Large numbers and useful estimates

A digit’s place decides its value

40 709 is forty thousand seven hundred and nine:

The zero in the thousands place and the zero in the tens place preserve those empty places. 47 009 has seven thousands instead, so it is greater than 40 709.

You can also build a numeral from exchanged units. Six ten-thousands and fourteen thousands make 60 000 + 14 000 = 74 000. Ten of those thousands form another ten-thousand; the amount stays the same when its representation changes.

Check your understanding 1 Keep the empty places when reading words

Write ninety thousand and sixty as a numeral. Then write 90 600 in words. Explain which place contains six in each number.

Your numeral, number words and place-value explanation, or work on paper

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The first number is 90 060, with six tens. The second is ninety thousand six hundred, with six hundreds. The empty places keep the six in its intended position; the two numbers are different amounts.

Compare 62 480, 62 408 and 62 840 from left to right. The ten thousands and thousands match. Eight hundreds makes 62 840 the greatest. Between the others, eight tens is greater than zero tens. Increasing order is 62 408, 62 480, 62 840.

For 28 500, 29 500, 30 500, the rule adds 1000. The next number is 31 500. Check the same step across the boundary instead of copying a pattern of written digits.

Rounding chooses a nearby multiple

To round 36 748 to the nearest 100, locate the multiples 36 700 and 36 800. The midpoint is 36 750. Our number is below that midpoint, so it is nearer 36 700:

The symbol ≈ means approximately equal, not exactly equal. The rounded number is an estimate; the original has not changed.

To the nearest 10, the same number is 36 750. To the nearest 1000, it is 37 000. Always state the place. At a halfway point, use the usual school convention of rounding these positive numbers to the higher multiple: 36 750 rounds to 36 800 to the nearest 100.

If a given number is already a multiple of the requested unit, rounding leaves it unchanged: 36 700 to the nearest hundred is still 36 700. In that case 36 700 = 36 700 is an exact equality. In contrast, 36 748 = 36 700 is false: rounding has changed the value, so connect these numbers with ≈.

Check your understanding 2 Keep exact and approximate statements distinct

Round 42 681 to the nearest ten, hundred and thousand. Then write a statement connecting 42 681 with its value rounded to the nearest hundred. Use = or ≈ as appropriate. If your keyboard has no ≈, you can type ~. Would the same relation be needed for 42 700 rounded to the nearest hundred?

Your three rounded values, statements and explanation, or work on paper

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The rounded values are 42 680, 42 700 and 43 000, respectively. 42 681 ≈ 42 700 to the nearest hundred. The two numbers are different, so an exact equals sign would make a false statement. Reversing the approximate statement, 42 700 ≈ 42 681, still connects the same two quantities.

42 700 is already a multiple of one hundred, so rounding leaves it unchanged: 42 700 = 42 700 is exact. The requested place and the original number together decide whether a value changes.

Check your understanding 3 Locate the neighbouring multiples

Round 58 460 to the nearest 1000.

Rounded number

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58 000. Rounding to the nearest hundred instead would give 58 500, which answers a different question.

Check your understanding 4 An estimate is not an exact count

A stadium attendance is reported as 24 000 to the nearest thousand. Could the exact attendance have been 23 700? Could it have been 24 700? Explain.

Your answer or drawing on paper

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23 700 rounds to 24 000, so it is possible. 24 700 rounds to 25 000, so it is not consistent with this report. A rounded report usually corresponds to many exact numbers, not just the printed multiple.

Use the estimate for a purpose

For an exact subtraction, keep the original numbers in your working. Use rounded numbers separately to judge whether the result has a sensible size. Do not replace exact information with an estimate unless the question asks you to.