A decimal is a way to write a fraction with a denominator of 10, 100, 1000 or another power of ten, using a decimal point instead of a fraction bar. The digits to the right of the point are tenths, hundredths, thousandths, and so on.
This page covers decimal place value, comparing and rounding, the four basic operations, and converting between decimals, fractions and percents, with worked examples and practice problems. Every example below was calculated and checked twice.
What is decimal place value?
Each position after the decimal point has a value ten times smaller than the one before it. The first position is tenths, the second is hundredths, the third is thousandths.
Take the number 3.456. The 4 is in the tenths place, the 5 is in the hundredths place, and the 6 is in the thousandths place. Read together, it’s “three and four hundred fifty-six thousandths,” because all three digits after the point combine into one value out of a thousand.
A decimal without a whole-number part, like 0.4, is often written .4, but including the leading zero makes it easier to read and less likely to be missed. This section covers place value naming only. It doesn’t cover scientific notation or repeating decimals, which are separate topics.
How do you compare and round decimals?
To compare two decimals, line up the decimal points and compare digit by digit from left to right, adding trailing zeros if one number has fewer digits. For example, 0.7 and 0.65: written as 0.70 and 0.65, the tenths digit (7 vs. 6) decides it, so 0.7 is greater than 0.65, even though 65 looks like the bigger number.
To round a decimal, look at the digit just past the place you’re rounding to. If it’s 5 or more, round the target digit up. If it’s 4 or less, leave it as is. Rounding 3.14159 to the nearest hundredth: the digit after the hundredths place (5) rounds the 1 up to 2, giving 3.14. Rounding 2.675 to the nearest tenth: the digit after the tenths place (7) rounds the 6 up to 7, giving 2.7. This section covers standard rounding. It doesn’t cover rounding conventions used in banking or scientific contexts, which sometimes round differently at exactly 5.
How do you add and subtract decimals?
Line up the decimal points, add zeros as placeholders so both numbers have the same number of digits after the point, then add or subtract as you would with whole numbers.
Example: 4.75 + 3.60 = 8.35. Adding the hundredths (5 + 0 = 5), then tenths (7 + 6 = 13, write 3, carry 1), then ones (4 + 3 + 1 = 8) gives 8.35.
Example: 8.20 – 3.45 = 4.75. Since 0 hundredths can’t subtract 5, borrow from the tenths: 10 – 5 = 5 hundredths, leaving 1 tenth. 1 – 4 needs another borrow from the ones, giving 11 – 4 = 7 tenths. 7 – 3 = 4 ones. Result: 4.75. This section covers lining up decimal points for addition and subtraction only. It doesn’t cover adding decimals to fractions directly, which requires converting first.
How do you multiply and divide decimals?
To multiply, ignore the decimal points, multiply the numbers as whole numbers, then count the total decimal places in both original numbers and place the point that many digits from the right in the answer.
Example: 2.5 × 1.2. Multiplying 25 × 12 = 300. Both numbers together have two decimal places (one each), so the answer is 3.00, or 3.
To divide, if the divisor has a decimal point, move it to the right until it’s a whole number, then move the decimal point in the number you’re dividing by the same number of places, and divide as usual.
Example: 6.4 ÷ 0.8. Moving both decimal points one place right turns this into 64 ÷ 8 = 8. This section covers multiplying and dividing by hand. It doesn’t cover long division with repeating decimal answers.
How do you convert between decimals, fractions and percents?
A decimal converts to a fraction by using the place value as the denominator. For example, 0.6 is 6 tenths, or 6/10, which reduces to 3/5. And 0.45 is 45 hundredths, or 45/100, which reduces to 9/20. See our fractions page for more on simplifying fractions.
A fraction converts to a decimal by dividing the numerator by the denominator. For example, 3/4 = 3 ÷ 4 = 0.75, and 1/8 = 1 ÷ 8 = 0.125.
A decimal converts to a percent by multiplying by 100 and adding a percent sign, which is the same as moving the decimal point two places right. So 0.75 becomes 75%, and 0.125 becomes 12.5%. Our percentages page covers percent calculations in more depth. This section covers converting whole and terminating decimals. It doesn’t cover repeating decimals like 1/3, which need a different method to write as an exact fraction.
Practice problems
- Order from least to greatest: 0.45, 0.5, 0.099
- Round 6.849 to the nearest tenth.
- 12.5 + 7.65 = ?
- 9.3 – 4.68 = ?
- 3.2 × 4 = ?
- 15.6 ÷ 4 = ?
- Convert 7/20 to a decimal and a percent.
- Convert 0.8 to a fraction in lowest terms.
Show answers and reasoning
- 0.099, 0.45, 0.5. Line up the decimal places: 0.099 has a 0 in the tenths place, 0.450 and 0.500 both have higher tenths digits, and 0.5 has a higher tenths digit than 0.45.
- 6.8. The digit after the tenths place is 4, so round down and drop it.
- 20.15. Hundredths: 0 + 5 = 5. Tenths: 5 + 6 = 11, write 1 carry 1. Ones: 2 + 7 + 1 = 10, write 0 carry 1. Tens: 1 + 1 = 2.
- 4.62. Borrowing across the decimal point: 30 – 8 = 22 hundredths (write 2, carry from tenths). Tenths become 2 (after borrow) – 6, borrow again: 12 – 6 = 6. Ones: 8 – 4 (after borrow) = 4.
- 12.8. 32 × 4 = 128, one decimal place total, so 12.8.
- 3.9. 156 ÷ 4 = 39, with one decimal place, so 3.9.
- 0.35, or 35%. 7 ÷ 20 = 0.35, and 0.35 × 100 = 35%.
- 4/5. 0.8 is 8/10, which reduces to 4/5.
For more number practice, visit our math hub or try a set of math worksheets covering this and other skills.
Sources
- Khan Academy. Free math courses and practice