Decimal Place Value and Rounding

Rounding starts with place value, not a guess about which digits look important. Find the place requested, inspect the next digit, and then decide whether the target digit changes.

Read the decimal places

The first digit to the right of the decimal point is the tenths place. The next is hundredths, followed by thousandths. In 8.374, the 3 is tenths, the 7 is hundredths, and the 4 is thousandths. A zero can hold a place even when it contributes no value, as in 8.04.

The rounding rule

1. Locate the requested place.
2. Look at the digit immediately to its right.
3. If that digit is less than 5, keep the target digit the same.
4. If that digit is 5 or more, add 1 to the target digit.
5. Drop the digits to the right, unless the problem asks you to retain fixed decimal places.

Examples

Round 8.374 to the nearest hundredth. The hundredths digit is 7; the next digit is 4, so the result is 8.37. Round 8.376 to the nearest hundredth: the next digit is 6, so increase 7 to 8 and write 8.38.

Carrying across a decimal place

If the target digit is 9 and must increase, carry to the place on its left. For example, 2.996 rounded to the nearest hundredth becomes 3.00. The value is three; the zeros show that the requested precision is hundredths.

Rounding whole numbers and decimals

The same rule works to the left or right of the decimal point. To round 6,482 to the nearest hundred, locate the 4 in the hundreds place and inspect the tens digit 8; increase the hundreds digit and replace later places with zeroes to get 6,500. To round 8.376 to the nearest hundredth, inspect the thousandths digit 6 and increase the hundredths digit, giving 8.38. The digit immediately after the target place controls the decision; do not inspect only the final digit of the number.

Carry when the target digit is nine

If the target digit is 9 and the next digit requires rounding up, regroup just as in addition. For 1.998 rounded to the nearest hundredth, the hundredths digit 9 increases, carrying through the tenths place and the ones place. The result is 2.00. The trailing zeroes communicate that the answer is expressed to the nearest hundredth, even though 2 and 2.00 have the same numerical value.

Keep the requested precision

If a problem asks for the nearest tenth, write one digit after the decimal; if it asks for the nearest hundredth, write two. In money or measurement contexts, trailing zeroes can be meaningful because they show the requested place. Do not round an intermediate calculation unless instructed. If you round 12.46 to 12.5 and then use 12.5 in another step, the final result may differ from calculating with 12.46 and rounding once.

Round negative numbers by magnitude

For ordinary school rounding to the nearest place, determine which neighboring values are closer on the number line. For example, −3.46 rounded to the nearest tenth is −3.5 because it lies closer to −3.5 than −3.4. Rounding a negative value is not the same as dropping digits toward zero. When a problem specifies a special tie-breaking convention, such as round-half-to-even, follow that convention instead of assuming a default.

Estimate to confirm the result

After rounding, compare the result with the original. A number rounded to the nearest tenth should differ by no more than about 0.05 under the usual nearest-value rule. Rounding 8.376 to the nearest tenth gives 8.4; 8.37 would be hundredth-level precision, not tenth-level. In a word problem, the appropriate place may come from the measurement or decision being made, not simply from the number of digits displayed.

Identify the place you are rounding to

Each position in a decimal has a value: tenths, hundredths, thousandths, and so on. To round to a stated place, locate that digit and inspect the digit immediately to its right. If the next digit is 5 or greater, increase the target digit by 1; if it is 4 or less, leave it unchanged. Then remove the remaining digits or replace them with zeros when working with a whole-number place.

For 8.376 rounded to the nearest hundredth, the hundredths digit is 7 and the thousandths digit is 6, so round up to 8.38. Rounded to the nearest tenth, look at the hundredths digit 7 and obtain 8.4. Keep trailing zeros when they communicate precision or are needed to align place values, as in 4.50 compared with 4.5.

Round whole numbers and negative values carefully

To round 3,482 to the nearest hundred, identify the hundreds digit 4 and inspect the tens digit 8. Increase the hundreds digit to 5 and change the following digits to zero: 3,500. To round to the nearest thousand, inspect the hundreds digit 4 and keep 3,000. The place requested determines which digit is checked.

For negative numbers, rounding direction can be described differently across conventions, especially for exact halfway values. Follow the method stated in the problem or course materials. For ordinary decimal rounding away from a halfway boundary, round the magnitude by the stated digit and retain the negative sign: −2.36 to the nearest tenth becomes −2.4. If a financial or statistical context specifies a tie-breaking convention, use that convention.

Use estimation to check calculations

Round numbers before a calculation when an estimate is requested, but use exact values for an exact answer. If 49.8 × 19.9 is estimated, 50 × 20 = 1,000 is reasonable. The exact product is 991.02. An estimate can show whether a misplaced decimal or arithmetic error produced an implausible result.

Do not round intermediate values too early in a multi-step problem because small rounding errors can compound. Keep several extra digits during calculations and round the final answer to the requested place. In measurement contexts, the stated precision may also reflect instrument accuracy, so avoid reporting more certainty than the data support.

• Find the target digit and inspect the digit immediately to its right.
• Keep or increase the target according to the next digit.
• Use place value to determine how many trailing zeros are needed.
• Follow any stated convention for negative values and halfway cases.
• Estimate first and round only the final result unless an estimate is requested.

Exam takeaway

Name the target place, inspect the digit immediately to its right, and carry when needed. Do not confuse the digit being rounded with the digit that determines the rounding.