Division (II): Remainder

1Summary 重點摘要

If not equal, there is remainder. 7 / 3 = 2 R1.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: Try sharing onceShare 7 sweets equally among 3 children: give 1 sweet to each child first. That round uses 3 sweets and leaves 7 − 3 = 4. Because 4 is still more than 3 children, the sharing can continue. Equal sharing means trying round after round until what is left is too small to give everyone one more. A common mistake is to stop after one round and take 1 sweet each as the answer. The test for continuing is simple: compare what is left with the number of people; 4 sweets are more than 3 people, so share again, until what remains is less than one sweet per person.
2. Step 2: Keep sharingGive 1 more sweet to each child. This round uses another 3 sweets, 6 in all, and only 7 − 6 = 1 sweet is left. One sweet is not enough for another round among 3 children, so the sharing must stop here; the shares cannot be forced equal any further. The 1 sweet left over is the remainder; it belongs to no one in particular and simply stays behind. A common mistake is to give the last sweet to someone, spoiling the equality so that one child has 3 and the others 2, or to throw the remainder away and ignore it. Write the remainder truthfully in the answer; it is part of the result.
3. Step 3: Write the divisionThe result is written as 7 ÷ 3 = 2 remainder 1. The quotient 2 says each child gets 2 sweets, and the remainder 1 says 1 sweet is left over. Both the quotient and the remainder must be written down; an answer with only the quotient is incomplete. A common mistake is to write only 7 ÷ 3 = 2 and drop the 'remainder 1', or to swap the roles and write 7 ÷ 3 = 1 remainder 2. To check: quotient 2 times divisor 3 is 6, plus remainder 1 gives exactly 7; if the swapped version cannot rebuild the original number, it is wrong.
4. Step 4: Remainder < divisorThe remainder must be smaller than the divisor. In 7 ÷ 3 the remainder is 1, and 1 is less than 3, so it is correct. If the remainder were 3 or more, what is left would still allow each person one more sweet, which means the quotient is too small and the sharing must be redone. For example, writing 13 ÷ 4 as 2 remainder 5 must be wrong because the remainder 5 exceeds the divisor 4; the correct answer is 3 remainder 1. A common mistake is a quotient chosen too small, leaving a remainder bigger than the divisor. After each answer, compare the remainder with the divisor before writing the final sentence; this small check blocks most errors.
5. Step 5: Check with multiplicationCheck the answer with multiplication: multiply the quotient by the divisor and add the remainder; the result should equal the dividend. For 7 ÷ 3 = 2 remainder 1, check with 2 × 3 = 6 and 6 + 1 = 7, exactly the original 7 sweets, so the quotient and the remainder are both right. A common mistake is to forget the remainder in the check, comparing only 2 × 3 = 6 with 7 and concluding wrongly that the answer is bad; the gap of 1 between 6 and 7 is exactly the leftover sweet. Learn the full formula: quotient × divisor + remainder = dividend; none of the four parts may be omitted.
6. Step 6: Use it in daily lifeRemainders are useful in daily life. For example, with 13 biscuits packed 4 to a bag, 13 ÷ 4 = 3 remainder 1, so 3 full bags can be packed with 1 biscuit left. Before answering, decide whether the question asks for the quotient or the remainder: 'how many full bags' needs the quotient 3; 'how many are left' needs the remainder 1; 'at least how many more biscuits fill one more bag' needs 4 − 1 = 3. A common mistake is to ignore what the question asks and put both quotient and remainder into every answer. A word problem must answer its own question: read the question sentence first, then decide whether to use the quotient, the remainder, or both together.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Leftovers become the remainder.
• Write it: 7 ÷ 3 = 2 remainder 1.
• Remainder is always less than the divisor.
• Check: quotient x divisor + remainder = total.
• Use times tables to find the best quotient.