Multiplying and Dividing Decimals

Multiplying decimals splits into two easy jobs. First multiply the digits and forget the decimal points completely. Then count how many decimal places the question contained, and give the answer exactly that many. Take zero point six times zero point four. Six times four is twenty-four. Zero point six has one decimal place and zero point four has one, so the answer needs two decimal places, giving zero point two four. The hundred square proves it: six columns by four rows is twenty-four little squares out of a hundred, which is twenty-four hundredths. Count the places carefully, because writing two point four instead of zero point two four is the single most common error in this topic.

The same two jobs handle bigger numbers. Three point two times one point five. Drop the points and work out thirty-two times fifteen. Thirty-two times ten is three hundred and twenty, thirty-two times five is one hundred and sixty, and those add to four hundred and eighty. Now count the places: one in three point two, one in one point five, so two places altogether. Four hundred and eighty becomes four point eight zero, which is just four point eight. Always estimate before you finish. Three point two is roughly three and one point five is roughly one and a half, so you expect something near four and a half. Four point eight is close, so the point is in the right place.

Multiplying or dividing by ten, a hundred or a thousand needs no working at all. The digits simply slide along the place-value columns while the decimal point stays where it is. Two point three five times ten sends every digit one column to the left, giving twenty-three point five. Times a hundred sends them two columns, giving two hundred and thirty-five. Dividing sends the digits the other way: forty-seven point two divided by ten is four point seven two, and eight point four divided by a hundred is nought point nought eight four. Count the zeros to know how far to move, and fill any empty column with a zero. Before you write anything, ask whether the answer should be bigger or smaller.

Dividing a decimal by an ordinary whole number is short division with one extra habit. Before you divide anything, put the answer's decimal point directly above the point in the number you are dividing. Now divide as usual and the point looks after itself. For seven point two divided by four: four into seven goes once with three left over, you carry that three across to make thirty-two tenths, and four into thirty-two goes eight. So the answer is one point eight. Always check by multiplying back — one point eight times four is seven point two, so it is right. That check takes ten seconds and catches almost every slip you can make.

Nobody wants to divide by a decimal, so change the question into one you already know. Take four point eight divided by nought point six. Multiply both numbers by ten and it becomes forty-eight divided by six, which is eight. That is allowed because a division is really a fraction, and scaling top and bottom by the same amount never changes the value, exactly as one half equals five tenths. So the rule is: move the point in the divider until it is a whole number, then move the other number by the same number of places. Nine point six divided by nought point four becomes ninety-six divided by four, which is twenty-four.