Compatible numbers: what they are (numbers that are easy to work with in your head and fit the operation), how to use them to estimate sums, differences, products and especially quotients, compatible numbers against rounding, worked examples from Grade 3 to 6, when the estimate is high or low, and a printable sheet with practice

"Compatible numbers" is searched about 4,100 times a month in the United States, almost always by a parent whose Grade 3 to 5 child has brought home a worksheet that says "estimate using compatible numbers", a phrase the parent's own school never used. The idea is older than the name: pick nearby numbers that make the sum easy, then do it in your head. This page gives the definition, how to pick compatible numbers for each of the four operations (division is where they matter most), the difference from rounding and front-end estimation, twelve worked examples, how to judge whether an estimate is high or low, the curriculum placements, and a printable sheet with practice.

Download the compatible numbers sheet with practice (PDF) (A4, the method for each operation with worked examples on page 1, the comparison with rounding and the practice set with answers on page 2, free).

What compatible numbers are

Compatible numbers are numbers that are easy to work with mentally and that suit the operation: they add to a round total, or one divides the other exactly, or their product is a known fact. You replace the numbers in a problem with compatible ones, compute, and the result is an estimate.

The point is not accuracy; it is a quick answer close enough to check the exact one. A child who estimates 3,268 ÷ 8 as about 400 will notice if long division produces 48 or 4,085.

Picking compatible numbers for each operation

Why division is the big one

Rounding breaks down in division. Round 47 ÷ 6 the ordinary way and you get 50 ÷ 6, which is no easier than the original. Compatible numbers fix it: ask "what multiple of 6 is near 47?" and the answer, 48, turns the problem into a times-table fact. The habit is:

1. Look at the divisor (round it to one digit if it has two: 42 → 40).
2. Run through the divisor's multiples until you reach one near the dividend (for 8 and 3,268: 8 × 4 = 32, so 3,200).
3. Divide. The estimate also tells you how many digits the quotient has (3,200 ÷ 8 = 400, three digits), which is the check that catches a misplaced digit in long division.

This is the Grade 4 and 5 skill the worksheets want (4.NBT.6 and 5.NBT.6 ask for estimation alongside the division), and it is the one that parents find hardest to explain because it is a choice, not a rule.

Compatible numbers against rounding and front-end estimation

Children may use any method unless the question names one; when it says "use compatible numbers", the answer must show numbers that divide or multiply cleanly, not just rounded ones.

Is the estimate high or low?

Say which way the estimate leans; the better worksheets ask.

• If you made a number bigger in an addition or multiplication, the estimate is high (25 × 48 = 1,200 is above 23 × 48).
• If you made the dividend bigger in a division, the estimate is high (48 ÷ 6 = 8 is above 47 ÷ 6). If you made the divisor bigger, the estimate is low (2,750 ÷ 50 = 55 is below 2,750 ÷ 42).
• If one number went up and another down (20 × 50 for 23 × 48), the direction is unclear, and the exact answer could be either side.

Twelve worked examples

1. 38 + 64: 40 + 60 = 100 (exact 102).
2. 153 + 249 + 97: 150 + 250 + 100 = 500 (exact 499).
3. 91 − 28: 90 − 30 = 60 (exact 63).
4. 4,012 − 1,988: 4,000 − 2,000 = 2,000 (exact 2,024).
5. 7 × 59: 7 × 60 = 420 (exact 413).
6. 32 × 28: 30 × 30 = 900 (exact 896).
7. 51 × 19: 50 × 20 = 1,000 (exact 969).
8. 65 ÷ 9: 63 ÷ 9 = 7 (exact 7.2).
9. 238 ÷ 4: 240 ÷ 4 = 60 (exact 59.5).
10. 5,230 ÷ 7: 4,900 ÷ 7 = 700 (exact 747; 5,600 ÷ 7 = 800 is the other choice, and the truth is between).
11. 1,856 ÷ 31: 1,800 ÷ 30 = 60 (exact 59.9).
12. A school buys 28 boxes of 36 pencils. About how many pencils? 30 × 35 or 28 × 35... choose 30 × 36 = 1,080, or 25 × 40 = 1,000 (exact 1,008).

The mistakes that cost marks

• Rounding both numbers in a division by the rounding rule and getting a harder problem (50 ÷ 6).
• Changing the divisor so much that the estimate is useless (47 ÷ 6 → 50 ÷ 10).
• Giving the exact answer when the question says estimate. The mark is for the method.
• Not writing the compatible numbers you chose. Show "48 ÷ 6 = 8", not just "8".
• Forgetting the decimal point after estimating with decimals (4.8 × 6.1 is about 30, not 300).

Where it sits in each curriculum

Practice (answers in the PDF)

Estimate each using compatible numbers, write the numbers you chose, and say whether your estimate is high or low.

1. 44 + 37 2. 683 − 297 3. 8 × 49 4. 42 × 19 5. 53 ÷ 9 6. 614 ÷ 7 7. 4,170 ÷ 6 8. 3,890 ÷ 48 9. 6.9 × 4.2 10. 49% of 61 11. A theatre has 38 rows of 52 seats. About how many seats? 12. 2,635 marbles are shared equally among 9 children. About how many each?

Download the sheet and the practice set (PDF)

Where to get help

A 1-on-1 online maths class in US after-school hours with a tutor who holds a master's in mathematics and teaches Grades 3 to 6 every week: estimation taught as the habit that checks every long calculation, with the times tables underneath it kept sharp. The Grade 5 maths page lists the units; the free 30-minute first class is a real lesson on this week's worksheet. Related pages: divisor, dividend, quotient and the divisibility rules and times tables tricks for the hard facts.

Questions parents ask

See how we teach this, 1-on-1 online →