Estimating sums and differences worksheet for practicing rounding and arithmetic estimation.

Let's solve the problem step by step based on the worksheet titled "Estimating sums and differences".

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Part 1: Round the numbers to the leading digit. Estimate the sum or difference.

We are to round each number to its leading digit, then estimate the sum or difference.

#### Rule:
- The leading digit is the first non-zero digit in a number.
- Round the number to that digit, keeping only one significant figure (i.e., round to the nearest multiple of that place value).

For example:
- 3,524 → leading digit is 3 (thousands), so round to 3,000
- 4,987 → leading digit is 4 (thousands), so round to 5,000

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Problem Set 1: Round to the leading digit

#### 1. 3,524 + 4,050
- 3,524 → 4,000 (rounded to nearest thousand)
- 4,050 → 4,000
- Estimate: 4,000 + 4,000 = 8,000

#### 2. 198,348 – 200,000
- 198,348 → 200,000 (leading digit 1, but it's in hundred thousands; round to nearest 100,000 → 200,000)
- 200,000 → 200,000
- Estimate: 200,000 – 200,000 = 0

#### 3. 11,862 + 7,000
- 11,862 → 10,000 (leading digit 1 in ten thousands → round to 10,000)
- 7,000 → 7,000
- Estimate: 10,000 + 7,000 = 17,000

#### 4. 43,801 – 44,000
- 43,801 → 40,000 (leading digit 4 in ten thousands → round to 40,000)
- 44,000 → 40,000? Wait — 44,000 has leading digit 4 in ten thousands → round to 40,000?
- Actually, 44,000 → leading digit is 4, which is in the ten thousands place → round to 40,000
- But wait: rounding to leading digit means we keep only the first digit and replace others with zeros.
- So 44,000 → 40,000
- 43,801 → 40,000
- Estimate: 40,000 – 40,000 = 0

Wait — this seems off. Let’s double-check:

But actually, 43,801 — leading digit is 4, in the ten thousands place. So we round to the nearest 10,000 → 40,000.

Similarly, 44,000 → also rounds to 40,000?

No! Wait — 44,000 is already at 44,000. Leading digit is 4 (ten thousands). Rounding to the nearest 10,000 would be 40,000 because 44,000 is closer to 40,000 than 50,000? No!

Wait — 44,000 is between 40,000 and 50,000. Since it's more than 45,000, it rounds up to 50,000?

But leading digit rounding usually means you just take the first digit and zero out the rest.

So:
- 43,801 → leading digit is 4 → round to 40,000
- 44,000 → leading digit is 4 → round to 40,000

So estimate: 40,000 – 40,000 = 0

But 44,000 is not rounded to 40,000 if we're using standard rounding rules — but here it's leading digit rounding, not standard rounding.

> Clarification: "Round to the leading digit" means round to the place value of the first digit.

So:
- 43,801 → first digit is 4 in ten thousands → round to nearest 10,000 → 40,000
- 44,000 → first digit is 4 in ten thousands → round to nearest 10,000 → 40,000 (since 44,000 < 45,000)

Wait — 44,000 is less than 45,000 → rounds down to 40,000

Yes.

So estimate: 40,000 – 40,000 = 0

But that feels odd. Let's move on.

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Now let's do the next set.

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Part 2: Round to the nearest 100. Estimate the sum or difference.

Now we round to the nearest 100, then estimate.

#### 1. 485 + 21,455
- 485 → nearest 100 = 500
- 21,455 → nearest 100 = 21,500
- Estimate: 500 + 21,500 = 22,000

#### 2. 2,524 – 1,525
- 2,524 → nearest 100 = 2,500
- 1,525 → nearest 100 = 1,500
- Estimate: 2,500 – 1,500 = 1,000

#### 3. 492,726 + 3,925,000
- 492,726 → nearest 100 = 492,700
- 3,925,000 → nearest 100 = 3,925,000 (already divisible by 100)
- Estimate: 492,700 + 3,925,000 = 4,417,700

But wait — the instruction says "round to the nearest 100", so both numbers should be rounded to the nearest 100.

But 3,925,000 is already a multiple of 100 → stays same.

So yes: 492,700 + 3,925,000 = 4,417,700

But maybe they expect estimation with rounding to the same magnitude? But no — just round each to nearest 100.

#### 4. 56,499 – 22,114
- 56,499 → nearest 100 = 56,500
- 22,114 → nearest 100 = 22,100
- Estimate: 56,500 – 22,100 = 34,400

#### 5. 962,276 – 65,850
- 962,276 → nearest 100 = 962,300
- 65,850 → nearest 100 = 65,900
- Estimate: 962,300 – 65,900 = 896,400

#### 6. 80,502 + 3,823,000
- 80,502 → nearest 100 = 80,500
- 3,823,000 → already multiple of 100 → stays 3,823,000
- Estimate: 80,500 + 3,823,000 = 3,903,500

#### 7. 9,234 – 4,850
- 9,234 → nearest 100 = 9,200
- 4,850 → nearest 100 = 4,900
- Estimate: 9,200 – 4,900 = 4,300

#### 8. 5,212 + 3,872
- 5,212 → nearest 100 = 5,200
- 3,872 → nearest 100 = 3,900
- Estimate: 5,200 + 3,900 = 9,100

#### 9. 3,212 + 3,800
- 3,212 → nearest 100 = 3,200
- 3,800 → already multiple of 100 → stays 3,800
- Estimate: 3,200 + 3,800 = 7,000

#### 10. 4,045 + 8,045
- 4,045 → nearest 100 = 4,000
- 8,045 → nearest 100 = 8,000
- Estimate: 4,000 + 8,000 = 12,000

#### 11. 15,297 – 108,911
- 15,297 → nearest 100 = 15,300
- 108,911 → nearest 100 = 108,900
- Estimate: 15,300 – 108,900 = –93,600

(But negative — okay, subtraction can result in negative.)

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Part 3: Write < or > in each problem

Compare two expressions without calculating exactly — use estimation.

#### 1. 329 + 495 ___ 800
- Estimate: 300 + 500 = 800
- But actual: 329 + 495 = 824 > 800
- So: >

#### 2. 11,540 – 6,130 ___ 6,000
- Estimate: 11,500 – 6,100 = 5,400
- 5,400 < 6,000 → so <

#### 3. 863 + 311 ___ 835
- 863 + 311 ≈ 860 + 310 = 1,170
- 1,170 > 835 → >

#### 4. 8,193 – 6,448 ___ 1,600
- 8,200 – 6,400 = 1,800
- 1,800 > 1,600 → >

#### 5. 43,924 – 12,445 ___ 50,000
- 43,924 – 12,445 ≈ 44,000 – 12,000 = 32,000
- 32,000 < 50,000 → <

#### 6. 634,572 + 197,536 ___ 850,000
- 634,572 ≈ 630,000; 197,536 ≈ 200,000 → total ≈ 830,000
- 830,000 < 850,000 → <

#### 7. 18,865 + 12,694 ___ 30,000
- 18,865 + 12,694 ≈ 19,000 + 13,000 = 32,000
- 32,000 > 30,000 → >

#### 8. 33,604 – 31,641 ___ 400,000
- 33,604 – 31,641 ≈ 2,000
- 2,000 < 400,000 → <

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✔ Final Answers

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#### Part 1: Round to leading digit

| Problem | Rounded | Estimate |
|--------|--------|---------|
| 3,524 + 4,050 | 4,000 + 4,000 | 8,000 |
| 198,348 – 200,000 | 200,000 – 200,000 | 0 |
| 11,862 + 7,000 | 10,000 + 7,000 | 17,000 |
| 43,801 – 44,000 | 40,000 – 40,000 | 0 |

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#### Part 2: Round to nearest 100

| Problem | Rounded | Estimate |
|--------|--------|---------|
| 485 + 21,455 | 500 + 21,500 | 22,000 |
| 2,524 – 1,525 | 2,500 – 1,500 | 1,000 |
| 492,726 + 3,925,000 | 492,700 + 3,925,000 | 4,417,700 |
| 56,499 – 22,114 | 56,500 – 22,100 | 34,400 |
| 962,276 – 65,850 | 962,300 – 65,900 | 896,400 |
| 80,502 + 3,823,000 | 80,500 + 3,823,000 | 3,903,500 |
| 9,234 – 4,850 | 9,200 – 4,900 | 4,300 |
| 5,212 + 3,872 | 5,200 + 3,900 | 9,100 |
| 3,212 + 3,800 | 3,200 + 3,800 | 7,000 |
| 4,045 + 8,045 | 4,000 + 8,000 | 12,000 |
| 15,297 – 108,911 | 15,300 – 108,900 | –93,600 |

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#### Part 3: Write < or >

1. 329 + 495 > 800
2. 11,540 – 6,130 < 6,000
3. 863 + 311 > 835
4. 8,193 – 6,448 > 1,600
5. 43,924 – 12,445 < 50,000
6. 634,572 + 197,536 < 850,000
7. 18,865 + 12,694 > 30,000
8. 33,604 – 31,641 < 400,000

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✔ Summary of Key Concepts

- Leading digit rounding: Round to the place value of the first digit (e.g., 3,524 → 4,000).
- Nearest 100: Round to the closest multiple of 100.
- Estimation: Helps check reasonableness of answers.
- Comparison: Use estimation to compare without exact calculation.

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