Adding, subtracting, multiplying and dividing decimals

A dressmaker has a roll of ribbon, eight point five metres. She cuts three point six five metres off for a sleeve, then works out what is left on paper, in neat columns. The paper says four point four nought metres. Then she runs the tape measure along what is actually left. Four point eight five. The ribbon and the paper disagree by nearly half a metre. And every subtraction on that paper is correct. The mistake happened before any subtracting started, in where she wrote the digits down.

Columns are the whole game. A digit is worth whatever its column says. A five in the tenths column means five tenths. Move it one column right and it means five hundredths. Ten times smaller. So when you stack two decimals to add or subtract, tenths must sit over tenths and hundredths over hundredths. One mark on the page guarantees it. The decimal point. That is your angle on this whole topic. The decimal points sit lined up, so every digit is in its right column.

Quick one for you, before any arithmetic. Two students set up eight point five take away three point six five. Student A writes eight point five with its five under the five of three point six five. Student B writes eight point five nought, points one above the other. Which set-up is ready to subtract? Look at both, then pick. I'll wait. The answer is B. In A, that five means five tenths but sits in the hundredths column, so the sum is wrong before it starts. B lines the points up, with a placeholder nought holding the hundredths column open. A's set-up is exactly how the dressmaker got four point four nought.

So here is her sum again, set up properly, one column at a time. Eight point five nought on top. Three point six five underneath. Points in a line, so every digit is over its own kind. Hundredths first. Nought take away five cannot be done, so borrow from the tenths, exactly as in a whole-number subtraction. Ten take away five is five. Tenths. That five is now a four. Four take away six cannot be done either, so borrow again, from the units. Fourteen take away six is eight. Units. The eight became a seven. Seven take away three is four. The point in the answer goes straight below the other two. Four point eight five metres. The method was never the problem. The columns were. Adding runs off the same set-up. Put the ribbon back together. Four point eight five plus three point six five, points in a line. Ten hundredths, write nought and carry one. Fifteen tenths, write five and carry one. Eight units. Eight point five nought, the whole roll again.

Money is kind to you here, and it is worth knowing why. Money is always written with two decimal places. Pounds, then tenths of a pound, then the pennies. So two money amounts already have the same number of places, and the columns line up on their own. Here is one for you. A market trader buys a box of mangoes for forty-five pounds sixty and sells them all for seventy-two pounds fifteen. Work out the trader's profit. Profit is what came in take away what went out. So it is seventy-two point one five take away forty-five point six nought. Pause here and work it through. I'll wait. The answer is twenty-six pounds fifty-five. Five take away nought is five. One take away six needs a borrow, and eleven take away six is five. One take away five borrows again, giving six. Then six take away four is two. Write it as money, with the pound sign and both places. Money keeps both decimal places even when the last digit is a nought. Two pounds forty is two point four nought.

Multiplying changes the rule. Lining the points up is the wrong thing to do here. You cover the point instead, multiply the digits as whole numbers, then put the point into the answer at the end by counting. A tile is nought point three five metres wide. Six go side by side. We want the total width. Point covered. Thirty-five times six. Five sixes are thirty, so write nought and carry three. Three sixes are eighteen, plus three is twenty-one. Two hundred and ten. Now the counting. Nought point three five has two digits after the point. Six has none. Two and none makes two, so the answer gets two digits after the point. Two point one nought metres. Why the counting works. Nought point three five is thirty-five divided by a hundred, so thirty-five times six came out a hundred times too big. Dividing by a hundred slides every digit two columns back to the right, past a point that never moved. This is the step to slow down on. Digits right, counting wrong, and you claim twenty-one metres of tiles, or nought point two one. One is a wall. The other, a hand span. Two point one nought and two point one are the same number, so for a length either is fine. Money is the one place that insists on both.

One operation left. Dividing has a single move in it, and then it behaves. By a whole number it is gentle. Seven point two divided by three. The digits stay in their columns, so the answer's point sits directly above the one below. Three into seven goes twice, remainder one. Three into twelve goes four. Two point four. By a decimal, the move appears. Four point eight divided by nought point six. Sharing into lots of nought point six is hard to picture, so make that number whole first. Multiply both by ten. The digits slide one column to the left, past a point that stays put. Four point eight becomes forty-eight. Nought point six becomes six. Forty-eight divided by six is eight. That is allowed because division asks how many of the second number fit inside the first, and scaling both by ten does not change that. Eight lots of nought point six fit into four point eight, and eight sixes fit into forty-eight.

Now the writing side, and how the working should look on the page. Exam specifications ask for the four operations applied to decimals, including formal written methods. That means the columns are on the paper, not in your head. Take a question like this. Eight point five metres of ribbon, three point six five cut off, how much is left. Four point eight five is the answer being looked for. Four point four nought is what unlined columns hand you instead. Two habits that cost nothing. Write the placeholder nought in before you subtract, so every column is full. And give the answer its units, because a bare number is not an amount. And on the multiplying side, every digit can be right while the point sits one place out, putting the answer out by a factor of ten. Count the places every time.

Everything you need for these two is already on the page. First, six point two take away three point four five. Then nought point seven times eight. Write the columns out both times. Take your time with those two. I'll wait. First, two point seven five. Six point two needs its placeholder nought to become six point two nought, then the points line up and you subtract. Second, five point six. Seven eights are fifty-six, and nought point seven has one digit after the point, so the answer has one too.

Before you go, the three moves back to you, in order. Adding and subtracting. Points in a line, and a placeholder nought wherever a column would sit empty. Multiplying. Cover the point, multiply the digits, then count how many decimal places went in and give the answer the same number. Dividing. If the number you divide by is a decimal, slide the digits until it is whole, slide the other number the same way, then divide as normal. Three lines to carry out of here. Points in a line to add and subtract. Count the places to multiply. Make the one you divide by whole. Underneath all three sits one idea. Where a digit sits is what it is worth, so the job is to get every digit into its right column.

Next in the chapter: Order of operations, brackets, powers, roots and BIDMAS.

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