A decimal is a number where the whole part and the fractional part are separated by a decimal point: 3.25 or 0.7. All operations with decimals — addition, subtraction, multiplication, and division — work almost the same way as with whole numbers; the main thing is to keep track of the decimal point. This page collects all the rules with column examples, a cheat sheet for moving the decimal point, a breakdown of common mistakes, and an interactive quiz for self-check.
Place values of a decimal
Each digit after the decimal point has its own name and weight:
- the first place — tenths (0.1);
- the second — hundredths (0.01);
- the third — thousandths (0.001) and so on.
Important property: if you add zeros to the right of the fractional part, the number doesn't change: 5.7 = 5.70 = 5.700. We'll use this property constantly — it lets us equalize the number of decimal places when adding, subtracting, and comparing.
All operations with decimals: quick cheat sheet
Each operation has its own key rule about the decimal point. The four cards below sum up the whole topic, and each operation is explained in detail further down, with examples.
Column addition and subtraction of decimals
Addition and subtraction follow the same rule:
- Write the numbers in a column so that the decimal points line up exactly — then matching place values end up above each other: tenths above tenths, hundredths above hundredths.
- If the numbers have a different number of decimal places, equalize them with zeros.
- Add or subtract as if they were whole numbers, ignoring the decimal points.
- Put the decimal point in the answer directly below the points of the original numbers.
A whole number can also be written as a decimal: 6 = 6.00 — this helps in examples like 6 − 2.35.
The decimal multiplication rule
Multiplication is the only operation where you don't need to line up the decimal points. The decimal multiplication rule has two steps:
- Multiply the numbers ignoring the decimal points — just like regular whole numbers.
- In the result, place the decimal point so that there are as many digits after it as there are decimal digits in both factors combined.
If the product has fewer digits than you need to mark off, add zeros on the left: 12 × 3 = 36, but for 0.12 × 0.3 you need to mark off three digits, so the answer is 0.036.
Quick self-check: when multiplying by a number less than one, the result should be smaller than the first factor. If it came out bigger, the decimal point is in the wrong place.
Dividing decimals
There are two cases here.
Dividing by a whole number. Do long division as usual, and place the decimal point in the quotient the moment the whole part of the dividend is finished: 8.46 ÷ 2 = 4.23.
Dividing by a decimal. First get rid of the decimal point in the divisor: move the decimal point in both the dividend and the divisor to the right by the same number of places — in other words, multiply both numbers by 10, 100, or 1000. The quotient doesn't change, and the division reduces to the first case.
Multiplying and dividing by 10, 100, 1000
The fastest operations with decimals are multiplying and dividing by a power of ten. No long division needed: just move the decimal point by as many places as there are zeros in 10, 100, or 1000. Not enough digits? Add zeros.
Comparing decimals
Decimals are compared place by place:
- First compare the whole parts: the number with the larger whole part is bigger.
- If the whole parts are equal, compare the tenths, then the hundredths, then the thousandths, until you find a difference.
- To avoid confusion, equalize the number of decimal places with zeros: comparing 5.7 and 5.68 is easier as 5.70 and 5.68 — seventy hundredths is more than sixty-eight, so 5.7 > 5.68.
The length of the number doesn't tell you anything: 0.5 is bigger than 0.4999, even though it has fewer digits.
Operations with decimals: worked examples
Example 1. Addition: 5.3 + 2.47
Step 1. Equalize the decimal places: 5.3 = 5.30.
Step 2. Write in a column, decimal point under decimal point.
Step 3. Add as whole numbers: 530 + 247 = 777.
Step 4. Put the decimal point below the points of the addends: 7.77.
Example 2. Subtracting from a whole number: 6 − 2.35
Step 1. Write 6 as a decimal: 6 = 6.00.
Step 2. Write in a column: 6.00 − 2.35.
Step 3. Subtract, borrowing from the higher places: 3.65.
Check. 3.65 + 2.35 = 6 — it checks out.
Example 3. Multiplication: 2.5 × 0.04
Step 1. Multiply, ignoring the decimal points: 25 × 4 = 100.
Step 2. Count the decimal digits: one in 2.5 and two in 0.04 — three total.
Step 3. Mark off three digits from the right: 0.100.
Step 4. Drop the trailing zeros in the decimal part: 0.1.
Order-of-magnitude check. 0.04 is less than one, so the answer should be smaller than 2.5 — correct.
Example 4. Dividing by a decimal: 1.44 ÷ 1.2
Step 1. Make the divisor a whole number: move the decimal point one place to the right in both numbers — you get 14.4 ÷ 12.
Step 2. Long-divide the whole part: 14 ÷ 12 = 1, remainder 2.
Step 3. The whole part is finished — place the decimal point in the quotient and bring down the 4: 24 ÷ 12 = 2.
Answer: 1.2. Check: 1.2 × 1.2 = 1.44.
Common mistakes with decimal operations
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When adding or subtracting, lining up numbers by the right edge, as if they were whole numbers: writing 3.25 + 4.8 in a column without lining up the decimal points gives 3.73.
Line up by the decimal point, not by the last digit: 4.8 = 4.80, so 3.25 + 4.80 = 8.05. Tenths must be above tenths, hundredths above hundredths.
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When multiplying, trying to line up the decimal points or placing the point in the product 'under the points' of the factors.
In multiplication, the decimal points aren't lined up at all. Multiply as whole numbers and mark off from the right as many digits as there are decimal digits in both factors combined.
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If the product has fewer digits than needed to mark off, dropping zeros: 0.12 × 0.3 = 0.36 instead of 0.036.
12 × 3 = 36 — that's only two digits, but you need to mark off three. Add the missing zeros on the left: 0.036.
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When dividing by a decimal, moving the decimal point only in the divisor: turning 4.2 ÷ 0.7 into 4.2 ÷ 7 and getting 0.6.
The decimal point moves in both numbers by the same number of places: 4.2 ÷ 0.7 = 42 ÷ 7 = 6. Otherwise the quotient shrinks tenfold.
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Comparing decimals by the number of digits: thinking 5.68 is bigger than 5.7 because it's written with more digits.
Equalize the places with zeros and compare place by place: 5.70 and 5.68 — in the tenths place 7 > 6, so 5.7 > 5.68.
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When multiplying or dividing by 10, 100, 1000, moving the decimal point in the wrong direction.
Multiplication makes the number bigger — the point moves right: 3.74 × 10 = 37.4. Division makes it smaller — the point moves left: 3.74 ÷ 10 = 0.374. The number of places equals the number of zeros.
Questions and answers
Which operations with decimals are taught in 5th grade?
In 5th grade, students cover all four operations — addition, subtraction, multiplication, and division of decimals — plus comparing and rounding them. Usually, column addition and subtraction come first, then multiplication and division by a whole number, and finally division by a decimal.
Does a decimal change if you add zeros to the right?
No: 5.7 = 5.70 = 5.700 — these are the same number. Adding zeros is used to equalize the number of decimal places when adding, subtracting, and comparing. But you can't insert a zero between digits or remove one from the middle of a number — that would change its value.
How do you round a decimal?
Find the place you want to round to and look at the next digit: if it's less than 5, the rounded digit stays the same (3.748 ≈ 3.7); if it's 5 or more, it increases by one (2.85 ≈ 2.9). All digits to the right are dropped.
How do you convert a fraction to a decimal?
Divide the numerator by the denominator: 3/4 = 3 ÷ 4 = 0.75. Not every fraction gives a terminating decimal: for example, 1/3 = 0.333… is an infinite repeating decimal. You get a terminating decimal when the denominator factors only into 2s and 5s.
What order do you follow for operations in expressions with decimals?
The order is the same as for whole numbers: first do what's in parentheses, then multiplication and division from left to right, then addition and subtraction from left to right. Writing numbers as decimals doesn't change the order of operations.
Can you divide a smaller decimal by a larger one?
Yes. If the dividend is smaller than the divisor, the quotient will simply be less than one: 0.3 ÷ 0.6 = 3 ÷ 6 = 0.5. In that case, you write 0 in the whole part of the quotient and continue dividing after the decimal point.
