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This page sets out the method for estimating in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Estimating topic page.
When you see 'estimate', 'approximately' or 'roughly', do not use exact values — round first.
When the question asks whether your estimate is too big or too small, look at which way each number was rounded.
Keep the first non-zero digit and turn the rest into zeros.
382 is about 400. 0.047 is about 0.05.
Estimate 19 × 21 as 20 × 20 = 400.
The estimate checks the size of an exact answer. It is not the exact answer.
Estimate 612 ÷ 28 as 600 ÷ 30 = 20.
Choose numbers that divide cleanly.
Estimate a square root by finding a nearby square.
√50 is a bit more than 7, because 7² = 49.
Rounding 19 to 10 makes a worse estimate than rounding to 20.
One significant figure is the usual request.
Write the rounded calculation, then the result.
An estimate with no rounded line does not show the method.
Start at the first digit that is not zero. That digit stays, and the next digit decides. 5 or more rounds up. 4 or less leaves it. 41.7 has first digit 4 and next digit 1, so it rounds down to 40. 0.48 has first non-zero digit 4 and next digit 8, so it rounds up to 0.5. 1.9 rounds up to 2.
Zeros after the first digit are placeholders, not extra significant figures. 41.7 is not 42 in this job. 42 is 41.7 rounded to the nearest whole number, which is a different task. One significant figure for a number in the forties is 40 or 50. Leading zeros in 0.048 are not significant, so 0.048 to 1 significant figure is 0.05.
Round every number that is being calculated with, including numbers in a denominator. Do this before any multiplying or dividing. Rounding the exact answer at the end is a different task. An estimate that still contains 41.7 has not been rounded yet.
Method: 1. Find the first non-zero digit of each number. 2. Use the next digit to decide whether that first digit rounds up. 3. Replace every number with its 1 significant figure value.
Replace each number, write the new calculation, then work it out. For (41.7 × 0.48)/1.9 the rounded line is (40 × 0.5)/2. First 40 × 0.5 = 20, then 20 ÷ 2 = 10. The estimate is 10. Show that line. A bare 10 does not show that each number was rounded.
Choose the order that stays easy. Multiplying by 0.5 halves, and dividing by 2 halves again, so 40 goes to 20 and then to 10. The exact value of the original calculation is a bit more than 10, and that is close enough for an estimate. You are not trying to recover the exact product.
Division needs a denominator you can divide by. 1.9 became 2, which divides 20 exactly. If a rounded denominator does not divide cleanly, check the significant-figure rounding again before you invent a different number. One significant figure is the usual request. Rounding 19 to 10 is too rough when 19 is next to 20.
Method: 1. Write the calculation again using only the rounded numbers. 2. Work out that simpler calculation. 3. The result of the rounded line is the estimate.
Compare each rounded number with the original. 78 rounds up to 80, and 0.46 rounds up to 0.5. Both factors got bigger, so 80 × 0.5 = 40 is larger than 78 × 0.46. The estimate 40 is an overestimate. The true product is 35.88, which sits under 40 and confirms the direction.
If the changes pull in opposite directions, do not claim a direction. In (41.7 × 0.48)/1.9, 41.7 went down, 0.48 went up, and 1.9 went up. A larger divisor makes a quotient smaller. Those effects compete, so 10 is the estimate and that is all you should say about its side of the true value.
Dividing by a smaller number makes the result bigger. If you round a denominator down, that part of the estimate is high, even if the numerator was rounded down as well. Track one change at a time. Only call the whole estimate an overestimate or an underestimate when every change agrees.
Method: 1. Compare each rounded number with the original number. 2. If every factor was rounded up, the estimate is over the true value. 3. Say over or under only when the rounding all pushes the same way.
Estimate a square root by a square you know. 7^2 = 49 and 6^2 = 36. The number 48 is just under 49, so √(48) is a little under 7. Using 7 in a product rounds that root up. It is not sensible to say √(48) is about 24, which is half of 48, or about 49, which is the square itself.
In an estimate, square only a rounded value so the square is exact. 3.1 to 1 significant figure is 3, and 3^2 = 9. That is the estimate of 3.1^2. The exact square is a different question: 3.1^2 = 9.61. Do not build an estimate out of a long decimal square. Round the base first, then square.
A product of two overestimates is an overestimate. √(48) × 19.6 uses 7 instead of a slightly smaller root, and 19.6 rounds up to 20. Then 7 × 20 = 140, and 140 is above the true product. If one factor had been rounded down, you would still give the estimate, but you would not claim a direction.
Method: 1. Find two square numbers that the value sits between. 2. The root sits between the roots of those squares. 3. Use the nearer square if you need to say whether the estimate is over or under.
An estimate question always tells you how to round, or leaves you to choose. "By rounding each number to the nearest 10" means 47 becomes 50 and 22 becomes 20, so 47 × 22 is estimated as 50 × 20 = 1000. "Round each number to 1 significant figure" means keep one non-zero digit, so 0.0618 becomes 0.06 and 5.97 becomes 6.
Follow the instruction exactly, because the method mark depends on it. Rounding to the nearest whole number would turn 0.48 into 0 and ruin a division. Write each rounded number down on its own line before you calculate with them. Examiners then see that the method is right even if the arithmetic slips, and you can see the question has become something you can do by hand.
Method: 1. Read the rounding instruction and underline it. 2. Round every number in the calculation, writing each rounded value. 3. Write the new calculation using only the rounded values. 4. Work it out by hand and state the estimate, using ≈ for estimates.
Tip: Never round a number to 0 when it will be divided by. Use 1 significant figure, so 0.49 becomes 0.5.
After rounding, a division by a decimal such as 0.5 or 0.2 is common. Dividing by 0.5 doubles the number, because there are two halves in every whole. Dividing by 0.2 multiplies by 5, and dividing by 0.1 multiplies by 10. So (8 × 30)/0.5 is 240/0.5 = 480.
If the working feels awkward, multiply the top and bottom of the fraction by 10 so the bottom becomes a whole number. 240/0.5 becomes 2400/5 = 480. Do the multiplications in the numerator first and keep the denominator separate. Then check the size: dividing by a number less than 1 must make the answer bigger, never smaller.
Method: 1. Round every number to 1 significant figure. 2. Multiply the rounded numbers on the top of the fraction. 3. Divide by the rounded bottom number, using doubling for 0.5 if it helps. 4. Check that dividing by less than 1 made the answer bigger.
"By rounding each number, show that his answer is incorrect" is a favourite. Someone has put numbers into a calculator and written down a wrong answer, often because of a slipped decimal point. You are not asked for the exact answer. You estimate, and show the estimate is far from the stated answer.
A full-mark answer has three parts: the rounded values, the estimate, and a sentence that compares. For example: "(38.4 + 61.7) ÷ 9.9 ≈ (40 + 60) ÷ 10 = 10, so 101.1 is too big." The comparison sentence is what earns the final mark. An estimate of 10 and a claimed answer of 101.1 differ by a factor of ten, which usually means the decimal point is in the wrong place.
Method: 1. Round each number as the question instructs and write the rounded calculation. 2. Work out the estimate. 3. Compare the estimate with the claimed answer. 4. Write a sentence saying the claimed answer is too big or too small, with the numbers.
Often the question adds "is your estimate bigger or smaller than the exact value?" Look at what each rounding did. Rounding a number that is multiplied up makes the estimate bigger, and rounding it down makes it smaller. A number on the bottom of a fraction does the opposite: rounding it down makes the estimate bigger.
The reason has to refer to the rounding. "Both numbers were rounded up so the estimate is bigger than the exact cost" earns the mark. If some numbers were rounded up and others down, you often cannot tell without working out the exact value, and you should say so. Contexts to expect are costs, journeys, areas and volumes with π replaced by 3, which is smaller than π.
Method: 1. List each number and write whether rounding made it bigger or smaller. 2. Decide whether each number is multiplied (on the top) or divided by (on the bottom). 3. Combine the effects to decide if the estimate is bigger or smaller. 4. Write the reason in a sentence that mentions the rounding.
Tip: Using π = 3 instead of 3.14 makes a circle estimate smaller than the exact value.
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