Multiplying and Dividing Decimals by 10, 100, 1000

Multiplying by ten makes every digit worth ten times as much, which means every digit moves one column to the left. Thirty-four point two: the three tens become three hundreds, the four ones become four tens, the two tenths become two ones. Three hundred and forty-two. It looks as if the decimal point jumped one place to the right; what actually happened is that every digit slid one place to the left, and the point stayed put. Seeing it that way is what stops the mistakes when zeros are involved.

Dividing by ten is the reverse: every digit moves one column to the right. Three hundred and forty-two divided by ten: the three hundreds become three tens, the four tens become four ones, the two ones become two tenths. Thirty-four point two. So when a question says Kim divided three hundred and forty-two by a number and got thirty-four point two, look at how far the digits moved — one column — and the number was ten.

A hundred is ten times ten, so multiplying by a hundred moves the digits two columns; a thousand, three. Count the zeros in the multiplier. Zero point seven five times a hundred: seventy-five. Two point five times a hundred: the two lands in the hundreds, the five in the tens, and the empty ones column gets a zero — two hundred and fifty. Times a thousand, two thousand five hundred, with two zeros filling in. Estimate the size to check: two point five times a hundred is about two hundred, so two hundred and fifty is right.

Dividing by a hundred or a thousand moves the digits right, and now the zeros fill in after the point. Five divided by a hundred: the five goes from the ones to the hundredths, with a zero in the tenths and a zero in the ones — zero point zero five. Six divided by a thousand is zero point zero zero six. Three point six divided by a hundred: the three goes to the hundredths, the six to the thousandths, zero point zero three six. And reversed: a number times a hundred is four thousand and seventy, so the number is four thousand and seventy divided by a hundred, forty point seven.

Every unit conversion in the metric system is one of these moves. Kilometres to metres is times a thousand; metres to centimetres, times a hundred; centimetres to millimetres, times ten; and the reverse directions divide. Two point three five kilometres is two thousand three hundred and fifty metres. Fifteen hundred centimetres is fifteen metres. Dollars to cents is times a hundred, kilograms to grams is times a thousand. The check is the direction: a bigger unit means a smaller number, so if the number grew when you went to a bigger unit, you moved the digits the wrong way. Quiz and exercises now.