Decimals: Adding and Subtracting

Everything about adding decimals comes down to one habit: line up the decimal points. When the points are under each other, tenths are under tenths and hundredths under hundredths, and the sum is the ordinary column method with a point dropped back in at the end. Three point six plus two point seven five: write three point six as three point six zero so it has the same number of places, add, and the answer is six point three five. The wrong version lines up the last digits instead, which puts six tenths above five hundredths, and gets a nonsense answer. Estimate with whole numbers to catch that: four plus three is about seven, and six point three five is close.

Subtraction adds one thing: pad the shorter number with zeros before you start, so that twelve point five becomes twelve point five zero next to four point eight five. Then trade as usual. Zero take five cannot be done, so trade from the tenths: ten take five is five. Four take eight cannot be done, so trade from the ones: fourteen take eight is six. One take four cannot be done, so trade from the tens: eleven take four is seven. Seven point six five. The point changes nothing about trading; one from the column to the left is always ten in this column. Estimate: thirteen minus five is eight, so seven point six five is right.

Money is decimals with exactly two places, and test questions like to hide a two-step problem in it. David and Sarah each bought a T-shirt and a hat and spent the same amount. David's T-shirt cost twenty-eight ninety and his hat twenty-one ten. Sarah's T-shirt cost thirty ninety-five. What did her hat cost? First find what David spent: twenty-eight ninety plus twenty-one ten is exactly fifty dollars. Sarah spent the same fifty, so her hat is fifty minus thirty ninety-five, which is nineteen dollars and five cents. Write the fifty down before you use it; the first answer is the ingredient for the second.

When the numbers are given in millions with a decimal, keep them that way. The population was eight point two seven million in 1950 and nineteen point one six million in 2000. The difference is nineteen point one six minus eight point two seven, which is ten point eight nine — ten point eight nine million. That is the answer unless the question asks for digits, in which case ten point eight nine million is ten million eight hundred and ninety thousand. Measurements behave the same way: two point three five metres plus zero point eight metres is three point one five metres, once the zero point eight is padded to zero point eight zero. The unit rides along; only the decimal arithmetic changes.

Chains of decimal moves are just the whole-number chains you already know, with the points lined up. A jug holds one point five litres; zero point six five is poured out, then zero point four is added. One point five zero minus zero point six five is zero point eight five; plus zero point four zero is one point two five litres. And backwards: Kai spends twelve eighty-five, then receives twenty dollars, and now has thirty-one sixty. Undo the twenty: eleven sixty. Undo the spending by adding it back: twenty-four forty-five. Pad every amount to two places first — twenty is twenty point zero zero — and the columns look after themselves. Quiz and exercises now.