Decimals are whole-number arithmetic with one extra job: putting the decimal point in the right place. Nearly every decimal mistake is a place-value mistake, not a times-table mistake. This guide covers all four operations, shows exactly where the point goes and why, and gives you a quick estimate check for each one.
Each place to the right of the point is worth a tenth of the place before it. In 3.07 there are 3 ones, 0 tenths and 7 hundredths. In 3.7 there are 3 ones and 7 tenths, which is much more. So 3.7 is bigger than 3.07, even though "07" looks like more digits.
The same idea settles a classic confusion: 0.45 is smaller than 0.5. Write them with the same number of places, 0.45 and 0.50, and compare 45 hundredths with 50 hundredths.
Adding zeros on the end of a decimal never changes its value: 2.5, 2.50 and 2.500 are the same number. That is useful when you line up numbers to add or subtract.
Line up the decimal points, fill any gaps with zeros, and add or subtract as normal.
In your head, counting up is often easier than borrowing with decimals. For 7 − 2.48, go 2.48 + 0.52 = 3.00, then 3.00 + 4.00 = 7.00, so the answer is 0.52 + 4 = 4.52. That is the same method shopkeepers use to give change, and it is explained in more detail in mental subtraction tips.
Ignore the decimal points, multiply as whole numbers, then put the point back. The answer has as many decimal places as the two numbers have together.
For 6.75 × 3.2: first 675 × 32 = 21600. Then count the decimal places: 6.75 has two and 3.2 has one, so the answer needs three. Counting three places from the right of 21600 gives 21.600, which is 21.6. So 6.75 × 3.2 = 21.6.
Check with an estimate: 6.75 is about 7 and 3.2 is about 3, and 7 × 3 = 21. The answer 21.6 is close, so the point is in the right place. If you had written 216 or 2.16, the estimate would catch it at once.
Small decimals work the same way: 4 × 3 = 12, and 0.4 × 0.3 has two decimal places, so 0.4 × 0.3 = 0.12. Notice that the answer is smaller than both numbers. Multiplying by a number less than 1 always makes things smaller.
Multiplying by 10, 100 or 1,000 just moves the digits: 3.6 × 100 = 360. For more ways to split up a multiplication, see multiply in minutes.
Dividing is easiest when the number you divide by is whole. So multiply both numbers by 10 (or 100) until it is. Multiplying both by the same amount doesn't change the answer, just as 6 ÷ 3 and 60 ÷ 30 are both 2.
The second example often surprises people: 6 ÷ 0.25 is bigger than 6. That is right. Dividing by a quarter asks "how many quarters fit into 6?", and there are 4 in each whole, so 24 in six wholes.
Check a division by multiplying back: 3.6 × 2.4 = 8.64, so 8.64 ÷ 2.4 = 3.6 is right.
Some decimals are easy fractions, and the fraction is often the faster route:
So 0.25 × 48 is a quarter of 48, and 0.25 × 48 = 12. And 0.125 × 64 is an eighth of 64, which is 8. These are also the same as 25% and 12.5%, which is why our quick percentage tricks lean on them, and the fraction to percentage table lists the common ones.
Round each number to one easy digit and work out a rough answer first. For 19.87 × 4.1, think 20 × 4 = 80. The exact answer is 81.467: 19.87 × 4 = 79.48, then 19.87 × 0.1 = 1.987, and 79.48 + 1.987 = 81.467. The estimate tells you the answer is about 80, so the decimal point can't be anywhere else.
Answers:
Decimals get easier when the whole-number facts underneath them are quick. Keep those sharp in practice mode or the Arithmetic Memory Game, then test yourself under time pressure with Against Time. If adding the pieces together is the slow part, quick addition tricks will help.
Count the decimal places in both numbers and add them up; the answer has that many. Then check with an estimate: 6.75 × 3.2 is about 7 × 3, so the answer must be about 21.
Dividing asks how many times one number fits into another. A number less than 1, like 0.25, fits into 6 many times, 24 in fact, so the answer is bigger than 6.
Line up the decimal points and add zeros to the shorter number, so 12.5 becomes 12.50. Then add as normal. The zeros don't change the value, they just keep the columns straight.