Line up the point to add and subtract, place the point to multiply, and turn the divisor whole to divide.
You already know how to add, subtract, multiply, and divide whole numbers. Decimals do not change those four skills — they only add one new question to each: where does the point go? To add or subtract, you line up the decimal point so matching places stack up, then work down the columns. To multiply, you ignore the point, multiply like whole numbers, and place the point at the end by counting how many decimal places the two factors had. To divide, you slide the point until the divisor is a whole number — and the problem turns into ordinary division. Along the way some divisions stop cleanly while others run on forever, and you will learn to round honestly and to tell the two apart.
Think about money. To add $1.25 and $0.40 you would never let the digits drift around — you would stack the prices so the dollars sit over the dollars and the cents over the cents. The decimal point is what keeps everything in its lane: line up the points and every place lines up automatically. Tenths land over tenths, hundredths over hundredths, ones over ones.
After that, there is nothing new. You add or subtract down each column, carrying or borrowing exactly as you do with whole numbers, and you bring the point straight down into the answer. If one number is short a place, fill it with a zero — that zero changes nothing (3.5 showed that 0.4 = 0.40), and it keeps the column honest.
To add or subtract decimals, line up the decimal point. Matching places stack up; pad short numbers with zeros; then add or subtract in columns and drop the point straight down.
Dial each number in hundredths, flip between + and −, and watch the points stay in one line.
Don't line up by the right edge the way you might with whole numbers. 0.5 + 0.05 is not 0.10 — line up the points and it is 0.50 + 0.05 = 0.55. The point, not the last digit, is the anchor.
Multiplying is where the rule changes — and where beginners most often go wrong. For multiplication you do not line up the points. Instead you pretend the points are gone, multiply the two numbers as if they were whole, and then put the point back at the very end.
Where does the point go? Count the decimal places in both factors and add. To find 1.2 × 0.3, first multiply 12 × 3 = 36. The first factor has one place, the second has one place, so the product has 1 + 1 = 2 places: count two from the right of 36 and the point lands to give 0.36. A quick reasonableness check agrees — a bit more than one whole times about a third should be about a third, and 0.36 is about a third.
To multiply decimals: multiply as whole numbers, then count the decimal places in both factors and place the point that many digits from the right. The product's places = the sum of the factors' places.
Pick two decimals. See the plain whole-number product, the place count, and where the point lands.
Lining up the points is for + and − only. When multiplying, the points can be anywhere — you count places at the end. And don't drop a leading zero too soon: 0.2 × 0.2 = 0.04, not 0.4, because two places means the 4 needs a placeholder zero in the tenths.
Dividing by a decimal feels awkward — how do you split something into "1.5 groups"? The trick is to make the divisor whole. Shift its point to the right until it is an ordinary whole number, then slide the dividend's point the same number of places. Sliding both points the same amount is multiplying top and bottom by the same power of ten, so the quotient does not change — it is just an equivalent division that is easier to do.
Take 4.5 ÷ 1.5. Move the divisor's point one place right to make it 15; move the dividend's point one place right too, to 45. Now it is the familiar 45 ÷ 15 = 3. And the check confirms it: 1.5 × 3 = 4.5.
To divide by a decimal: shift the divisor's point right until it is whole, slide the dividend's point the same number of places, then divide as usual. Moving both points the same amount keeps the quotient the same.
Choose how far to shift. The divisor must become a whole number — and the dividend slides the same amount.
Not every division ends. Ask for 2 ÷ 3 and the digits keep coming: 0.666… forever. When that happens you stop at as many places as the job needs and round. Rounding to two places means deciding whether the hundredths digit stays or ticks up — and that decision is made by looking at the very next digit, the one you are about to drop.
The rule is round half up: if the next digit is 5 or more, round up; if it is 4 or less, leave it. For 2 ÷ 3 = 0.666…, rounding to two places, the next digit (the third 6) is 6, which is 5 or more, so the hundredths tick up: 0.66 becomes 0.67. Because we threw a little away, we write the result with the "about equal" sign: 2 ÷ 3 ≈ 0.67.
To round a decimal, look only at the next digit — the first one you are dropping. If it is 5 or more, round up; if it is 4 or less, keep the digit as is. A rounded answer is written with ≈, not =.
Pick a division and how many places to keep. The highlighted digit is the one that decides.
Round by the next digit only — not by the whole tail and not by the digit after next. To round 0.0649 to two places you look at the 4 in the thousandths place (the next one after hundredths), so 0.0649 rounds to 0.06, not 0.07 — even though "49" feels close to 50.
Every fraction is a division (the bar is a division sign), so every fraction can be written as a decimal — but the long division can end in one of two ways. Sometimes the remainder hits zero and the decimal terminates: 1 ÷ 4 = 0.25, finished. Other times a remainder you have already seen comes back, which means the same digits must repeat forever — a repeating decimal: 1 ÷ 3 = 0.333…, the 3 going on without end.
We mark the part that repeats with a bar (or a dot) over it: 0.3 with a bar over the 3 means the 3 repeats. The repeating block can be long: 2 ÷ 7 = 0.285714… repeats a whole six-digit block 285714 over and over. Why must it eventually repeat? Because dividing by 7 can only ever leave the remainders 1 through 6 — once a remainder returns, the cycle begins again.
A fraction's decimal either terminates (the remainder reaches 0) or repeats forever (a remainder comes back, restarting the cycle). Mark the repeating block with a bar, as in 0.3̇.
Pick a fraction. See the long-division digits, whether it terminates or repeats, and its 2-place rounded value.
5 ÷ 8: the long division gives 0.625 and the remainder reaches 0, so it terminates. 5 ÷ 12: it gives 0.41666…, where only the 6 repeats — written 0.41 with a bar over the 6.
Each of the four operations keeps its whole-number method and adds one question about the point.
| Operation | What to do with the point | Example |
|---|---|---|
| Add / subtract | Line up the points; pad with zeros; bring the point straight down. | 1.25 + 0.40 = 1.65 |
| Multiply | Multiply as whole numbers; the answer's places = sum of the factors' places. | 1.2 × 0.3 = 0.36 |
| Divide | Shift the divisor's point until whole; slide the dividend's point the same amount. | 4.5 ÷ 1.5 = 3 |
| Round | Look at the next digit; 5 or more rounds up; write the result with ≈. | 2 ÷ 3 ≈ 0.67 |
| Terminate / repeat | Remainder reaches 0 → terminates; remainder returns → repeats (bar the block). | 1∕4 = 0.25, 1∕3 = 0.3̇ |
Write 2.7 as 2.70 so both have hundredths. Then 2.70 + 0.45 = 3.15. (Lining up the points puts the 7 tenths over the 4 tenths.)
Pad to 5.20, then borrow down the columns: 5.20 − 1.86 = 3.34.
Ignore the points: 6 × 7 = 42. One place plus two places is three places, so count three from the right: 0.042. The leading zeros are needed as placeholders.
Make the divisor whole: shift one place to turn 0.4 into 4, and slide the dividend one place to 36. Now 36 ÷ 4 = 9. Check: 0.4 × 9 = 3.6. ✓
Keep two places: 1.16, then look at the next digit. It is 6 (5 or more), so round up to 1.17. Write it with ≈: 7 ÷ 6 ≈ 1.17.
3 ÷ 8 = 0.375 terminates (the remainder reaches 0). 4 ÷ 9 = 0.444… = 0.4̇ repeats (the 4 never stops). So 3∕8 terminates and 4∕9 repeats.
Six questions to lock it in. Tap the answer you think is right.
This lesson covers the four operations on decimals and is built around CCSS 5.NBT.B.7 (add, subtract, multiply, and divide decimals to hundredths, using place value and the relationship between operations) and 5.NBT.A.4 (use place-value understanding to round decimals). The closing section on terminating versus repeating decimals reaches forward to 7.NS.A.2.d (convert a rational number to a decimal that terminates or eventually repeats). A helpful habit to reinforce at home: for addition and subtraction the decimal point is the anchor (line it up), but for multiplication the point is placed at the very end by counting places — keeping those two moves distinct prevents the most common decimal mistakes.