Why can dividing by a decimal be rewritten using whole numbers?

The numbers do change, but they change by the same factor, so their ratio—the division result—does not. Think of division as asking how many groups of the divisor fit into the dividend. If you scale both amounts equally, the number of groups stays the same.

For instance, 4.2 ÷ 0.6 asks how many groups of 0.6 fit into 4.2. If you measure both amounts in tenths instead of ones, 4.2 becomes 42 tenths and 0.6 becomes 6 tenths. The question is now 42 ÷ 6, which is 7. The unit changed from whole units to tenths, but the number of groups did not. More formally, multiplying numerator and denominator by the same nonzero number leaves a fraction’s value unchanged. [S1]

Use the divisor to choose the multiplier: count how many places its decimal point must move right to make it a whole number. Multiply both the dividend and divisor by the matching power of ten, then divide.

Worked example: 4.2 ÷ 0.6

1. The divisor, 0.6, has one decimal place. Moving its point one place right gives 6, so use 10.
2. Multiply both numbers by 10: 4.2 × 10 = 42 and 0.6 × 10 = 6.
3. Rewrite the division: 4.2 ÷ 0.6 = 42 ÷ 6.
4. Divide: 42 ÷ 6 = 7.

So 4.2 ÷ 0.6 = 7. This is a change to the written calculation, not a change to its value. The method is to make the divisor a whole number and move the dividend’s decimal point the same number of places. [S1]

Add zeros as placeholders when needed, and apply the same scaling to both numbers. For example, in 1.2 ÷ 0.04, the divisor needs two places to become a whole number. Multiply both numbers by 100: 1.2 becomes 120, and 0.04 becomes 4. Then 120 ÷ 4 = 30.

The zero in 1.2 → 120 is necessary because 1.2 × 100 is 120. You can check the transformation by multiplying each original number by 100; they must both use the same multiplier. This method makes the divisor a whole number while preserving the quotient. [S1]

Check that the decimal point moved the same number of places in both numbers, and that both were multiplied by the same power of ten. Then check the result by reversing the division: quotient × divisor should equal the original dividend, allowing for any rounding if the answer is not exact.

For 4.2 ÷ 0.6 = 7, the reverse check is 7 × 0.6 = 4.2. A common mistake is changing 4.2 ÷ 0.6 into 42 ÷ 0.6: only one number was scaled, so that is a different ratio. Keep the original expression nearby until the transformed one is complete.