Adding and Subtracting Decimals

Adding decimals is the column adding you already know, with one extra rule: line up the decimal points. That rule is really a place value rule, because tenths can only be added to tenths and units to units. Write twelve point seven five, then write three point four underneath so that the two points sit in the same column, and put a point in the answer line directly below them. Then work from the right as usual. Five hundredths plus nothing is five. Seven tenths plus four tenths is eleven tenths, so write one and carry one. Two units plus three units plus the carried one is six. The answer is sixteen point one five.

The two numbers you are adding often have different numbers of decimal places, and that is where mistakes creep in. The cure is to make them match by writing zeros on the end. Look at the pictures: four tenths shaded, and forty hundredths shaded, cover exactly the same amount of the whole. Each tenth simply split into ten smaller pieces. So nought point four and nought point four zero are the same number, and a zero written on the end of a decimal changes nothing at all. Now four point nought six plus one point nine is easy: write the second number as one point nine zero, and every column has a digit in it.

Subtracting uses the same set up. Line up the points, fill the short columns with zeros, then work from the right. Take nought point seven minus nought point three five. Written as nought point seven zero, the hundredths column has a zero and you need to take five from it, so you trade. Borrow one of the seven tenths, which leaves six tenths, and that borrowed tenth is worth ten hundredths. Now ten take five is five, and six tenths take three tenths is three tenths, so the answer is nought point three five. The one thing you must never do is turn a column upside down to dodge the trade.

What if one number has no decimal point at all? Every whole number has one hiding at its right hand end, so five is five point nought, and it is also five point nought nought. Write those zeros in and the question becomes ordinary. Five point nought nought minus two point six nought needs a trade, and it gives two point four. The same trick handles twenty minus thirteen point four five: write twenty as twenty point nought nought and subtract to get six point five five. Two habits protect you here. Estimate first, because five minus three is about two, and check by adding your answer back to the number you subtracted.

Money is where you meet decimals most often, and money always uses two decimal places. Suppose you spend thirty seven dollars eighty five and pay with a fifty. Estimate first: fifty take thirty eight is about twelve dollars. Then set it out as fifty point nought nought minus thirty seven point eight five, which gives twelve dollars fifteen. Measurements work the same way. Three pieces of timber measuring one point two five, nought point five and two point seven five metres come to four point five metres once you write them all with two decimal places. Estimate, keep the points in one column, and check a subtraction by adding your answer back.