Decimal ÷ Decimal

1Summary 重點摘要

0.8 ÷ 0.2 = 4. Shift to integers.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: Look at the divisorBefore computing 0.6 ÷ 0.2, look at what the divisor is. Here the divisor 0.2 is a decimal, and column division is awkward to run with it directly, so the first step is turning the divisor into a whole number. Why make it whole? Because the steps of whole-number division are the most familiar and the least error-prone, and turning a new problem into an old one is a classic strategy of mathematics. Once the divisor is whole, everything after follows the familiar whole-number method. A common error is pressing on without the conversion, and once the decimal points tangle, everything fails; before starting, check whether the divisor is whole, and handle it first if not.
2. Step 2: Multiply both by 10The divisor 0.2 has one decimal place, so multiply it by 10 to get 2; the dividend must be multiplied by the same number, 0.6 × 10 = 6. The expression becomes 6 ÷ 2. Why must both sides be multiplied by the same number? Because scaling only one side changes the quotient, and the expression is no longer the original; scaling both keeps the quotient unchanged, and indeed 0.6 ÷ 0.2 and 6 ÷ 2 both give 3. A common error is multiplying only the divisor, leaving 6 ÷ 0.2, a completely different question; remember that the decimal points move together, and by exactly the same number of places.
3. Step 3: Compute the quotientNow divide as whole numbers: 6 ÷ 2 = 3, and this 3 is the quotient of the original 0.6 ÷ 0.2. Because the converted expression holds no decimals, there is no need to place a decimal point in the quotient at all; write it down directly, far simpler than the original layout. Why does the quotient need no point? Because the invariance of the quotient guarantees it is identical before and after the change, and the whole purpose of the change is to carry the decimal-point trouble away. A common error is spoiling a good answer by adding a point, writing 0.3; check 0.2 × 3 = 0.6 and it is clear that 3 is correct.
4. Step 4: Why it worksWhen the dividend and the divisor are multiplied by the same nonzero number, the quotient does not change: 6 ÷ 2 = 60 ÷ 20 = 600 ÷ 200, all giving 3. This property, the invariance of the quotient, is the foundation that turns decimal division into whole-number division. Why does it hold? Think in terms of portions: 6 contains 3 portions of 2; scale every portion up tenfold and it becomes 60 containing 3 portions of 20, and the number of portions is untouched. With this understood, moving the decimal points feels secure. A common error is scaling the sides by different amounts, or attempting to divide by 0; the divisor can never be 0, and every transformation must avoid it.
5. Step 5: Two-decimal divisorsIf the divisor has two decimal places, as in 7.2 ÷ 0.04, multiply both numbers by 100: 0.04 × 100 = 4 and 7.2 × 100 = 720, so the expression becomes 720 ÷ 4 = 180. How far right the points move depends on the divisor: shift enough places to make it whole. Why look at the divisor rather than the dividend? Because the goal of the conversion is a whole divisor, and the dividend only needs to move the same number of places. A common error is moving the dividend by a different amount, such as turning 7.2 into just 72; the shifts must match exactly, and after moving, first confirm the divisor is whole.
6. Step 6: Check with the original numbersAfter dividing, verify by multiplication, and use the original decimals: 3 × 0.2 = 0.6, exactly the dividend; 180 × 0.04 = 7.2, also the dividend. Why check with the original numbers? Because only the question's own numbers test the scaling and the division together; verifying with the converted numbers alone would miss any slip made during the conversion. For instance, if 3 × 0.2 did not equal 0.6, the shifting would need to be examined again. A common error is checking with the wrong version of the numbers, or forgetting to check altogether; make multiply-back a habit and decimal division stays safe.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Decimal ÷ decimal: make the divisor whole first
• Multiply BOTH numbers by the same 10 or 100
• The quotient is unchanged: 0.6 ÷ 0.2 = 6 ÷ 2
• Shift as many places as the divisor has decimal places
• Verify by multiplying the original decimals back