Counting Decimal Places
Multiply as if the points were not there, then put the point back. The answer has as many decimal places as the two factors have together.
Adding and subtracting decimals is exactly like whole numbers as long as the decimal points are lined up, because then tenths sit under tenths and hundredths under hundredths. Multiplying is done as whole numbers first; the decimal point is placed afterwards by counting decimal places.
Ignore the point: . There was one decimal place, so the answer is .
Line up the points and write as , so both numbers have hundredths. Hundredths: . Tenths: . Ones: . Sum: .
Lining decimals up by their right-hand ends instead of by the decimal point: is , not . Pad with zeros () so every number has the same number of decimal places.
is close to , so the answer should be near — and is. A misplaced point would give or , both clearly wrong.
Find and .
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, and has one decimal place, so (sense check: about ). For the subtraction write as : .
In Plain Terms & More Examples
Multiply as if the points were not there, then count how many decimal places the two numbers had together and put the point back that far from the right.
Ignore the point: . One decimal place, so the answer is .
Ignore the points: . Two decimal places, so the answer is .
You buy notebooks at each and pay with a bill. Step 1: . Step 2: the change is .
Counting only one factor's decimal places. is not : taking of something shrinks it, so the answer must be smaller than , and is bigger. Two factors with one place each give two places: .
Find and .
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with two decimal places: . with two decimal places: , which is half of , as it should be.
Dividing Decimals
To divide by a decimal, move the point in BOTH numbers the same number of places until the divisor is whole. becomes . The answer does not change because you multiplied both by the same power of ten.
Moving the point one place in both numbers multiplies both by , and a division does not change when both numbers are scaled by the same amount. asks how many halves fit in ; asks how many s fit in . It is the same question drawn ten times bigger, and the answer, , stays put.
Move two places: .
The divisor is already whole, so divide as usual and keep the point where it is. is tenths, and , so the answer is tenths: . Check: .
Moving the point in only one number. Turning into gives , but halves fit into many times, not less than once. Whatever you do to the divisor, do to the number being divided: .
Shift the point, divide, and check by multiplying back.
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Shift one place: . Check: .
Word Problems
Finding the total of several equal amounts means multiply. Sharing a total equally, or finding the price of one, means divide. Estimate first, so you know roughly what the answer should be.
The estimate decides where the decimal point goes. If kg of apples cost , one kilogram costs a bit less than , so is believable and or are not.
Apples cost for kg. Per kilogram: .
Fabric costs per metre. How much do m cost? Multiply without the points: . The factors have decimal places, so the product is : . The area model shows the same four pieces: .
Reading money with the wrong place value. , two dollars and eighty cents. A calculator may show , and that is fine, but of a dollar is cents, not cents.
A car uses liters of fuel per kilometer. How much fuel for a km trip?
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: liters. (Two decimal places in the factors, two in the product: .)
Adding and Subtracting Decimals
Write the numbers in columns with the decimal points lined up, fill empty places with zeros, then add or subtract place by place from the right, regrouping just as with whole numbers.
Lining up the points lines up the place values: tenths under tenths, hundredths under hundredths. Writing as changes nothing, because tenths and hundredths are nothing, but it gives you digits to regroup from.
Write as . Hundredths: . Tenths: ; write , carry . Ones: . Tens: . Sum: .
Write as . The hundredths need a trade: take from the ones (leaving ) to make tenths, then of those tenths (leaving ) to make hundredths. Now subtract: , , . Answer: . Check: .
Round: . Then compute: , close to . An answer like or would show at once that a point was misplaced.
Copying the decimal part down. For some people subtract the whole numbers, , and copy the to get . But the whole of is being taken away, so its must be subtracted too: .
Find , then check by adding.
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Write . Hundredths: trade tenth, . Tenths: needs a trade from the ones, . Ones: needs a trade from the tens, . Answer: . Check: .
Why It Works
A decimal with two places is a whole number of hundredths: hundredths. So hundredths — which is why lining up the points (lining up the hundredths) makes ordinary addition work. For multiplication, : the denominators multiply, so the decimal places ADD ( places).
Do not read a proof the way you read a rule. Cover the last line, ask what the previous line forces, and only then uncover it. If a step feels like magic, that is the step to work by hand with a concrete number.