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Estimate Sums and Differences: Round First, Then Check

Round to the nearest ten or hundred to estimate sums and differences, compare front-end estimates, and decide whether an exact answer is reasonable.

Grades 3–5 · 32 practice questions · Original material
Learning objectiveEstimate sums and differences by rounding each number to the same place, compare rounding with front-end estimation, tell whether an estimate is too high or too low, decide when an estimate is safe and when an exact answer is needed, and use an estimate to judge whether an exact answer is reasonable.

The idea

An estimate is a quick “about” answer. Suppose a class collects 312 cans on Monday and 489 cans on Tuesday (invented numbers). Rounding gives 300 + 500 = 800, so the class collected about 800 cans. The exact total is 801, so the estimate was very close, and it took only a moment.

Estimating has three steps. First, round each number to the same place, such as the nearest ten or the nearest hundred. Next, do the easy calculation with the rounded numbers, keeping their zeros: 3 hundreds + 5 hundreds is 8 hundreds, or 800. Last, compare: is the exact answer close to the estimate?

The place you round to changes how close you get. For 347 + 228, rounding to the nearest hundred gives 300 + 200 = 500, but rounding to the nearest ten gives 350 + 230 = 580. The exact sum is 575. Rounding to hundreds is quicker; rounding to tens keeps more detail and usually lands closer.

Front-end estimation is another quick method: keep the first digit of each number and turn the rest into zeros. For 389 + 297 it gives 300 + 200 = 500, but the exact sum is 686. When you add, front-end always comes out too low because the dropped digits are never added back. It is fast, but rounding is usually closer.

Sometimes you can tell which way an estimate is off. When you add and every number is rounded up, the estimate is too high (an overestimate); when every number is rounded down, it is too low (an underestimate). Subtraction is trickier: rounding the number you take away up makes the estimate smaller. If you need to be sure you have enough money, round every price up.

An estimate is a check on size. If 812 − 395 should be about 800 − 400 = 400, an answer of 583 tells you to look again; the exact answer is 417. But an estimate cannot prove an answer is exactly right: 799 and 801 are both close to 800. When every person needs a chair or every cent counts, calculate the exact answer.

Worked examples

Estimating sums and differences by rounding, compared with front-end estimationThree number lines. Line 1 runs from 300 to 500. 312 rounds down to 300 and 489 rounds up to 500, so 312 plus 489 is about 300 plus 500, which is 800; the exact sum is 801. Line 2 runs from 400 to 800 and compares estimates for 389 plus 297: the front-end estimate 500 is far too low, the rounding estimate 700 is close, and the exact sum is 686. Line 3 runs from 300 to 700 and checks 812 minus 395: the estimate is 800 minus 400, which is 400. The exact answer 417 is close to the estimate, but the answer 583 is far away, so it is not reasonable. An estimate checks the size of an answer; it cannot prove the answer is exactly right.Estimate: round first, then check the sizeOriginal schematic, drawn to the labelled values.1 · Round each number, then add: 312 + 489300350400450500312489300 + 500 = 800exact: 8012 · Compare estimates for 389 + 297400500600700800front-end 500rounding 700exact 686front-end: 300 + 200rounding: 400 + 300Rounding is closer.3 · Is it reasonable? 812 − 395300400500600700estimate 400417 ✓583 ✗ too far800 − 400 = 400583: check againAn estimate checks the size of an answer. It cannot prove an answer is exactly right.
Original schematic, drawn to the labelled values. Round each number to the same place, then calculate with the friendly numbers. Front-end estimation drops digits, so for sums it is always too low. A close estimate shows an answer is reasonable in size; it does not prove the answer is exact.

Example 1 · Round, then add

About how many cans is 312 + 489?

  1. Round each number to the nearest hundred: 312 → 300 (tens digit 1) and 489 → 500 (tens digit 8).
  2. Add the rounded numbers: 300 + 500 = 800.
  3. Compare with the exact sum: 312 + 489 = 801, which is close to 800.

About 800 cans.

Example 2 · Hundreds or tens?

Estimate 347 + 228 two ways.

  1. Nearest hundred: 300 + 200 = 500.
  2. Nearest ten: 350 + 230 = 580.
  3. The exact sum is 575. The nearest-ten estimate is 5 away; the nearest-hundred estimate is 75 away.

Rounding to tens is closer; rounding to hundreds is quicker.

Example 3 · Front-end or rounding?

Estimate 389 + 297 with front-end estimation and with rounding.

  1. Front-end keeps the first digits: 300 + 200 = 500.
  2. Rounding to the nearest hundred: 400 + 300 = 700.
  3. The exact sum is 686. Front-end dropped 89 and 97, almost 200 altogether.

Rounding (700) is much closer than front-end (500).

Example 4 · Is the answer reasonable?

Maya says 812 − 395 = 583. Should she check?

  1. Round: 812 → 800 and 395 → 400.
  2. Estimate: 800 − 400 = 400.
  3. 583 is almost 200 away from 400, so check. She took the smaller digit from the bigger one in each column; regrouping gives 417.

Yes. 583 is not reasonable; 417 is.

Example 5 · Is there enough money?

A family has $20 (invented prices). Items cost $4.79, $6.49 and $7.85. Is $20 enough?

  1. Round every price up to the next whole dollar: $5 + $7 + $8 = $20.
  2. Every price went up, so the real total must be $20 or less.
  3. Check: $4.79 + $6.49 + $7.85 = $19.13.

Yes. Rounding every price up gives a safe estimate.

Common mistakes

These are tempting because they use familiar operations. Check what the question is actually measuring.

Treating the estimate as the answer.

Why this fails: An estimate is “about”. If a question asks for the exact total, 800 is not an answer to 312 + 489; 801 is.

Losing the zeros.

Why this fails: 300 + 500 is 3 hundreds + 5 hundreds = 8 hundreds, which is 800, not 8 or 80. Keep the place value of the rounded numbers.

Using front-end when the dropped digits are big.

Why this fails: For 389 + 297, front-end gives 500 but the exact sum is 686. Front-end is always too low for sums like this; rounding gives 700.

Thinking “close to the estimate” means correct.

Why this fails: 799 and 801 are both close to 800, but only 801 is right. Estimates catch big mistakes, not small ones.

Expecting rounding up to make a difference too big.

Why this fails: In 630 − 270 ≈ 600 − 300 = 300, the number taken away went up, so the estimate is too small; the exact answer is 360.

Trusting an estimate when you need enough.

Why this fails: 287 + 145 rounds to 300 + 100 = 400, but 432 people need chairs. When you must have enough, round up or calculate exactly.

Foundation

Foundation practice

Round each number, keep the zeros and estimate.

Estimate a sum

1. Round 312 and 489 to the nearest hundred, then add. What is the estimate?

  1. 700
  2. 800
  3. 900
  4. 801
Nearest ten

2. Estimate 46 + 38 by rounding each number to the nearest ten.

  1. 90
  2. 70
  3. 84
  4. 80
Estimate a difference

3. Estimate 73 − 28 by rounding each number to the nearest ten.

  1. 50
  2. 45
  3. 40
  4. 100
Front-end

4. Front-end estimation keeps the first digit of each number and makes the other digits zeros. What is the front-end estimate of 487 + 362?

  1. 900
  2. 849
  3. 800
  4. 700
Is it reasonable?

5. Ava says 527 + 368 = 8815. Is that reasonable?

  1. No. The estimate is about 500 + 400 = 900, so a four-digit answer is far too big.
  2. Yes, because it uses the right digits.
  3. Yes, because both numbers are more than 300.
  4. No, because the answer should be exactly 900.
Over or under?

6. You estimate 48 + 37 as 50 + 40 = 90. Is 90 more or less than the exact sum?

  1. Less: both numbers were rounded down, so 90 is an underestimate.
  2. Exactly the same: rounding does not change the sum.
  3. You cannot tell without working out the exact sum.
  4. More: both numbers were rounded up, so 90 is an overestimate.
Keep the zeros

7. Jun rounds 312 + 489 to 3 hundreds + 5 hundreds. What is the estimate?

  1. 8
  2. 80
  3. 800
  4. 8000
Estimate a difference

8. Estimate 912 − 487 by rounding each number to the nearest hundred.

  1. 500
  2. 400
  3. 425
  4. 1400

Practice

Practice practice

Choose the rounding place, compare methods and check reasonableness.

Closest estimate

9. The exact sum of 389 + 297 is 686. Which estimate is closest?

  1. 500 (front-end: 300 + 200)
  2. 600 (400 + 200)
  3. 690 (round to the nearest ten: 390 + 300)
  4. 700 (round to the nearest hundred: 400 + 300)
Nearest hundred

10. Round each number to the nearest hundred to estimate 1489 + 2310.

  1. 3000
  2. 3800
  3. 3799
  4. 3700
Real context

11. A school library has 634 fiction and 279 nonfiction books (invented numbers). Round to the nearest hundred: about how many books are there?

  1. 900
  2. 800
  3. 1000
  4. 913
Spot the error

12. Maya says 812 − 395 = 583. Is that reasonable?

  1. Yes, because 583 is less than 812.
  2. Yes, because the estimate is 800 − 300 = 500.
  3. No, because the answer must be more than 812.
  4. No. The estimate is 800 − 400 = 400, so 583 is too far off.
Over or under?

13. You estimate 630 − 270 as 600 − 300 = 300. The exact answer is 360. Why is the estimate too small?

  1. 270 was rounded up, so the estimate must be too big.
  2. The number being taken away was rounded up, so more was subtracted.
  3. 630 was rounded down, so the estimate must be too big.
  4. Rounding always makes a difference too small.
Enough money?

14. Kim has $12 (invented prices). She wants items that cost $3.85, $2.60 and $4.90. She rounds every price up to the next whole dollar: $4 + $3 + $5 = $12. Does she have enough?

  1. No. The estimate equals her money, so she will be short.
  2. Yes, because rounding down gives $9.
  3. Yes. Every price was rounded up, so the real total is $12 or less.
  4. You cannot tell without adding the exact prices.
Front-end

15. When you add two numbers that are not already round, a front-end estimate is …

  1. an overestimate
  2. always exactly right
  3. always closer than rounding
  4. an underestimate, because the dropped digits are never added
Choose the rounded numbers

16. Round each number to the nearest hundred. Which estimate for 451 + 336 is correct?

  1. 500 + 300 = 800
  2. 400 + 300 = 700
  3. 500 + 400 = 900
  4. 5 + 3 = 8

Think

Think practice

Decide what an estimate can and cannot tell you.

Choose the place

17. Sam says 4567 − 1892 = 3335. Which estimate best shows that Sam should check his work?

  1. 5000 − 2000 = 3000, so 3335 looks fine.
  2. 4600 − 1900 = 2700, so 3335 is too far off.
  3. 4000 − 1000 = 3000, so 3335 looks fine.
  4. 4567 − 1892 cannot be estimated.
What can an estimate tell you?

18. Two students add 312 + 489. One says 799 and one says 801. Both are close to the estimate 800. What does the estimate tell you?

  1. Both answers are right.
  2. Neither answer can be right.
  3. It cannot choose between them; you must calculate exactly.
  4. The bigger answer is right.
Which is furthest off?

19. For which sum is the front-end estimate furthest from the exact answer?

  1. 210 + 320
  2. 405 + 501
  3. 120 + 130
  4. 290 + 390
When an estimate is not enough

20. A concert will have 287 students and 145 parents (invented numbers). Rounding to the nearest hundred gives 300 + 100 = 400. Why is setting out 400 chairs a problem?

  1. 145 was rounded down a lot, so 400 is less than the 432 people coming.
  2. 400 is more than enough, because 287 was rounded up.
  3. Estimates are always too big.
  4. Nothing is wrong; 400 is close enough.
Check with an estimate

21. Which answer to 703 − 296 is reasonable?

  1. 407
  2. 593
  3. 999
  4. 4007
Over or under?

22. 352 rounds up to 400 and 449 rounds down to 400. The estimate for 352 + 449 is 800, and the exact sum is 801. What does this show?

  1. The estimate is always too high when any number rounds up.
  2. The estimate is always too low when any number rounds down.
  3. When one number rounds up and the other rounds down, the estimate could be too high or too low.
  4. Rounding cannot be used when the numbers round different ways.
Explain

23. Explain why front-end estimation gives 500 for 389 + 297 when the exact sum is 686. How could you get a closer estimate?

True or false?

24. True or false: if every price is rounded up, the estimated total is at least the real total.

  1. False
  2. True
  3. True only for two items
  4. True only for whole-dollar prices

Challenge

Challenge practice

Work backwards, spot hidden underestimates and choose a strategy.

Work backwards

25. Two whole numbers each round to 300 (nearest hundred). What is the largest their exact sum could be?

  1. 700
  2. 600
  3. 500
  4. 698
Work backwards

26. Two whole numbers each round to 300 (nearest hundred). What is the smallest their exact sum could be?

  1. 500
  2. 498
  3. 502
  4. 600
Estimate a bigger sum

27. Round each number to the nearest hundred to estimate 2785 + 1460 + 3215.

  1. 7000
  2. 7460
  3. 7500
  4. 7300
Check with an estimate

28. Which answer to 6048 − 2975 is reasonable?

  1. 4933
  2. 3073
  3. 9023
  4. 373
Hidden underestimate

29. Lina has $10 (invented prices). She wants items that cost $2.45, $3.40 and $4.35. Rounding to the nearest dollar gives $2 + $3 + $4 = $9. Does she have enough?

  1. Yes. The estimate is $9, so she will have $1 left.
  2. No. Every price was rounded down, so the real total is more than $9; it is $10.20.
  3. Yes, because rounding always gives a safe answer.
  4. No, because rounding up gives $12, which is more than $10.
Tens or hundreds?

30. Estimate 247 + 438 to the nearest ten and to the nearest hundred. The exact sum is 685. Which estimate is closer?

  1. Nearest ten: 250 + 440 = 690
  2. Nearest hundred: 200 + 400 = 600
  3. They are equally close.
  4. Front-end: 200 + 400 = 600
Choose a strategy

31. You need a quick check of 498 + 503. Which estimate is easiest and closest?

  1. 400 + 500 = 900
  2. 490 + 500 = 990
  3. 500 + 500 = 1000
  4. 498 + 503 cannot be estimated
Work backwards

32. Kai rounded two whole numbers to the nearest ten and got 70 + 20 = 90. What are the smallest and largest the exact sum could be?

Answer key

  1. 1. 800
    312 rounds to 300 and 489 rounds to 500, so the estimate is 300 + 500 = 800. 700 chops 489 down to 400 instead of rounding it, 900 rounds 312 up even though its tens digit is 1, and 801 is the exact answer, not an estimate.
  2. 2. 90
    46 rounds to 50 and 38 rounds to 40, so 50 + 40 = 90. 70 keeps only the tens digits (40 + 30), which is front-end estimation, 84 is the exact sum, and 80 rounds 46 down even though its ones digit is 6.
  3. 3. 40
    73 rounds to 70 and 28 rounds to 30, so 70 − 30 = 40. 50 chops 28 down to 20, 45 is the exact difference, and 100 adds the rounded numbers instead of subtracting them.
  4. 4. 700
    Front-end keeps 400 and 300, so the estimate is 700. 900 is the rounding estimate (500 + 400), 849 is the exact sum, and 800 rounds one number but uses front-end for the other.
  5. 5. No. The estimate is about 500 + 400 = 900, so a four-digit answer is far too big.
    Ava wrote 7 + 8 = 15 without regrouping, so her digits ran together. An estimate of 900 shows that 8815 cannot be right; the exact sum is 895. Having familiar digits does not make an answer reasonable, being more than 300 says almost nothing, and the estimate is close to the answer, not the exact answer.
  6. 6. More: both numbers were rounded up, so 90 is an overestimate.
    48 went up to 50 and 37 went up to 40, so the estimate is bigger than the exact sum, 85. Neither number was rounded down, rounding does change the sum, and when both numbers are rounded up you can tell without the exact answer.
  7. 7. 800
    3 hundreds + 5 hundreds = 8 hundreds, which is 800. 8 and 80 lose the hundreds place value, and 8000 adds an extra zero, so it is 8 thousands.
  8. 8. 400
    912 rounds to 900 and 487 rounds to 500, so 900 − 500 = 400. 500 chops 487 down to 400, 425 is the exact difference, and 1400 adds the rounded numbers instead of subtracting.
  9. 9. 690 (round to the nearest ten: 390 + 300)
    690 is only 4 away from 686. Rounding to the nearest hundred gives 700, which is 14 away and still useful, but rounding to tens keeps more detail. 600 rounds 297 down although it is almost 300, and front-end 500 is 186 too low.
  10. 10. 3800
    1489 rounds to 1500 (tens digit 8) and 2310 rounds to 2300, so 1500 + 2300 = 3800. 3000 rounds to the nearest thousand instead, 3799 is the exact sum, and 3700 chops 1489 to 1400 instead of rounding it.
  11. 11. 900
    634 rounds to 600 and 279 rounds to 300, so about 900 books. 800 keeps only the first digits (600 + 200), 1000 rounds 634 up even though its tens digit is 3, and 913 is the exact total, not the rounded estimate.
  12. 12. No. The estimate is 800 − 400 = 400, so 583 is too far off.
    Rounding gives 400, and 583 is almost 200 away, so Maya should check. She took the smaller digit from the bigger one in each column; the exact answer is 417. Being less than 812 is true of many wrong answers, 395 rounds to 400, not 300, and a difference is smaller than the starting number, not bigger.
  13. 13. The number being taken away was rounded up, so more was subtracted.
    Taking away 300 instead of 270 removes 30 extra, and starting at 600 instead of 630 removes another 30, so the estimate is 60 too small. In subtraction, rounding the number you take away up makes the answer smaller, not bigger, and rounding does not always make a difference too small: 680 − 220 ≈ 700 − 200 = 500, which is more than the exact 460.
  14. 14. Yes. Every price was rounded up, so the real total is $12 or less.
    Each rounded price is at least the real price, so the real total cannot be more than $12; it is $11.35. An estimate equal to her money is still safe when every price was rounded up. Rounding down only shows the total is more than $9, and you do not need the exact total to know she has enough.
  15. 15. an underestimate, because the dropped digits are never added
    Front-end throws away the digits after the first one, and those digits are never added back, so the estimate is too low. It cannot be too high or exact for these numbers, and it is often much further off than rounding: 389 + 297 gives 500 instead of 686.
  16. 16. 500 + 300 = 800
    451 rounds to 500 (tens digit 5) and 336 rounds to 300 (tens digit 3), so 800. 400 + 300 chops 451 instead of rounding, 500 + 400 rounds 336 the wrong way, and 5 + 3 = 8 drops the hundreds place value.
  17. 17. 4600 − 1900 = 2700, so 3335 is too far off.
    Rounding to the nearest hundred gives 2700, and 3335 is more than 600 away. The exact answer is 2675. The nearest-thousand estimate and the front-end estimate are both 3000, which is too rough to catch this mistake. Any whole-number difference can be estimated.
  18. 18. It cannot choose between them; you must calculate exactly.
    An estimate catches big mistakes, not small ones. Both answers are close to 800, so you need the exact sum: 801. They cannot both be right, one of them is right, and bigger is not a rule.
  19. 19. 290 + 390
    Front-end gives 200 + 300 = 500 for 290 + 390, but the exact sum is 680, so it is 180 too low. The other sums are 30, 6 and 50 too low. Front-end is weakest when the digits it drops are big.
  20. 20. 145 was rounded down a lot, so 400 is less than the 432 people coming.
    287 went up by 13 but 145 went down by 45, so the estimate is 32 too low and some people would have no chair. Rounding one number up does not make the whole estimate safe, estimates can be too big or too small, and when everyone needs a seat, close is not enough.
  21. 21. 407
    700 − 300 = 400, and 407 is close to that. 593 takes the smaller digit from the bigger one in each column, 999 adds instead of subtracting, and 4007 has place-value digits out of order, so it is about ten times too big.
  22. 22. When one number rounds up and the other rounds down, the estimate could be too high or too low.
    352 went up by 48 and 449 went down by 49, so they almost cancel and the estimate ends up 1 too low. With mixed rounding you cannot predict the direction without looking at the sizes. Rounding still gives a useful estimate.
  23. 23. Front-end drops 89 and 97, which is almost 200 that never gets added. Rounding to the nearest hundred gives 400 + 300 = 700, and rounding to the nearest ten gives 390 + 300 = 690, which are much closer.
    Strong answers say which digits were dropped, why that makes the estimate low, and show a rounding estimate.
  24. 24. True
    Each rounded price is the same as or more than its real price, so adding them gives at least the real total. This works for any number of items and for prices with cents.
  25. 25. 698
    The largest whole number that rounds to 300 is 349, because 350 rounds up to 400. So 349 + 349 = 698. 700 uses 350, which rounds to 400, 600 is only the estimate, and 500 is the smallest possible sum (250 + 250).
  26. 26. 500
    The smallest whole number that rounds to 300 is 250, because halfway rounds up. So 250 + 250 = 500. 498 uses 249, which rounds to 200, 502 skips the halfway numbers, and 600 is only the estimate.
  27. 27. 7500
    2785 → 2800, 1460 → 1500 and 3215 → 3200, so the estimate is 7500. 7000 rounds to the nearest thousand instead, 7460 is the exact sum, and 7300 chops 2785 and 1460 instead of rounding them.
  28. 28. 3073
    6000 − 3000 = 3000, and 3073 is close. 4933 takes the smaller digit from the bigger one in each column, 9023 adds instead of subtracting, and 373 has lost a digit, so it is about ten times too small.
  29. 29. No. Every price was rounded down, so the real total is more than $9; it is $10.20.
    All three prices lost cents when they were rounded, so the estimate is too low; the exact total, $10.20, is more than $10. Rounding is not always safe, and the round-up estimate of $12 only shows the total is $12 or less, so it cannot prove she is short.
  30. 30. Nearest ten: 250 + 440 = 690
    690 is 5 away and 600 is 85 away. 247 is close to the halfway mark 250, so rounding it to 200 loses almost 50. Front-end also gives 600 here, and the two estimates are not equally close.
  31. 31. 500 + 500 = 1000
    Both numbers are very close to 500, so 1000 is quick and only 1 away from the exact 1001. 900 is front-end and 101 too low, 990 rounds 498 the wrong way, and every sum like this can be estimated.
  32. 32. Smallest 80 (65 + 15). Largest 98 (74 + 24).
    Numbers from 65 to 74 round to 70 and numbers from 15 to 24 round to 20. Halfway numbers round up, so 75 and 25 are not allowed.

Related practice

Rounding to 10, 100 and 1000: then estimate and checkMental addition and subtraction: choose a shortcutMoney problems: total cost, change and a reasonableness checkMath hub