Multiplying and Dividing Decimals

Look at the place values in that equation. A decimal place is a digit position to the right of the decimal point. The factor $0.3$ has one decimal place, so it equals $3$ divided by $10$. The factor $0.12$ has two decimal places, so it equals $12$ divided by $100$.

The numerators (top numbers) multiply to give the whole-number product, $3 \times 12 = 36$. The denominators (bottom numbers) multiply to give $10 \times 100 = 1000$, and thousandths take three decimal places. So the decimal places of the two factors add: $1 + 2 = 3$. Each decimal place in a factor is one division by $10$, and the product carries all of them.

That gives you a method. Multiply the factors as if they were whole numbers. Then place the decimal point in that whole-number product so that it has as many decimal places as the two factors have together. Keep every digit while you count, including any ending zeros. If there are too few digits, put zeros in front, as in $0.036$.

Now turn to division. In $1.2 \div 0.3$, the dividend $1.2$ is the number you are dividing, the divisor $0.3$ is the number you divide by, and the quotient is the result. A divisor can be any number except zero. For example, $1.2 \div 0$ has no answer, because no number times $0$ equals $1.2$.

The division $1.2 \div 0.3$ asks how many groups of $0.3$ fit into $1.2$. Suppose you pour $1.2$ liters of juice into cups that each hold $0.3$ liters. Name both amounts in tenths of a liter: $1.2$ liters is $12$ tenths and $0.3$ liters is $3$ tenths. Asking how many groups of $3$ tenths fit into $12$ tenths is the same as asking how many groups of $3$ fit into $12$. So the quotient is $4$, and you fill $4$ cups.

Notice that the quotient $4$ is larger than the dividend $1.2$. That happens whenever you divide a positive number by a positive divisor less than $1$, because more than one group fits into each whole.

Naming both numbers in tenths did the same job as multiplying both by $10$, because $1.2 \times 10 = 12$ and $0.3 \times 10 = 3$. Here is why the quotient did not change. The quotient $4$ is the number that makes $4 \times 0.3 = 1.2$ true. Multiply both sides by $10$, which keeps the equation true, and group the $10$ with the $0.3$. You get $4 \times (0.3 \times 10) = 1.2 \times 10$, which is $4 \times 3 = 12$. So $4$ is also the answer to $12 \div 3$.

This works whenever the divisor is not zero: multiply the dividend and the divisor by the same nonzero number (any number except zero), and the quotient stays the same.

To divide by a nonzero decimal, use a power of ten ($10$, $100$, $1000$ and so on) that turns the divisor into a whole number, because you already know how to divide by a whole number. Use $10$ when the divisor has one decimal place, $100$ for two, and $1000$ for three.

Only the divisor has to become a whole number. The table that follows shows the pattern. In its last row the dividend stays a decimal: $1.2 \div 3$ is $12$ tenths divided by $3$, which is $4$ tenths, or $0.4$.

The takeaway: you multiply and divide decimals with whole-number arithmetic, and place value tells you where the decimal point goes. To multiply, give the whole-number product as many decimal places as the two factors have together, counting before you drop any ending zeros. To divide by a decimal that is not zero, multiply the dividend and the divisor by the same power of ten so that the divisor becomes a whole number. The quotient does not change.

Next, you will estimate sums, differences, products and quotients of decimals, and you will use all four operations to solve word problems.