What Is Rounding?

Rounding means replacing a number with a nearby value that is simpler to use.

For example:

6.83

rounded to the nearest whole number is:

7

The rounded number is not exactly equal to the original number.

Instead:

6.83 ≈ 7

The symbol:

≈

means approximately equal to.

Rounding is useful when an exact value is unnecessary or when we want to estimate a calculation quickly.

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Why Do We Round Numbers?

Imagine a journey is exactly:

397.8 km

In conversation, someone might say:

about 400 km

The rounded value communicates the approximate distance more simply.

We use rounding when:

• estimating costs
• estimating distances
• predicting calculation results
• summarizing measurements
• checking answers
• communicating data at an appropriate level of precision

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Decimal Place Values

Before rounding decimals, identify the place values.

Consider:

24.6837

The digits represent:

• 2 tens
• 4 ones
• 6 tenths
• 8 hundredths
• 3 thousandths
• 7 ten-thousandths

If we are asked to round this number, we first identify the rounding place.

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Common Rounding Places

Decimals are commonly rounded to the nearest:

• whole number
• tenth
• hundredth
• thousandth

For example:

7.486

could be rounded differently depending on the required precision.

Nearest whole number:

7

Nearest tenth:

7.5

Nearest hundredth:

7.49

Nearest thousandth:

7.486

The requested place value matters.

---

The Basic Rounding Rule

To round a decimal:

Step 1: Identify the place value you are rounding to.

Step 2: Look at the digit immediately to its right.

Step 3: If that digit is 0, 1, 2, 3, or 4, keep the rounding digit unchanged.

Step 4: If that digit is 5, 6, 7, 8, or 9, increase the rounding digit by 1.

Step 5: Remove the digits to the right.

A common memory rule is:

0–4: keep

5–9: round up

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Rounding to the Nearest Whole Number

Consider:

8.3

The ones digit is:

8

Look at the tenths digit:

3

Since 3 is less than 5, keep the 8.

Therefore:

8.3 ≈ 8

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Another Whole-Number Example

Round:

8.7

to the nearest whole number.

Look at the tenths digit:

7

Since 7 is 5 or greater, increase the ones digit.

Therefore:

8.7 ≈ 9

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Why the Rule Works

Rounding is really about determining which benchmark is closer.

Consider:

4.2

It lies between:

4 and 5

The distance from 4.2 to 4 is:

0.2

The distance from 4.2 to 5 is:

0.8

Therefore, 4.2 is closer to 4.

So:

4.2 ≈ 4

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Rounding on a Number Line

Consider:

6.7

It lies between:

6 and 7

The halfway point is:

6.5

Since 6.7 lies above the halfway point, it is closer to 7.

Therefore:

6.7 ≈ 7

The number line helps explain why 5 is important in the rounding rule.

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Rounding to the Nearest Tenth

Consider:

4.36

We want to round to the nearest tenth.

The tenths digit is:

3

Look one place to the right, at the hundredths digit:

6

Since:

6 ≥ 5

increase the tenths digit from 3 to 4.

Therefore:

4.36 ≈ 4.4

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Another Tenths Example

Round:

7.82

to the nearest tenth.

Tenths digit:

8

Hundredths digit:

2

Since:

2 < 5

keep the tenths digit.

Therefore:

7.82 ≈ 7.8

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Number Line for Tenths

Consider:

3.47

To round to the nearest tenth, compare it with:

3.4 and 3.5

The halfway point is:

3.45

Since:

3.47 > 3.45

3.47 is closer to 3.5.

Therefore:

3.47 ≈ 3.5

---

Rounding to the Nearest Hundredth

Consider:

5.273

We want the nearest hundredth.

Hundredths digit:

7

Look at the thousandths digit:

3

Since:

3 < 5

keep the 7.

Therefore:

5.273 ≈ 5.27

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Another Hundredths Example

Round:

5.278

to the nearest hundredth.

Hundredths digit:

7

Thousandths digit:

8

Since:

8 ≥ 5

increase 7 to 8.

Therefore:

5.278 ≈ 5.28

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Rounding to the Nearest Thousandth

Consider:

12.4837

We want the nearest thousandth.

Thousandths digit:

3

Look at the ten-thousandths digit:

7

Since:

7 ≥ 5

increase the 3 to 4.

Therefore:

12.4837 ≈ 12.484

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Worked Example 1

Round:

18.746

to the nearest whole number.

Look at the tenths digit:

7

Round up.

18.746 ≈ 19

---

Worked Example 2

Round:

18.746

to the nearest tenth.

Tenths digit:

7

Hundredths digit:

4

Keep the 7.

18.746 ≈ 18.7

---

Worked Example 3

Round:

18.746

to the nearest hundredth.

Hundredths digit:

4

Thousandths digit:

6

Round up.

18.746 ≈ 18.75

Notice that the same number can produce different rounded values depending on the requested place value.

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When Rounding Causes Regrouping

Sometimes rounding causes a digit to become 10.

Consider:

3.98

rounded to the nearest tenth.

The tenths digit is:

9

The hundredths digit is:

8

So the 9 rounds up.

But:

9 + 1 = 10

Therefore, regroup:

3.98 ≈ 4.0

to the nearest tenth.

---

Another Regrouping Example

Round:

9.997

to the nearest hundredth.

Hundredths digit:

9

Thousandths digit:

7

Round up.

This causes regrouping through several places.

Therefore:

9.997 ≈ 10.00

to the nearest hundredth.

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Precision Matters

Compare:

4

4.0

4.00

Numerically, these have the same value.

However, in measurement contexts they can communicate different levels of precision.

For example:

4.00 m

suggests the measurement has been reported to the nearest hundredth of a metre.

This becomes particularly important in science and engineering.

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What Is Estimation?

Estimation means finding an approximate answer rather than an exact answer.

Suppose:

4.87 + 3.12

Instead of calculating exactly, round:

4.87 ≈ 5

3.12 ≈ 3

Then:

5 + 3 = 8

So:

4.87 + 3.12 ≈ 8

The exact answer is:

7.99

Our estimate is very close.

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Estimation and Rounding

Rounding is one of the most common ways to estimate calculations.

For example:

12.78 + 6.14

Round to whole numbers:

13 + 6

Estimate:

19

Exact:

18.92

The estimate gives us a useful idea of the expected answer.

---

Estimating Decimal Addition

Estimate:

7.86 + 4.21

Round to whole numbers:

7.86 ≈ 8

4.21 ≈ 4

Then:

8 + 4 = 12

So the estimate is:

12

Exact calculation:

7.86 + 4.21 = 12.07

The estimate is reasonable.

---

Estimating Decimal Subtraction

Estimate:

15.73 − 8.16

Round:

15.73 ≈ 16

8.16 ≈ 8

Then:

16 − 8 = 8

Exact calculation:

15.73 − 8.16 = 7.57

The exact answer is reasonably close to the estimate.

---

Estimating Decimal Multiplication

Estimate:

6.18 × 3.9

Round:

6.18 ≈ 6

3.9 ≈ 4

Then:

6 × 4 = 24

Exact calculation:

6.18 × 3.9 = 24.102

Our estimate of:

24

is very close.

---

Estimating Decimal Division

Estimate:

19.8 ÷ 4.1

Use nearby compatible numbers:

20 ÷ 4 = 5

Therefore:

19.8 ÷ 4.1 ≈ 5

The exact quotient is approximately:

4.83

So the estimate is reasonable.

---

Compatible Numbers

Sometimes standard rounding is not the easiest estimation strategy.

Instead, use compatible numbers: nearby values that are easy to calculate mentally.

For example:

24.7 ÷ 4.9

Rather than simply rounding each value according to a fixed place, notice:

24.7 ≈ 25

and:

4.9 ≈ 5

Then:

25 ÷ 5 = 5

So:

24.7 ÷ 4.9 ≈ 5

---

Front-End Estimation

Another strategy is front-end estimation.

This focuses on the largest place values first.

For:

43.72 + 28.46

use:

40 + 20 = 60

Then consider the remaining amounts to improve the estimate.

A better estimate might be:

44 + 28 = 72

The exact answer is:

72.18

Front-end estimation can be useful when a quick approximation is more important than high precision.

---

Overestimates and Underestimates

An estimate can be slightly above or below the exact answer.

Suppose:

4.2 + 6.3

Round both to whole numbers:

4 + 6 = 10

Exact:

10.5

The estimate is lower than the exact answer.

This is an underestimate.

---

Overestimate Example

Consider:

4.8 + 6.7

Round:

5 + 7 = 12

Exact:

11.5

The estimate is higher than the exact answer.

This is an overestimate.

---

Checking Reasonableness

One of the most important uses of estimation is checking whether an exact calculation makes sense.

Suppose someone calculates:

6.24 × 3.1 = 193.44

Estimate:

6 × 3 = 18

The exact answer should be somewhere near:

18 or 20

An answer of:

193.44

is far too large.

Therefore, the calculation is unreasonable.

The actual answer is:

19.344

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Another Reasonableness Check

Suppose:

42.6 ÷ 6.1

A student calculates:

69.8

Estimate:

42 ÷ 6 = 7

The quotient should be around:

7

not:

70

So the answer is unreasonable.

This type of error often comes from incorrect decimal placement.

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Decimal Placement and Estimation

Estimation is especially useful for detecting incorrectly placed decimal points.

Suppose:

3.8 × 4.2

We know:

4 × 4 ≈ 16

So the answer should be near:

16

If a calculator entry or written calculation gives:

159.6

or:

1.596

we know the decimal point is probably misplaced.

The exact answer is:

15.96

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How Accurate Should an Estimate Be?

The appropriate precision depends on the purpose.

Suppose an item costs:

$49.95

If you simply want a quick idea of the price:

about $50

is useful.

If you are calculating whether you have enough money to pay exactly, an estimate may not be sufficient.

The context determines how much precision is needed.

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When Estimates Are Appropriate

Estimates are useful when:

• making quick comparisons
• planning a budget
• predicting costs
• checking calculations
• estimating travel distances
• planning quantities
• interpreting measurements
• checking calculator results
• exact values are unavailable
• an approximate answer is sufficient

---

When Exact Answers Are Needed

An estimate may not be enough when:

• paying an exact bill
• calculating medication quantities
• performing precise scientific measurements
• manufacturing components to exact specifications
• reporting financial records
• calculating final assessment scores
• exact legal or contractual values are required

The required precision depends on the situation.

---

Real-World Application: Shopping

Suppose you buy:

• an item for $12.89
• an item for $7.35
• an item for $19.76

You want to know approximately how much money you need.

Round:

$12.89 ≈ $13

$7.35 ≈ $7

$19.76 ≈ $20

Estimate:

$13 + $7 + $20 = $40

The exact total is:

$40.00

In this case, the estimate happens to match the exact total.

---

Real-World Application: Budgeting

Suppose your weekly expenses are:

$48.75

$32.40

$21.95

$17.80

For quick planning, round:

49 + 32 + 22 + 18

Estimate:

$121

The exact total is:

$120.90

The estimate is useful for planning.

---

Real-World Application: Travel

A journey has three sections:

48.7 km

31.4 km

22.8 km

Estimate:

49 + 31 + 23

= 103 km

Exact distance:

102.9 km

So describing the trip as:

about 103 km

is reasonable.

---

Real-World Application: Measurement

A room measures:

5.92 m × 3.87 m

Estimate the area.

Round:

5.92 ≈ 6

3.87 ≈ 4

Then:

6 × 4 = 24

Estimated area:

24 m²

Exact area:

5.92 × 3.87 = 22.9104 m²

The estimate tells us the area should be around:

23–24 m²

---

Real-World Application: Fuel

A car uses approximately:

7.8 L

of fuel per 100 km.

For a quick estimate over:

400 km

think:

about 8 L per 100 km

There are:

4 groups of 100 km

So estimated fuel use:

8 × 4 = 32 L

Using the original rate:

7.8 × 4 = 31.2 L

The estimate is useful for travel planning.

---

Real-World Application: Science

Suppose three measured masses are:

12.47 g

8.96 g

6.62 g

For a quick estimate:

12.47 ≈ 12.5

8.96 ≈ 9.0

6.62 ≈ 6.6

Then:

12.5 + 9.0 + 6.6 = 28.1 g

Depending on the purpose, using tenths may provide a more useful estimate than rounding everything to whole numbers.

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Choosing the Best Level of Precision

Suppose we want to estimate:

18.46 + 7.82

Rounding to whole numbers:

18 + 8 = 26

Rounding to tenths:

18.5 + 7.8 = 26.3

Exact:

26.28

Both estimates are reasonable.

The second estimate is more precise, but it also requires slightly more work.

A good estimate balances:

speed and usefulness

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Worked Example 1: Rounding

Round:

14.672

to the nearest tenth.

Tenths digit:

6

Hundredths digit:

7

Round up:

14.672 ≈ 14.7

---

Worked Example 2: Rounding

Round:

14.672

to the nearest hundredth.

Hundredths digit:

7

Thousandths digit:

2

Keep the 7.

14.672 ≈ 14.67

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Worked Example 3: Estimating Addition

Estimate:

28.74 + 16.19

Round to whole numbers:

29 + 16 = 45

Exact:

44.93

Therefore:

45

is a good estimate.

---

Worked Example 4: Estimating Subtraction

Estimate:

63.82 − 28.14

Round:

64 − 28 = 36

Exact:

35.68

The estimate is reasonable.

---

Worked Example 5: Estimating Multiplication

Estimate:

9.72 × 5.13

Use:

10 × 5 = 50

Exact:

49.8636

So:

50

is an excellent estimate.

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Worked Example 6: Estimating Division

Estimate:

61.4 ÷ 9.8

Use compatible numbers:

60 ÷ 10 = 6

Exact quotient is approximately:

6.27

Therefore, the estimate is reasonable.

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Worked Example 7: Checking Reasonableness

A student claims:

8.12 + 4.73 = 128.5

Estimate:

8 + 5 = 13

An answer near:

13

is expected.

Therefore:

128.5

is unreasonable.

Correct calculation:

8.12 + 4.73 = 12.85

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Worked Example 8: Practical Estimation

A restaurant bill contains:

$18.95

$12.40

$7.75

$5.10

Estimate:

19 + 12 + 8 + 5

= $44

Exact:

$44.20

Therefore, about:

$44

is a useful estimate.

---

Worked Example 9: Choosing Precision

A scientist records:

12.6847 cm

Rounded to the nearest:

whole number: 13 cm

tenth: 12.7 cm

hundredth: 12.68 cm

thousandth: 12.685 cm

The appropriate result depends on the precision required.

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Worked Example 10: Comparing Estimate and Exact Answer

Calculate:

24.86 × 3.12

Estimate:

25 × 3 = 75

Exact:

24.86 × 3.12 = 77.5632

The exact answer is reasonably close to the estimate.

Therefore, the result appears sensible.

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Estimation Before Using a Calculator

Estimation is valuable even when calculators are available.

Before entering:

48.72 × 6.14

estimate:

50 × 6 = 300

If the calculator returns:

299.0208

the result is reasonable.

If it returns:

29.90208

you should check your input.

A calculator can perform arithmetic, but it does not know whether you entered the intended calculation.

---

Estimation Is Not Guessing

An estimate should be based on mathematical reasoning.

A random guess:

"Maybe the answer is about 50."

is not the same as an estimate.

A mathematical estimate explains why:

24.8 + 26.1 ≈ 25 + 26 = 51

The estimate is supported by nearby values and a calculation.

---

Common Mistakes

Mistake 1: Looking at the wrong digit

To round to the hundredths place, look at the:

thousandths digit

not the tenths digit.

---

Mistake 2: Changing digits before the rounding place

Only the rounding digit may increase by 1.

Other digits to its left remain unchanged unless regrouping occurs.

---

Mistake 3: Thinking 5 always means the whole number increases

The 5 affects only the digit in the place being rounded.

---

Mistake 4: Forgetting regrouping

For:

7.98

rounded to the nearest tenth:

7.98 ≈ 8.0

not:

7.10

---

Mistake 5: Treating an estimate as an exact answer

Use:

≈

rather than:

=

when writing an approximate value.

---

Mistake 6: Using too much or too little precision

Rounding:

$19.97

to:

$0

would be far too rough for most shopping situations.

Choose a useful level of precision.

---

Mistake 7: Accepting unreasonable calculator answers

Always compare the exact result with an approximate result.

---

Error Analysis

A student rounds:

6.483

to the nearest hundredth and writes:

6.5

This is incorrect.

The hundredths digit is:

8

The digit immediately to its right is:

3

Since 3 is less than 5, keep the 8.

Therefore:

6.483 ≈ 6.48

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Another Error Analysis

A student calculates:

7.9 × 4.2 = 331.8

Estimate:

8 × 4 = 32

The calculated answer:

331.8

is about ten times too large.

Correct:

7.9 × 4.2 = 33.18

Estimation helps identify the decimal-placement error.

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A Reliable Rounding Strategy

When rounding:

Step 1: Identify the requested place value.

Step 2: Underline or identify the digit in that place.

Step 3: Look at the digit immediately to its right.

Step 4: If it is 0–4, keep the rounding digit.

Step 5: If it is 5–9, increase the rounding digit by 1.

Step 6: Remove digits to the right.

Step 7: Check whether the rounded result makes sense.

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A Reliable Estimation Strategy

When estimating a calculation:

Step 1: Examine the numbers.

Step 2: Decide how accurate the estimate needs to be.

Step 3: Round or choose compatible numbers.

Step 4: Perform the simpler calculation.

Step 5: State that the result is approximate.

Step 6: If an exact answer is available, compare it with the estimate.

Step 7: Investigate if the exact result is unexpectedly far from the estimate.

---

Did You Know?

Rounding always involves a trade-off between simplicity and precision.

Consider:

17.4836

We could report it as:

17

17.5

17.48

17.484

Each rounded value is easier to communicate than the original to a different degree, but each also preserves a different amount of information.

There is therefore no single "best" place to round every number.

The appropriate precision depends on why the number is being used.

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Key Terms

• Rounding: Replacing a number with a nearby value at a specified place value.
• Estimate: Approximate value based on mathematical reasoning.
• Approximate: Close to the exact value but not necessarily equal to it.
• Exact value: Precise result of a calculation or measurement.
• Place value: Value of a digit based on its position.
• Tenths: First decimal place.
• Hundredths: Second decimal place.
• Thousandths: Third decimal place.
• Compatible numbers: Nearby numbers chosen because they are easy to calculate with.
• Front-end estimation: Estimating primarily using the largest place values.
• Overestimate: Estimate greater than the exact value.
• Underestimate: Estimate less than the exact value.
• Reasonableness: Whether an answer makes sense based on the original quantities.
• Precision: Level of detail with which a value is expressed.
• Benchmark: Familiar or convenient value used for comparison.

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Key Rules

To round to a specified place:

0, 1, 2, 3, 4 → keep the rounding digit

5, 6, 7, 8, 9 → increase the rounding digit by 1

Look only at the digit immediately to the right of the place being rounded.

Examples:

4.32 ≈ 4.3 to the nearest tenth

4.37 ≈ 4.4 to the nearest tenth

8.264 ≈ 8.26 to the nearest hundredth

8.268 ≈ 8.27 to the nearest hundredth

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Key Takeaways

• Rounding replaces a number with a nearby value that is easier to use.
• The requested place value determines how a decimal should be rounded.
• Common rounding positions include whole numbers, tenths, hundredths, and thousandths.
• To round, examine the digit immediately to the right of the rounding place.
• Digits 0–4 cause the rounding digit to remain unchanged.
• Digits 5–9 cause the rounding digit to increase by 1.
• Number lines help explain rounding as choosing the nearest benchmark.
• Rounding can sometimes cause regrouping, such as 3.98 ≈ 4.0 to the nearest tenth.
• Estimation provides an approximate result rather than an exact answer.
• Rounding and compatible numbers are useful estimation strategies.
• Addition, subtraction, multiplication, and division with decimals can all be estimated.
• Estimation is useful for checking the reasonableness of exact calculations.
• Estimation is especially valuable for detecting misplaced decimal points.
• Estimates can be overestimates or underestimates.
• The amount of rounding should match the purpose of the calculation.
• Some situations require exact answers, while others only require useful approximations.
• Estimation is mathematical reasoning, not random guessing.
• Calculator answers should still be checked using estimation.
• Rounding and estimation are widely used in shopping, budgeting, travel, measurement, science, engineering, and everyday decision-making.
• A strong decimal calculation includes not only an accurate answer but also an understanding of whether that answer is reasonable.