands place (7).
• Since 7 is “5 or more,” we round up. Add 1 to the ten-thousands place.
• Result: 40,000

Question 5.
Round off each of the following to the nearest tens, hundreds and thousands.
(i) 864
(ii) 1249
(iii) 54, 547
(iv) 68, 076
(v) 56, 293
(vi) 7, 293
(vii) 89, 24, 379

Solution :

(i) 864

• Nearest Ten: Look at 4 (less than 5) $\rightarrow$ 860
• Nearest Hundred: Look at 6 (5 or more) $\rightarrow$ 900
• Nearest Thousand: Look at 8 (5 or more) $\rightarrow$ 1,000

(ii) 1,249

• Nearest Ten: Look at 9 (5 or more) $\rightarrow$ 1,250
• Nearest Hundred: Look at 4 (less than 5) $\rightarrow$ 1,200
• Nearest Thousand: Look at 2 (less than 5) $\rightarrow$ 1,000

(iii) 54,547

• Nearest Ten: Look at 7 (5 or more) $\rightarrow$ 54,550
• Nearest Hundred: Look at 4 (less than 5) $\rightarrow$ 54,500
• Nearest Thousand: Look at 5 (5 or more) $\rightarrow$ 55,000

(iv) 68,076

• Nearest Ten: Look at 6 (5 or more) – 68,080
• Nearest Hundred: Look at 7 (5 or more) – 68,100
• Nearest Thousand: Look at 0 (less than 5) – 68,000

(v) 56,293

• Nearest Ten: Look at 3 (less than 5)- 56,290
• Nearest Hundred: Look at 9 (5 or more) – 56,300
• Nearest Thousand: Look at 2 (less than 5) – 56,000

(vi) 7,293

• Nearest Ten: Look at 3 (less than 5) – 7,290
• Nearest Hundred: Look at 9 (5 or more) – 7,300
• Nearest Thousand: Look at 2 (less than 5) – 7,000

(vii) 89,24,379

• Nearest Ten: Look at 9 (5 or more) – 89,24,380
• Nearest Hundred: Look at 7 (5 or more) -$ 89,24,400
• Nearest Thousand: Look at 3 (less than 5) – 89,24,000

Question 6.
Round off the following to the nearest tens ;
(i) ₹ 562
(ii) 837 m
(iii) 545 cm
(iv) ₹ 27

Solution :

(i) ₹ 562

• Ones digit: 2 (less than 5).
• Action: Round down.
• Result: ₹ 560

(ii) 837 m

• Ones digit: 7 (5 or more).
• Action: Round up. Add 1 to the tens digit (3 becomes 4).
• Result: 840 m

(iii) 545 cm

• Ones digit: 5 (5 or more).
• Action: Round up. Add 1 to the tens digit (4 becomes 5).
• Result: 550 cm

(iv) ₹ 27

• Ones digit: 7 (5 or more).
• Action: Round up. Add 1 to the tens digit (2 becomes 3).
• Result: ₹ 30

Question 7.
List all the numbers which can be round off to 30.

Solution :

The numbers are: 25, 26, 27, 28, 29, 30, 31, 32, 33, and 34.

Question 8.
List all the numbers which can be rounded off to 50.

Solution :

The numbers are: 45, 46, 47, 48, 49, 50, 51, 52, 53, and 54.

Question 9.
Write the smallest and the largest numbers which are rounded off to 80.

Solution :

Smallest: 75

Greatest: 84

Question 10.
Write the smallest and the largest numbers which are rounded off to 130.

Solution :

Smallest Number = 125

Largest Number = 134

EXERCISE 2(B)

Question 1.
Estimate the sum of each pair of numbers to the nearest ten :
(i) 67 and 44
(ii) 34 and 87
(iii) 23 and 66
(iv) 78 and 18
(v) 96 and 55
(vi) 76 and 62
(vii) 457 and 175
(viii) 474 and 173
(ix) 527 and 267

Solution :

(i) 67 and 44

• 67 rounds to 70 (7 is 5 or more)
• 44 rounds to 40 (4 is less than 5)
• Estimated Sum: 70 + 40 = 110

(ii) 34 and 87

• 34 rounds to 30
• 87 rounds to 90
• Estimated Sum: 30 + 90 = 120

(iii) 23 and 66

• 23 rounds to 20
• 66 rounds to 70
• Estimated Sum: 20 + 70 =90

(iv) 78 and 18

• 78 rounds to 80
• 18 rounds to 20
• Estimated Sum: 80 + 20 =100

(v) 96 and 55

• 96 rounds to 100 (9 tens + 1 ten = 10 tens)
• 55 rounds to 60
• Estimated Sum: 100 + 60 = 160

(vi) 76 and 62

• 76 rounds to 80
• 62 rounds to 60
• Estimated Sum: 80 + 60 = 140

(vii) 457 and 175

• 457 rounds to 460
• 175 rounds to 180
• Estimated Sum: 460 + 180 =640

(viii) 474 and 173

• 474 rounds to 470
• 173 rounds to 170
• Estimated Sum: 470 + 170 = 640

(ix) 527 and 267

• 527 rounds to 530
• 267 rounds to 270
• Estimated Sum: 530 + 270 = 800

Question 2.
Estimate the sum of each pair of numbers to the nearest hundred :
(i) 336 and 782
(ii) 546 and 342
(iii) 270 and 495
(iv) 4280 and 5295
(v) 4230 and 2410
(vi) 30047 and 39287

Solution :

(i) 336 and 782

• 336: Tens digit is 3. Round down to 300.
• 782: Tens digit is 8. Round up to 800.
• Estimated Sum: 300 + 800 = 1,100

(ii) 546 and 342

• 546: Tens digit is 4. Round down to 500.
• 342: Tens digit is 4. Round down to 300.
• Estimated Sum: 500 + 300 = 800

(iii) 270 and 495

• 270: Tens digit is 7. Round up to 300.
• 495: Tens digit is 9. Round up to 500.
• Estimated Sum: $300 + 500 = \mathbf{800}$

(iv) 4280 and 5295

• 4280: Tens digit is 8. Round up to 4,300.
• 5295: Tens digit is 9. Round up to 5,300.
• Estimated Sum: 4,300 + 5,300 = 9,600

(v) 4230 and 2410

• 4230: Tens digit is 3. Round down to 4,200.
• 2410: Tens digit is 1. Round down to 2,400.
• Estimated Sum: 4,200 + 2,400 =6,600

(vi) 30047 and 39287

• 30047: Tens digit is 4. Round down to 30,000.
• 39287: Tens digit is 8. Round up to 39,300.
• Estimated Sum: 30,000 + 39,300 =69,300

Question 3.
Estimate the sum of the following pair of numbers to the nearest thousand:
(i) 53826 and 36455
(ii) 56802 and 22475

Solution :

(i) 53,826 and 36,455

• 53,826: The hundreds digit is 8 (5 or more). Round up to 54,000.
• 36,455: The hundreds digit is 4 (less than 5). Round down to 36,000.
• Estimated Sum: 54,000 + 36,000 = 90,000

(ii) 56,802 and 22,475

• 56,802: The hundreds digit is 8 (5 or more). Round up to 57,000.
• 22,475: The hundreds digit is 4 (less than 5). Round down to 22,000.
• Estimated Sum: 57,000 + 22,000 = 79,000

Question 4.
Estimate the following differences correct to nearest ten :
(i) 82 – 27
(ii) 96 – 36
(iii) 508 – 248

Solution :

(i) 82 – 27

• 82: The ones digit is 2 (less than 5). Round down to 80.
• 27: The ones digit is 7 (5 or more). Round up to 30.
• Estimated Difference: 80 – 30 = 50

(ii) 96 – 36

• 96: The ones digit is 6 (5 or more). Round up to 100.
• 36: The ones digit is 6 (5 or more). Round up to 40.
• Estimated Difference: 100 – 40 = 60

(iii) 508 – 248

• 508: The ones digit is 8 (5 or more). Round up to 510.
• 248: The ones digit is 8 (5 or more). Round up to 250.
• Estimated Difference: 510 – 250 =260

Question 5.
Estimate each difference to the nearest hundred :

(i) 769 – 314
(ii) 856 – 687
(iii) 6352 – 2086

Solution :

(i) 769 – 314

• 769: The tens digit is 6 (5 or more). Round up to 800.
• 314: The tens digit is 1 (less than 5). Round down to 300.
• Estimated Difference: 800 – 300 = 500

(ii) 856 – 687

• 856: The tens digit is 5 (5 or more). Round up to 900.
• 687: The tens digit is 8 (5 or more). Round up to 700.
• Estimated Difference: 900 – 700 = 200

(iii) 6352 – 2086

• 6352: The tens digit is 5 (5 or more). Round up to 6,400.
• 2086: The tens digit is 8 (5 or more). Round up to 2,100.
• Estimated Difference: 6,400 – 2,100 = 4,300

Question 6.
Estimate each difference to the nearest thousand:
(i) 45974 – 38766
(ii) 76003 – 48399

Solution :

(i) 45,974 – 38,766

• 45,974: The hundreds digit is 9 (5 or more). Round up. (Add 1 to 5 in the thousands place).
  • Rounded value: 46,000
• 38,766: The hundreds digit is 7 (5 or more). Round up. (Add 1 to 8 in the thousands place).
  • Rounded value: 39,000
• Estimated Difference: 46,000 – 39,000 = 7,000

(ii) 76,003 – 48,399

• 76,003: The hundreds digit is 0 (less than 5). Round down. (The 6 stays the same).
  • Rounded value: 76,000
• 48,399: The hundreds digit is 3 (less than 5). Round down. (The 8 stays the same).
  • Rounded value: 48,000
• Estimated Difference: 76,000 – 48,000 = 28,000

Question 7.
Estimate each of the following products by rounding of each number to the nearest ten :
(i) 49 x 52
(ii) 63 x 38
(iii) 27 x 54
(iv) 53 x 85
(v) 74 x 67
(vi) 25 x 33

Solution :

(i) 49 * 52

• 49 rounds to 50 (9 is 5 or more).
• 52 rounds to 50 (2 is less than 5).
• Estimated Product: 50 * 50 = 2,500

(ii) 63 * 38

• 63 rounds to 60 (3 is less than 5).
• 38 rounds to 40 (8 is 5 or more).
• Estimated Product: 60 * 40 = 2,400

(iii) 27 * 54

• 27 rounds to 30 (7 is 5 or more).
• 54 rounds to 50 (4 is less than 5).
• Estimated Product: 30 * 50 = 1,500

(iv) 53 * 85

• 53 rounds to 50 (3 is less than 5).
• 85 rounds to 90 (5 is 5 or more).
• Estimated Product: 50 * 90 =4,500

(v) 74 * 67

• 74 rounds to 70 (4 is less than 5).
• 67 rounds to 70 (7 is 5 or more).
• Estimated Product: 70* 70 = 4,900

(vi) 25 * 33

• 25 rounds to 30 (5 is 5 or more).
• 33 rounds to 30 (3 is less than 5).
• Estimated Product: 30 *30 = 900

Question 8.
Estimate each of the following products by rounding off each number to the nearest
hundred :
(i) 477 x 213
(ii) 624 x 236
(iii) 333 x 247
(iv) 537 x 283
(v) 382 x 127

Solution :

(i) 477 * 213

• 477: Tens digit is 7. Round up to 500.
• 213: Tens digit is 1. Round down to 200.
• Estimated Product: 500 * 200 = 1,00,000

(ii) 624 * 236

• 624: Tens digit is 2. Round down to 600.
• 236: Tens digit is 3. Round down to 200.
• Estimated Product: 600 * 200 = 1,20,000

(iii) 333 * 247

• 333: Tens digit is 3. Round down to 300.
• 247: Tens digit is 4. Round down to 200.
• Estimated Product: 300 * 200 = 60,000

(iv) 537 * 283

• 537: Tens digit is 3. Round down to 500.
• 283: Tens digit is 8. Round up to 300.
• Estimated Product: 500 * 300 = 1,50,000

(v) 382 * 127

• 382: Tens digit is 8. Round up to 400.
• 127: Tens digit is 2. Round down to 100.
• Estimated Product: 400 * 100 = 40,000

Question 9.
Estimate each of the following products by rounding off the first number correct to
nearest ten and the other number correct to nearest hundred :
(i) 28 x 287
(ii) 432 x 128
(iii) 48 x 165
(iv) 72 x 258
(v) 83 x 664
(vi) 44 x 250

Solution :

(i) 28 * 287

• 28 rounds to 30 (nearest ten).
• 287 rounds to 300 (nearest hundred).
• Estimated Product: 30 * 300 = 9,000

(ii) 432 * 128

• 432 rounds to 430 (nearest ten).
• 128 rounds to 100 (nearest hundred).
• Estimated Product: 430 *100 = 43,000

(iii) 48 * 165

• 48 rounds to 50 (nearest ten).
• 165 rounds to 200 (nearest hundred).
• Estimated Product: 50 * 200 = 10,000

(iv) 72 * 258

• 72 rounds to 70 (nearest ten).
• 258 rounds to 300 (nearest hundred).
• Estimated Product: 70 * 300 = 21,000

(v) 83 * 664

• 83 rounds to 80 (nearest ten).
• 664 rounds to 700 (nearest hundred).
• Estimated Product: 80 * 700 = 56,000

(vi) 44 * 250

• 44 rounds to 40 (nearest ten).
• 250 rounds to 300 (nearest hundred).
• Estimated Product: 40 * 300 = 12,000

Question 10.
Estimate each of the following quotients by converting each number to the nearest ten :
(i) 87 ÷ 28
(ii) 84 ÷ 23
(iii) 77 ÷ 22(iv) 198 ÷ 24
(v) 355 ÷ 26
(vi) 444 ÷ 42
(vii) 843 ÷ 33

Solution :

(i) 87 ÷ 28

• 87 rounds to 90 (7 is more than 5).
• 28 rounds to 30 (8 is more than 5).
• Calculation: 90 / 30 is the same as 9 / 3.
• Estimated Quotient: 3

(ii) 84 ÷ 23

• 84 rounds to 80 (4 is less than 5).
• 23 rounds to 20 (3 is less than 5).
• Calculation: 80 / 20 is the same as 8 / 2.
• Estimated Quotient: 4

(iii) 77 ÷ 22

• 77 rounds to 80 (7 is more than 5).
• 22 rounds to 20 (2 is less than 5).
• Calculation: 80 / 20 is the same as 8 / 2.
• Estimated Quotient: 4

(iv) 198 ÷ 24

• 198 rounds to 200 (8 is more than 5).
• 24 rounds to 20 (4 is less than 5).
• Calculation: 200 / 20 is the same as 20 / 2.
• Estimated Quotient: 10

(v) 355 ÷ 26

• 355 rounds to 360 (5 is the “round up” boundary).
• 26 rounds to 30 (6 is more than 5).
• Calculation: 360 / 30 is the same as 36 / 3.
• Estimated Quotient: 12

(vi) 444 ÷ 42

• 444 rounds to 440 (4 is less than 5).
• 42 rounds to 40 (2 is less than 5).
• Calculation: 440 / 40 is the same as 44 / 4.
• Estimated Quotient: 11

(vii) 843 ÷ 33

• 843 rounds to 840 (3 is less than 5).
• 33 rounds to 30 (3 is less than 5).
• Calculation: 840 / 30 is the same as 84 / 3.
• Estimated Quotient: 28