Dividing when the divisor has a decimal

0:00A journey of one hundred and twenty-six kilometres, finished in two and a half hours. What was the average speed? It is a division. Distance divided by time: one hundred and twenty-six divided by two point five. Nothing strange about it, and you can probably guess the answer is near fifty. But look hard at that divisor first. Every division so far has shared a number out among a whole number of parts.

0:32This one asks us to share among two and a half. And that is a real obstacle, not just an awkward number. Long division works by regrouping. A hundred that will not share becomes ten tens, and every step hands out a whole number of something to each part. So try to start. Take one hundred, and share it among two and a half equal parts. There is an answer. It is forty.

1:01But that is not what the procedure records. It counts the parts out — one, two, three — and there is no two-and-a-halfth part to hand anything to. The method does not stall halfway through. It cannot begin at all. So the fix cannot be an extra step at the end. The obstacle has to go first. Here is the move. Write the divisor as a fraction. Two point five is twenty-five tenths. Twenty-five over ten.

1:34And dividing by a fraction is multiplying by that fraction turned upside down. So our division becomes one hundred and twenty-six times ten over twenty-five. One thousand two hundred and sixty, over twenty-five. That is a division we can do. Twenty-five is a counting number, so the regrouping has parts to count. Fifty each, ten left over, which becomes one hundred tenths, four tenths each, nothing remaining. Fifty point four kilometres an hour.

2:11Now look at what that actually did to the two numbers. One hundred and twenty-six became one thousand two hundred and sixty. Two point five became twenty-five. Both of them multiplied by ten. And that is the whole rule. But why is it allowed? A division is a fraction, and a fraction does not change when top and bottom are multiplied by the same thing. That is not a special rule for decimals. It is the same equivalence that lets you write one half as two quarters.

2:46So the quotient does not move. Multiplying both by ten was not arithmetic done to the answer — it was rewriting the question. That was checked on more than seven thousand cases, and the quotient never once shifted. Try it on a pair built to be compared. Four point six eight divided by one point three. And four point six eight divided by nought point one three. Same dividend, same digits in the divisor, one extra decimal place.

3:22One point three has one place, so ten clears it, and the division becomes forty-six point eight over thirteen. Nought point one three has two places, so it takes a hundred, and becomes four hundred and sixty-eight over thirteen. Both landed on the same divisor, thirteen — which is the point of the pair. Forty-six point eight over thirteen is three point six. Four hundred and sixty-eight over thirteen is thirty-six.

3:55The same digits again, ten times apart, because the second divisor was ten times smaller. There is a wrong version of this rule worth naming, because it is the commonest one. It says: move the point in the divisor until it is whole, and leave the dividend alone. It sounds nearly right. The divisor was the problem, so the divisor is what you fix. But a fraction is only unchanged when both parts are multiplied.

4:28Change the bottom alone and you have not rewritten the question, you have asked a different one. How different? Exactly the factor you scaled by. Ten, or a hundred — never a little bit off. Which is the good news and the bad news together. It is never right by accident, and the answer is never close enough to look wrong. Now something the scaling makes easy to see, and otherwise hard to believe.

4:59One hundred and twenty-eight divided by four is thirty-two. No surprise there. One hundred and twenty-eight divided by nought point four is three hundred and twenty. Larger than the number we started with. Read the division as a question and it stops being strange. How many fours fit inside one hundred and twenty-eight? Thirty-two. Now how many four-tenths fit inside it? A much smaller piece, so of course more of them fit.

5:31Dividing does not make things smaller. Dividing by something bigger than one makes things smaller. That is a different sentence, and only the second one is true. Which raises a question about a sentence everybody has been told. A quotient is always less than the number you started with. Stay inside the counting numbers, where it is supposed to live, and it is still not quite right. Divide by one. The answer is the number you started with, unchanged.

6:06Not less. Equal. Every dividend has that exception, and it is always the same divisor causing it. One is not an awkward case to be swept aside. It is the pivot the whole thing turns on. Above one, the quotient falls. Below one, it rises. And at one, it stands still. So let us build the table that sentence needed and never had. Three rows, because there are exactly three things a divisor can be.

6:39Divisor above one: the quotient comes out below the dividend. One hundred and twenty-eight over four is thirty-two. Divisor exactly one: the quotient equals the dividend. One hundred and twenty-eight over one is one hundred and twenty-eight. Divisor between nought and one: the quotient comes out above the dividend. One hundred and twenty-eight over nought point four is three hundred and twenty. Three rows, no exceptions, and one of them is the row people forget.

7:17This holds for every positive dividend — checked across ten thousand of them. And notice what decides it. Not how the divisor is written, not how many digits follow its point. Only which side of one it sits. Take one dividend and move only the divisor's point. Two point four six divided by one point five is one point six four. Divided by nought point one five, sixteen point four.

7:50Divided by nought point nought one five, one hundred and sixty-four. One, six, four every time, stepping up one place as the divisor steps down one. And here is the question that makes the whole idea testable. Is twenty-four point six divided by one point five the same problem as two point four six divided by nought point one five? They look completely different. But both are two hundred and forty-six over fifteen.

8:24They are not two problems that happen to agree. They are one division written twice, and the answer to both is sixteen point four. Which means one division can be worth a great many. Suppose you know that seven hundred and fifty-six divided by thirty-six is twenty-one. Then seventy-five point six divided by thirty-six is two point one, because the dividend shrank and the divisor did not. Seven hundred and fifty-six divided by three point six is two hundred and ten, because this time the divisor shrank.

9:05Seven point five six divided by nought point three six is twenty-one again — both shrank by a hundred, so nothing moved. And seventy-five point six divided by nought point nought three six is two thousand one hundred. None of those needed dividing. Each is the first division with its point walked somewhere new. Scale the top and the answer follows it. Scale the bottom and the answer goes the other way.

9:38Push that to its limit and you get something worth seeing. Take the digits one five one seven with the point in five different places, and five divisors built the same way from three seven. Twenty-five divisions in the grid. Every single one of them has the digits four and one. Because every one of them is one thousand five hundred and seventeen over thirty-seven, which is forty-one, with the point moved.

10:09And there are not twenty-five different answers. There are nine. Forty-one turns up five times, four point one four times, four thousand one hundred three times, out to the corners, which appear once each. That is not a coincidence, and not something you would notice by filling the grid in. The answer depends only on how far the top moved minus how far the bottom moved, so cells agree whenever those differences match.

10:40One division, twenty-five questions, nine answers, and no dividing after the first.