When calculating 0.3 × 0.4, you learn to say "3×4=12, and since there are two digits after the decimal point, that's 0.12." But why does the decimal point land exactly there? The answer lies in fractions. 0.3 is 3/10 and 0.4 is 4/10, so 0.3 × 0.4 is really 3/10 × 4/10 = 12/100, which is 0.12. The rule of "adding up the number of decimal places" is really just the fraction multiplication rule hiding underneath, with denominators growing to 10, 100, 1000…
Draw a square with 10 columns and 10 rows, 100 squares in total. Shade 3 columns (=0.3) and 4 rows (=0.4), and the overlapping region is 3×4=12 squares. Since the whole grid is 100 squares, the overlap is 12/100, which is 0.12. When you multiply two decimals, the number of decimal places in the result equals the sum of the decimal places in the two numbers being multiplied (a tenths place × a tenths place gives a hundredths place — that is, 1+1=2 places).
Decimal division gets especially confusing when the divisor is a decimal. Looking at 1.2 ÷ 0.3 directly doesn't give you much of a feel for it, but multiply both numbers by 10, and it becomes 12 ÷ 3 = 4. Why does the answer stay the same? Because division is a "ratio." Scale both numbers up or down by the same factor, and that ratio (the quotient) never changes — just like enlarging or shrinking a photo keeps its width-to-height ratio the same. So whenever the divisor is a decimal, it's much easier to first multiply both numbers by 10 (for one decimal place) or 100 (for two decimal places) to turn the divisor into a whole number, and then calculate.
Multiplying and dividing decimals comes up all the time in real life: "If a 1.5 L carton of milk is split into 0.3 L cups, how many cups is that?" is division, while "How far do you walk in 0.6 hours at 4.5 km/h?" is multiplication. On our activity page, you can use a slider to see decimal multiplication play out on a 100-square grid, and watch division turn into whole-number division with a bar model.