Division with a remainder: round up or down

A baker has one thousand two hundred and fifty cookies. She packs them into boxes of 8. What is the maximum number of boxes of 8 cookies she can pack? Try it first. Pause if you like. Divide one thousand two hundred and fifty by 8. Work from left to right. 8 times 1 is 8, so it fits. 8 times 2 is 16, too big for 12. So 1 is the biggest that fits. Write 1 in the hundreds place. 1 times 8 is 8. 12 minus 8 leaves 4 hundreds. Rename the 4 hundreds as 40 tens. Bring down the 5 tens. Now we have 45 tens. 8 times 5 is 40, so it fits. 8 times 6 is 48, too big for 45. So 5 is the biggest that fits. Write 5 in the tens place. 5 times 8 is 40. 45 minus 40 leaves 5 tens. Rename the 5 tens as 50 ones. Bring down the 0 ones. Now we have 50 ones. 8 times 6 is 48, so it fits. 8 times 7 is 56, too big for 50. So 6 is the biggest that fits. Write 6 in the ones place. 6 times 8 is 48. 50 minus 48 leaves 2. One thousand two hundred and fifty divided by 8 is one hundred and fifty-six, remainder 2. Only the end of the run is shown here. There are one hundred and fifty-six full boxes, with 2 cookies left over. Can those 2 cookies fill another box of 8? The 2 cookies stay outside the full boxes. They do not make another full box. The maximum number of full boxes is one hundred and fifty-six. We round down. For full boxes, leave the remainder out.

A school orders one thousand and seventy exercise books. The books are delivered in cartons. Each carton can hold up to 9 books. What is the minimum number of cartons needed to deliver all the books? Try it first. Pause if you like. Divide one thousand and seventy by 9. Again, work from left to right. 9 times 1 is 9, so it fits. 9 times 2 is 18, too big for 10. So 1 is the biggest that fits. Write 1 in the hundreds place. 1 times 9 is 9. 10 minus 9 leaves 1 hundred. Rename the 1 hundred as 10 tens. Bring down the 7 tens. Now we have 17 tens. 9 times 1 is 9, so it fits. 9 times 2 is 18, too big for 17. So 1 is the biggest that fits. Write 1 in the tens place. 1 times 9 is 9. 17 minus 9 leaves 8 tens. Rename the 8 tens as 80 ones. Bring down the 0 ones. Now we have 80 ones. 9 times 8 is 72, so it fits. 9 times 9 is 81, too big for 80. So 8 is the biggest that fits. Write 8 in the ones place. 8 times 9 is 72. 80 minus 72 leaves 8. One thousand and seventy divided by 9 is one hundred and eighteen, remainder 8. Only the end of the run is shown. There are one hundred and eighteen full cartons, and 8 books left over. Must those 8 books be delivered too? One hundred and eighteen plus 1 is one hundred and nineteen. Those 8 books need one more carton. The minimum number of cartons needed is one hundred and nineteen. Full groups: leave the remainder out. Holding everything: the remainder needs one more group.