When the sharing will not come out even, whatever is left over is the remainder
Fill the bags
There are 37 sweets, and they go into bags of 5. Fill one bag, then the next, and keep going until you run out of sweets.
Seven bags fill up, and they hold 7 × 5 = 35 sweets. That leaves 2 sweets. They go into an eighth bag, but 2 sweets cannot fill a bag of 5, so the eighth bag stays short by 3.
Seven full bags of 5, and a last bag with 2 sweets in it and 3 empty places.
The same thing as a division
Putting 37 sweets into bags of 5 is the division 37 ÷ 5. The number of full bags is the answer, and the sweets left over are the remainder: 37 ÷ 5 = 7 remainder 2, which is also written 7 r 2.
Work it out the way long division does. 5 goes into 37 seven times, because 7 × 5 = 35 and 8 × 5 = 40 is too many. Subtract to find what is left: 37 − 35 = 2.
Multiplying back checks it. Seven bags of 5 and 2 more make 7 × 5 + 2 = 35 + 2 = 37, which is every sweet.
7 × 5 = 35 is written under the 37, and 37 − 35 = 2 is the remainder.
Always smaller than the divisor
The 2 sweets are the remainder because they are too few to fill another bag. If 5 or more were left over, another bag of 5 would fill, and the division would not be finished.
So when you divide by 5, the remainder is always 0, 1, 2, 3 or 4. A remainder of 5 or more means the answer above it is too small: go back and fit in another 5.
Add items one at a time until the remainder resets
Add sweets one at a time. Each one goes into the short bag, and the remainder goes up: 2, then 3, then 4. The next sweet fills the bag, so there is one more full bag and the remainder goes back to 0. It never reaches 5.
Two slips
The remainder is not the number of bags. For 37 ÷ 5, the 7 counts the full bags and the 2 counts the sweets left over. When a question asks for the remainder, the answer is 2.
The remainder is not the number still needed to fill the next bag either. That is 3, the gap from 37 up to 40. The remainder is what is left over from 37 after 35, and 37 − 35 = 2.
Patterns and page numbers · A Repeating String of Beads: the 147th Bead, and the Blue Beads in the First 200
QuestionA string of decorative beads is arranged in a repeating pattern: 2 Red, 3 Blue, 1 Yellow, 2 Green, followed by 2 Red, 3 Blue, 1 Yellow, 2 Green, and so on. (a) What is the color of the 147th bead? (b) How many Blue beads are there in the first 200 beads?
1.One block is red, red, blue, blue, blue, yellow, green, green: 8 beads, 3 of them blue.
One block: 2 red, 3 blue, 1 yellow, 2 green. Eight beads, three of them blue. 2.(a) 147 ÷ 8 = 18 remainder 3: the first 18 × 8 = 144 beads are 18 whole blocks, and 3 beads are left over.
(a) 147 ÷ 8 = 18 blocks with 3 over. 3.Count those 3 beads into the next block: red, red, blue. The 147th bead is blue.
Count 3 into the next block: blue. Bead 147 is blue. 4.(b) 200 ÷ 8 = 25 with nothing left over: the first 200 beads are exactly 25 whole blocks.
(b) 200 beads is 200 ÷ 8 = 25 whole blocks. 5.Each block has 3 blue beads, so there are 25 × 3 = 75 blue beads.
Each block has 3 blue: 25 × 3 = 75.
Answer: (a) Blue; (b) 75 Blue beads
Watch out for
- Reading a remainder of 0 as the first bead of a block. A remainder of 0 means the bead ends a whole block: it is the 8th bead, green.
- Starting the blue beads at the 2nd place. Write the block out in order; the blue beads are its 3rd, 4th and 5th beads.
Your turn
1.41 ÷ 4: what is the remainder?