Lesson 23: Decimal Division

§Text Passage

1Summary 重點摘要

Decimal division: divisor has decimal, multiply both by 10100 to make divisor whole. e.g. 1.5 ÷ 0.3 = 15 ÷ 3 = 5.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Block 1: Decimal ÷ Whole NumberA decimal divided by a whole number is the basic case; the point in the quotient lines up with the point in the dividend. Example: 8.4 ÷ 4 = 2.1 — divide the whole part first (8 ÷ 4 = 2), then the decimal part. Dropping the point to write 21 is completely wrong, so check the point afterwards. Another example: 9.6 ÷ 6 — the whole part gives 9 ÷ 6 = 1 remainder 3; the leftover 3 joins the 6 tenths to make 36 tenths, and 36 ÷ 6 = 6, so the quotient is 1.6; check 1.6 × 6 = 9.6.
2. Block 2: Whole ÷ DecimalWhen the divisor is a decimal, multiply divisor and dividend by 10 or 100 to make the divisor whole. Example: for 6 ÷ 0.5, the divisor 0.5 becomes 5 and the dividend 6 becomes 60, so 6 ÷ 0.5 = 60 ÷ 5 = 12, meaning twelve halves fit into 6. When the divisor is a two-place decimal such as 0.25, multiply both by 100: 5 ÷ 0.25 = 500 ÷ 25 = 20, meaning exactly twenty 0.25s fit into 5 — verify with 0.25 × 20 = 5.
3. Block 3: Decimal ÷ DecimalHowever many decimal places the divisor has, multiply both numbers by that many tens. Example: for 5.2 ÷ 0.13, the divisor 0.13 has 2 decimal places, so multiply both by 100, giving 520 ÷ 13 = 40. The question becomes a whole-number division, and the quotient is unchanged by this move. Another example: 7.2 ÷ 0.008 — the divisor has 3 decimal places, so multiply both by 1000, giving 7200 ÷ 8 = 900; the smaller the divisor, the larger the quotient, so estimate the order of size first.
4. Block 4: Not Exact, Pad ZerosIf the division does not finish, pad zeros after the decimal point and keep going. Example: for 1 ÷ 4, one whole cannot give 4 tenths a first digit, so regroup as 10 tenths: that gives 0.2 with 0.2 left; pad another zero and continue, ending with 1 ÷ 4 = 0.25. In the same way, 1 ÷ 8 = 0.125. Padding zeros never changes the dividend's size. For example, 7 ÷ 2 = 3.5: after the whole part is divided, place the decimal point in the quotient first, then pad zeros and continue; each padded zero regroups what remains into smaller units.
5. Block 5: Divide: Bigger or Smaller?Look at the divisor to tell whether the result shrinks or grows: dividing by a number greater than 1, such as ÷2, makes it smaller; dividing by a number less than 1, such as ÷0.5, makes it bigger. Example: 10 ÷ 0.5 = 20, twice the original. This pattern quickly checks whether an answer is sensible. Another example: 18 ÷ 1.2 — the divisor is greater than 1, so the result should be less than 18; 180 ÷ 12 = 15 is indeed smaller. Estimate first and mistakes become easy to notice.
6. Block 6: Word Problem: SharingExample: 3 classmates share $75.6 equally — how much each? Work out 75.6 ÷ 3: the whole part gives 75 ÷ 3 = 25, and the decimal part gives 0.6 ÷ 3 = 0.2, so each gets $25.2. Equal-sharing questions use division, and the answer should state each person's share clearly. Check with multiplication: 25.2 × 3 = 75.6, matching the total; after any sharing question, multiply one person's share back by the headcount — the quickest check there is.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Divisor must be whole number
• Divisor has decimal: multiply both by 10100
• Not exact: pad zeros
• ÷ < 1 → number gets bigger
• Quotient decimal point aligns with dividend