Math CentralQuandaries & Queries


Question from deepak, a parent:

find the smallest and the greatest numbers which when rounded off to the nearest thousand, become 6000.

Hi Deepak.

When we "round numbers off", we use the principle that a 5 or more causes the number to "round up" and 4 or less causes it to "round down". Here are some examples:

38.721, round to the nearest whole number: The number to the right of the whole number is a seven. Since 7 >= 5, we round up. So this becomes 38+1 = 39.

803, round to the nearest 10. The number to the right of the tens column is the ones column and that is a 3. Since 3 < 5, we round down, so we just change everything to the right into zeros: answer is 800.

80500, round to the nearest 1000. The number to the right of the rounding digit is a 5. Since 5 >= 5, we round up. So 80000 is the answer.

Let's see what the smallest and greatest numbers are which when rounded to the nearest thousand become 80000. We know that the number we examine is the digit to the right of the thousands digit, and that is the digit in the hundreds place. To round down, this digit must be a four or less. What is the largest number that starts out 804(something)(something)? It is the number where all the rest of the digits are 9, so the greatest number which rounds to 80000 is 80499. The smallest number is the one that rounds up to 80000. So that means a 5 in the hundreds column. If it rounds up, then we would be adding 1 to the thousands place. That means the smallest number that rounds up to 80000 is the smallest number that starts out with a 79, then a 5, then two more digits: 795(something)(something). So this time
since we are looking for the smallest number, the somethings must be zeros. That means the smallest number rounding up to 80000 in the thousands column is 79500.

Use these examples and explanation to find your answer for your problem.

Hope this helps,
Stephen La Rocque.

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