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Decimals: Place Value, Operations and Conversions

Decimal place value, comparing and rounding, all four operations, and converting to fractions and percents, with checked answers.

Place value chart showing the number 347.258 split into hundreds, tens, ones, tenths, hundredths and thousandths
LearningHaven original graphic.

A decimal is a number written with a decimal point to show parts of a whole. Some decimals end, like 0.75, and some repeat forever, like 0.333… The digits after the point are tenths, hundredths, thousandths and so on, and each place is 10 times smaller than the one before it (Math Is Fun).

This page covers decimal place value, reading decimals aloud, comparing and rounding, the four basic operations, converting between decimals, fractions and percents, and terminating and repeating decimals. It ends with practice problems and worked answers.

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.

To read a decimal aloud, say the whole part, say “point,” then say each digit after the point. Math Is Fun says, “We read digits after the decimal point one digit at a time” (Math Is Fun). So 17.591 is “seventeen point five nine one.”

To show place value, say “and” for the point and name the last place. A college tutoring handout says to “write the decimal part as if it were a whole number” and then name its last place. So 549.238 is “five hundred forty-nine and two hundred thirty-eight thousandths” (college tutoring handout).

Math Is Fun reads 45.6 as “forty-five and six tenths.” It also splits $2.75 into 2 dollars, 7 tenths of a dollar (70 cents) and 5 hundredths of a dollar (5 cents) (Math Is Fun).

A decimal without a whole-number part, like 0.4, is often written .4. Including the leading zero makes it easier to read and less likely to be missed. This section covers place value naming and reading aloud only. It doesn’t cover scientific notation, which is a separate topic. Repeating decimals have their own section below.

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. Add trailing zeros if one number has fewer digits. For example, compare 0.7 and 0.65. Written as 0.70 and 0.65, the tenths digits are 7 and 6. So 0.7 is greater than 0.65, even though 65 looks like the bigger number.

Number line from 0.60 to 0.80 with points at 0.65 and 0.70 showing that 0.7 is greater
On a number line, 0.7 (written 0.70) sits to the right of 0.65, so it is the greater number. LearningHaven original graphic.

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 (college tutoring handout). The Grade 5 standard on rounding reads, “Use place value understanding to round decimals to any place” (Common Core State Standards Initiative).

Example: Round 3.14159 to the nearest hundredth. The hundredths digit is 4. The digit after it is 1, which is less than 5, so the 4 stays. The answer is 3.14.

Example: Round 2.675 to the nearest tenth. The tenths digit is 6. The digit after it is 7, which is 5 or more, so the 6 rounds up to 7. The answer is 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, then add or subtract as you would with whole numbers. The zeros give both numbers the same number of digits after the point. Our subtraction page covers subtraction with whole numbers.

Example: 4.75 + 3.60 = 8.35. Add the hundredths first (5 + 0 = 5). Then add the tenths (7 + 6 = 13, so write 3 and carry 1). Then add the ones (4 + 3 + 1 = 8). The sum is 8.35.

Example: 8.20 − 3.45 = 4.75. Since 0 hundredths can’t subtract 5, borrow 1 tenth from the tenths place. Now 10 − 5 = 5 hundredths, leaving 1 tenth. Next, 1 tenth can’t subtract 4, so borrow 1 one from the ones place. Now 11 − 4 = 7 tenths, leaving 7 ones. Then 7 − 3 = 4 ones. The result is 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 and multiply the numbers as whole numbers. Then count the total decimal places in both original numbers. Place the point that many digits from the right in the answer (college tutoring handout).

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, first check the divisor, which is the number you divide by. If it has a decimal point, move the point right until the divisor is a whole number. Then move the point in the dividend, which is the number being divided, the same number of places to the right. Divide as usual (college tutoring handout). Our division page covers division with whole numbers.

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. When a division never ends, such as 1 ÷ 3, the answer is a repeating decimal, covered below.

Rule summary: Every operation on this page rests on decimal place value. To add or subtract, line up the decimal points. To multiply, count the decimal places in both numbers. To divide, make the divisor a whole number first.

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 (Mathematics LibreTexts). For example, 3/4 = 3 ÷ 4 = 0.75, and 1/8 = 1 ÷ 8 = 0.125. If the division never ends, the decimal repeats.

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 converts decimals that end. It doesn’t show how to write a repeating decimal like 0.333… as an exact fraction, which needs a different method.

What are terminating and repeating decimals?

A terminating decimal ends after a set number of digits, like 0.75 (Mathwords). A repeating decimal has a digit or block of digits that repeats forever (Wolfram MathWorld). The fraction 1/3 is 0.333…, and the 3 repeats without end.

Every fraction gives one type or the other. The Grade 7 standards say, “the decimal form of a rational number terminates in 0s or eventually repeats” (Common Core State Standards Initiative). A rational number is a number you can write as a fraction of two integers, where the bottom number is not zero.

You can predict the type before you divide. First write the fraction in lowest terms. Then look at the bottom number. If its only prime factors are 2 and 5, the decimal ends. If any other prime factor appears, such as 3 or 7, the decimal repeats (Mathematics LibreTexts). A prime factor is a prime number that divides evenly into another number.

  • 7/20 ends. Since 20 = 2 × 2 × 5, 7/20 = 0.35.
  • 1/6 repeats. Since 6 = 2 × 3, 1/6 = 0.1666…
  • 1/7 repeats. The block 142857 repeats forever, so 1/7 = 0.142857142857…

Common mistake: thinking that a 3 in the bottom number always means the decimal repeats. That happens when you skip the lowest-terms step. The fraction 9/12 has a 3 in its bottom number, but 9/12 reduces to 3/4, and 3/4 = 0.75 ends. So reduce first, then check the prime factors (Mathematics LibreTexts).

Try this: Which of 1/4, 9/12 and 1/6 end? Divide to check. Answer: 1/4 = 0.25 and 9/12 = 0.75 end. The fraction 1/6 = 0.1666… repeats.

This section does not cover irrational numbers, which neither end nor repeat (Wolfram MathWorld).

Practice problems

  1. Order from least to greatest: 0.45, 0.5, 0.099
  2. Round 6.849 to the nearest tenth.
  3. 12.5 + 7.65 = ?
  4. 9.3 − 4.68 = ?
  5. 3.2 × 4 = ?
  6. 15.6 ÷ 4 = ?
  7. Convert 7/20 to a decimal and a percent.
  8. Convert 0.8 to a fraction in lowest terms.
  9. Without dividing, decide whether 3/8 and 5/12 end or repeat. Then divide to check.
Show answers and reasoning
  1. 0.099, 0.45, 0.5. Write each number with three decimal places: 0.099, 0.450 and 0.500. Compare the tenths digits first. Since 0 is less than 4 and 4 is less than 5, the order is 0.099, then 0.45, then 0.5.
  2. 6.8. The tenths digit is 8. The digit after it is 4, which is less than 5, so the 8 stays and the rest drops.
  3. 20.15. Write 12.5 as 12.50. 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. 4.62. Write 9.3 as 9.30. Hundredths: 0 − 8 needs a borrow, so 10 − 8 = 2 and the tenths drop from 3 to 2. Tenths: 2 − 6 needs a borrow, so 12 − 6 = 6 and the ones drop from 9 to 8. Ones: 8 − 4 = 4.
  5. 12.8. 32 × 4 = 128, one decimal place total, so 12.8.
  6. 3.9. 156 ÷ 4 = 39, with one decimal place, so 3.9. Check: 4 × 3.9 = 15.6.
  7. 0.35, or 35%. 7 ÷ 20 = 0.35, and 0.35 × 100 = 35%.
  8. 4/5. 0.8 is 8/10, which reduces to 4/5.
  9. 3/8 ends and 5/12 repeats. Both fractions are already in lowest terms. The bottom number of 3/8 is 2 × 2 × 2, so 3/8 = 0.375. The bottom number of 5/12 is 2 × 2 × 3, and the 3 means it repeats, so 5/12 = 0.41666…

For more number practice, visit our math hub or try a set of math worksheets covering this and other skills.

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