Lesson 22: Decimal Multiplication

§Text Passage

1Summary 重點摘要

Decimal multiplication: multiply as whole numbers, count total decimal places. e.g. 2.5 × 0.4 = 1.00 = 1.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Block 1: Decimal × Whole NumberTo multiply a decimal by a whole number, multiply as integers first, then place the decimal point. Example: for 2.5 × 4, work out 25 × 4 = 100; since 2.5 has 1 decimal place, the product needs 1 too, written 10.0, which simplifies to 10. Trailing zeros may be removed without changing the size. Another example: 3.06 × 5 — work out 306 × 5 = 1530; since 3.06 has 2 decimal places, the product is 15.30, simplified to 15.3, and removing the trailing zero changes nothing.
2. Block 2: Decimal × DecimalDecimal times decimal works the same way. Example: for 1.2 × 0.3, work out 12 × 3 = 36; 1.2 has 1 decimal place and 0.3 has 1, making 2 in total, so the product is 0.36. Count the places from the right of the product towards the left before placing the point. Another example: 0.5 × 0.8 — work out 5 × 8 = 40, count 2 decimal places, giving 0.40, that is 0.4; a meaning check: 0.8 of 0.5 is less than 0.5, so the result is sensible.
3. Block 3: Pad ZerosIf the whole-number product has fewer digits than the required decimal places, pad zeros on the left. Example: for 0.12 × 0.3, work out 12 × 3 = 36; the factors have 3 decimal places in total (2 and 1), but 36 has only 2 digits, so the answer is 0.036. Recheck the count after padding. Another example: 0.3 × 0.04 — work out 3 × 4 = 12, count 3 decimal places in total, and pad to 0.012; a check: 0.012 is smaller than 0.04, matching the expectation of shrinking.
4. Block 4: Mental MathA handy mental shortcut: multiplying by 0.1 is the same as dividing by 10, and multiplying by 0.01 is dividing by 100. Example: 500 × 0.1 = 50 and 500 × 0.01 = 5. The idea is point-shifting: times 0.1 moves the point one place left, times 0.01 two places. The reverse holds too: 0.07 × 1000 = 70, because multiplying by 1000 shifts the point three places right; pad zeros when digits run out — 7 becomes 700 and the result is 70.
5. Block 5: Multiply: Bigger or Smaller?Look at the multiplier to tell whether the result grows or shrinks: multiplying by a number greater than 1, such as ×2, makes it bigger; multiplying by a number less than 1, such as ×0.5, makes it smaller. Example: 10 × 0.5 = 5, only half of the original. Estimating this way checks the size of the answer. Another example: 24 × 1.5 — the multiplier is greater than 1, so the result should exceed 24; 24 × 1.5 = 36, exactly 24 plus its own half of 12, so estimate and calculation agree.
6. Block 6: Word Problem: ShoppingExample: apples cost $5.5 each — how much for 3? Work out 5.5 × 3: first 55 × 3 = 165, then move the point one place left to get 16.5, so the cost is $16.5. With money, check the decimal places at the end: one place means tenths of a dollar, two places means cents. For 3.5 kg, work out 5.5 × 3.5: first 55 × 35 = 1925, the factors carry 2 decimal places in total, giving $19.25; money answers are generally written to two decimal places.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Decimal multiply: as whole numbers first
• Answer dp = sum of both factors
• Pad zeros if not enough digits
• × 0.1 = ÷ 10
• × < 1 → number gets smaller