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๐Ÿ”ข Real Numbers โ€ข Lesson 16 of 19

Estimation and Approximation

Learn how to round numbers accurately, estimate calculations and decide whether an answer is sensible.

๐Ÿ”ข Real Numbers โœ๏ธ Worked examples included ๐Ÿง  Step-by-step learning

๐ŸŽฌ See an estimate form

Round to convenient values before estimating.

๐ŸŽฏ By the end of this lesson...

You should be able to:

๐Ÿค” Think About This...

๐Ÿ›’

Shopping without a calculator

You want to buy three items costing:

Item 1 US$4.87
Item 2 US$9.62
Item 3 US$6.18

You do not need the exact total immediately.

You only need to know whether US$25 is enough.

US$4.87 โ‰ˆ US$5

US$9.62 โ‰ˆ US$10

US$6.18 โ‰ˆ US$6
Estimated total = 5 + 10 + 6 = US$21
US$25 should be enough. Estimation gives a quick, sensible answer without requiring the exact calculation.

1๏ธโƒฃ Exact and Approximate Values

An exact value gives the complete value without rounding.

An approximate value is close to the exact value but has been rounded or estimated.

Exact value

23 รท 7 = 3.285714285714...

Approximate value

23 รท 7 โ‰ˆ 3.29
The symbol โ‰ˆ means โ€œapproximately equal to.โ€

Do not use the ordinary equals sign when the value has been rounded.

2๏ธโƒฃ Approximation and Estimation

Approximation

Replacing a value with another value that is close to it.

8.763 โ‰ˆ 8.8

The original number is known, but it is written less precisely.

Estimation

Using convenient approximate numbers to predict the answer to a calculation or situation.

49 ร— 21 โ‰ˆ 50 ร— 20
โ‰ˆ 1000
Approximation changes a value into a nearby, simpler value.

Estimation uses approximate values to obtain a sensible answer.

๐Ÿ’ก Did You Know?

People use estimation every day, often without realising it.

โฐ Estimating travel time
๐Ÿ›’ Estimating shopping costs
๐Ÿ—๏ธ Estimating materials
๐ŸŒพ Estimating crop yield
๐Ÿ‘ฅ Estimating crowd size
โ›ฝ Estimating fuel required

3๏ธโƒฃ The Basic Rounding Rule

To round a number, look at the digit immediately to the right of the place being rounded.

0, 1, 2, 3 or 4

Round down

Keep the rounding digit unchanged.

5, 6, 7, 8 or 9

Round up

Increase the rounding digit by 1.

Identify the rounding digit
โ†“
Look at the next digit
โ†“
0โ€“4 Leave it unchanged
5โ€“9 Add 1

4๏ธโƒฃ Rounding Whole Numbers to a Place Value

A whole number may be rounded to the nearest ten, hundred, thousand or another stated place value.

Thousands 4
Hundreds 7
Tens 6
Ones 3
Number: 4763

5๏ธโƒฃ Rounding to the Nearest Ten

Example 1: Round 4763 to the nearest ten

The tens digit is 6.

Look at the ones digit:

4 7 6 3

Since 3 is less than 5, keep the tens digit unchanged.

Replace the digits after the tens place with zero.

4763 rounded to the nearest ten is 4760.

Example 2: Round 4787 to the nearest ten

The ones digit is 7, so increase the tens digit by 1.

4787 rounded to the nearest ten is 4790.

6๏ธโƒฃ Rounding to the Nearest Hundred

Example 3: Round 6437 to the nearest hundred

The hundreds digit is 4.

The next digit is the tens digit, which is 3.

3 < 5

Keep the hundreds digit unchanged and replace later digits with zeros.

6437 rounded to the nearest hundred is 6400.

Example 4: Round 6782 to the nearest hundred

The hundreds digit is 7 and the next digit is 8.

8 โ‰ฅ 5
6782 rounded to the nearest hundred is 6800.

7๏ธโƒฃ Rounding to the Nearest Thousand

Example 5: Round 34 482 to the nearest thousand

The thousands digit is 4.

The next digit is 4, so round down.

34 482 rounded to the nearest thousand is 34 000.

Example 6: Round 34 782 to the nearest thousand

The next digit is 7, so round up.

34 782 rounded to the nearest thousand is 35 000.

๐Ÿง  StudyNest Method: Rounding to a Place Value

Step 1: Locate the place value named in the question.

Step 2: Underline or identify that digit.

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

Step 4: If the next digit is 0โ€“4, leave the rounding digit unchanged.

Step 5: If the next digit is 5โ€“9, increase the rounding digit by 1.

Step 6: Replace every whole-number digit to the right with zero.

8๏ธโƒฃ What Are Decimal Places?

Decimal places count digits after the decimal point.

27
.
4 1st decimal place
8 2nd decimal place
6 3rd decimal place
2 4th decimal place
Number: 27.4862

9๏ธโƒฃ Rounding to Decimal Places

Example 7: Round 27.4862 to 2 decimal places

The second decimal digit is 8.

Look at the third decimal digit, which is 6.

6 โ‰ฅ 5

Increase 8 by 1.

27.4862 rounded to 2 decimal places is 27.49.

Example 8: Round 3.14159 to 3 decimal places

Keep 3 decimal digits:

3.141

The next digit is 5, so round the third decimal digit up.

3.14159 rounded to 3 decimal places is 3.142.

Example 9: Round 8.3041 to 2 decimal places

The second decimal digit is 0.

The next digit is 4, so the 0 remains unchanged.

The answer is 8.30.
The final zero must be included because the question asks for 2 decimal places.

๐Ÿง  StudyNest Method: Decimal Places

Step 1: Begin counting immediately after the decimal point.

Step 2: Locate the required decimal place.

Step 3: Look at the next digit.

Step 4: Apply the 0โ€“4 or 5โ€“9 rounding rule.

Step 5: Remove all digits after the required place.

Step 6: Keep ending zeros when they are needed to show the requested number of decimal places.

๐Ÿ”Ÿ What Are Significant Figures?

Significant figures begin at the first non-zero digit.

Start counting from the first digit that is not zero.
5832

5 is the first significant figure.

0.00472

4 is the first significant figure.

70.3

7 is the first significant figure.

Significant figures do not always begin immediately after the decimal point. They begin at the first non-zero digit.

1๏ธโƒฃ1๏ธโƒฃ Which Zeros Are Significant?

Leading zeros

0.0047

The zeros before 4 only locate the decimal point. They are not significant.

Zeros between non-zero digits

1007

Both zeros are significant.

Ending decimal zeros

4.500

The ending zeros show precision and are significant.

Ending whole-number zeros

4500

Without more information, their significance may be unclear. Standard form removes the ambiguity.

For example:

4.5 ร— 10ยณ has 2 significant figures.

4.500 ร— 10ยณ has 4 significant figures.

1๏ธโƒฃ2๏ธโƒฃ Counting Significant Figures

Example 10: How many significant figures are in 0.00608?

Ignore the leading zeros.

Significant digits: 6, 0 and 8
0.00608 has 3 significant figures.

Example 11: How many significant figures are in 12.040?

Begin at 1. Every digit from that point is significant.

1, 2, 0, 4, 0
12.040 has 5 significant figures.

1๏ธโƒฃ3๏ธโƒฃ Rounding to Significant Figures

Example 12: Round 5832 to 2 significant figures

The first two significant digits are 5 and 8.

The next digit is 3, so round down.

5832 โ†’ 5800
5832 to 2 significant figures is 5800.

Example 13: Round 7864 to 3 significant figures

The first three significant digits are 7, 8 and 6.

The next digit is 4, so round down.

7864 to 3 significant figures is 7860.

Example 14: Round 0.004768 to 2 significant figures

Start at the first non-zero digit, which is 4.

The first two significant digits are 4 and 7.

The next digit is 6, so increase 7 to 8.

0.004768 to 2 significant figures is 0.0048.

๐Ÿง  StudyNest Method: Significant Figures

Step 1: Ignore leading zeros and locate the first non-zero digit.

Step 2: Count significant digits from that point.

Step 3: Stop at the number of significant figures requested.

Step 4: Look at the next digit.

Step 5: Apply the rounding rule.

Step 6: For a whole number, replace later digits with zeros.

Step 7: For a decimal, remove later digits while keeping the decimal point in its correct position.

1๏ธโƒฃ4๏ธโƒฃ Decimal Places and Significant Figures Compared

Decimal places

Count digits after the decimal point.

0.004768 to 2 d.p.
= 0.00

Significant figures

Begin counting at the first non-zero digit.

0.004768 to 2 s.f.
= 0.0048
Decimal places and significant figures are not interchangeable. Read the instruction carefully.

1๏ธโƒฃ5๏ธโƒฃ Choosing a Suitable Degree of Accuracy

The most suitable accuracy depends on how the number will be used.

๐Ÿ‘ฅ

Population

A population may be reported to the nearest hundred or thousand.

๐Ÿ’ฐ

Money

Money is commonly written to 2 decimal places when cents are used.

๐Ÿ“

Length

A length may be measured to the nearest metre, centimetre or millimetre depending on the purpose.

โš–๏ธ

Mass

Medicine requires greater precision than estimating the mass of a bag of maize.

More decimal places or significant figures give greater precision, but unnecessary precision can be misleading.

1๏ธโƒฃ6๏ธโƒฃ Estimating a Sum

Example 15: Estimate 48.7 + 31.6 + 19.4

Round each number to 1 significant figure.

48.7 โ‰ˆ 50
31.6 โ‰ˆ 30
19.4 โ‰ˆ 20
Estimated total = 50 + 30 + 20
Estimated answer = 100.

1๏ธโƒฃ7๏ธโƒฃ Estimating a Product

Example 16: Estimate 49.2 ร— 20.7

Round each value to 1 significant figure.

49.2 โ‰ˆ 50
20.7 โ‰ˆ 20
50 ร— 20 = 1000
49.2 ร— 20.7 is approximately 1000.

1๏ธโƒฃ8๏ธโƒฃ Estimating a Quotient

Example 17: Estimate 598 รท 19.8

Choose nearby numbers that are easy to divide.

598 โ‰ˆ 600
19.8 โ‰ˆ 20
600 รท 20 = 30
598 รท 19.8 is approximately 30.

1๏ธโƒฃ9๏ธโƒฃ Estimating a More Complex Calculation

Example 18

Estimate:

(19.7 ร— 4.12) รท 0.203

Round each number to 1 significant figure.

19.7 โ‰ˆ 20
4.12 โ‰ˆ 4
0.203 โ‰ˆ 0.2
(20 ร— 4) รท 0.2
80 รท 0.2 = 400
The estimated answer is 400.

๐Ÿง  StudyNest Method: Estimating a Calculation

Step 1: Identify all the numbers in the calculation.

Step 2: Round each number to a convenient degree of accuracy, usually 1 significant figure unless instructed otherwise.

Step 3: Rewrite the complete calculation using the rounded values.

Step 4: Calculate mentally or using simple working.

Step 5: Use the approximately-equal symbol.

Step 6: Check that the estimated answer is of a sensible size.

2๏ธโƒฃ0๏ธโƒฃ Estimation as a Calculator Check

Example 19

A learner enters:

39.8 ร— 5.12

The calculator displays:

2037.76

Estimate first:

39.8 โ‰ˆ 40
5.12 โ‰ˆ 5
40 ร— 5 = 200
An answer near 200 is expected, so 2037.76 is not sensible. The learner probably entered the calculation incorrectly.
The exact answer is 203.776, which agrees with the estimate.

๐ŸŒ Real-Life Example: Estimating a Journey

Example 20

A journey is 287 km long.

A driver travels at an average speed of about 58 km/h.

Estimate the journey time.

287 km โ‰ˆ 300 km
58 km/h โ‰ˆ 60 km/h
Time โ‰ˆ 300 รท 60
The journey should take approximately 5 hours.

๐ŸŒ Real-Life Example: Attendance

Example 21

A stadium contains 19 684 spectators.

Give a suitable approximate attendance for a news report.

19 684 โ‰ˆ 20 000
A reporter might say:

โ€œApproximately 20 000 spectators attended.โ€

๐ŸŒ Real-Life Example: Money

Example 22

A bill is calculated as US$18.476.

Money is normally given to the nearest cent.

Round to 2 decimal places.

US$18.476 โ‰ˆ US$18.48.

๐Ÿง  The StudyNest Accuracy Strategy

Step 1: Identify what the question requires:

place value, decimal places, significant figures or an estimate.

Step 2: Locate the correct rounding digit.

Step 3: Look at only the next digit.

Step 4: Apply the 0โ€“4 or 5โ€“9 rule.

Step 5: Write the answer in the format requested.

Step 6: Use โ‰ˆ when the answer is approximate.

Step 7: Ask whether the answer is sensible in its context.

๐ŸŽฏ Exam Tips

For decimal places, count from the decimal point.
For significant figures, begin at the first non-zero digit.
Look at only the digit immediately after the required rounding place.
Keep ending zeros when they show the requested accuracy, such as 8.30 to 2 decimal places.
Estimation questions commonly use 1 significant figure unless another degree of accuracy is stated.
Do not round numbers too early in a calculation unless the question asks for an estimate.
Use an estimate to check whether a calculator answer has the correct size.

โš  Common Mistakes

Mistake 1:

Looking at several digits instead of only the next digit when rounding.
Mistake 2:

Counting significant figures from the decimal point instead of the first non-zero digit.
Mistake 3:

Removing a final zero that is needed to show the requested number of decimal places.
Mistake 4:

Using the equals sign for a rounded value instead of the approximately equal sign.
Mistake 5:

Rounding every number to the nearest whole number even when that makes the estimate unsuitable.
Mistake 6:

Giving an estimate with more detail than the original information justifies.
Mistake 7:

Accepting a calculator answer without checking whether it is sensible.

๐ŸŽฎ Your Turn!

๐ŸŸข Foundation Questions

  1. Round 374 to the nearest ten.
  2. Round 685 to the nearest ten.
  3. Round 4638 to the nearest hundred.
  4. Round 9752 to the nearest thousand.
  5. Round 45 381 to the nearest ten thousand.
  6. Round 7.386 to 1 decimal place.
  7. Round 12.745 to 2 decimal places.
  8. Round 9.0048 to 3 decimal places.

๐Ÿ”ต Developing Questions

  1. State the number of significant figures in 0.00408.
  2. State the number of significant figures in 12.030.
  3. Round 5837 to 2 significant figures.
  4. Round 76 492 to 3 significant figures.
  5. Round 0.006873 to 2 significant figures.
  6. Round 3.9951 to 3 significant figures.
  7. Estimate 62.1 + 38.7 + 19.6.
  8. Estimate 39.4 ร— 5.2.
  9. Estimate 804 รท 19.7.
  10. Estimate:

    (29.8 ร— 6.12) รท 0.301

๐Ÿ”ด Challenge Questions

  1. Round 0.00056789 to 3 significant figures.
  2. Round 987 654 to 2 significant figures.
  3. A calculator displays 48.675923. Give the answer to 3 decimal places.
  4. A mass is measured as 0.008046 kg. Give it to 3 significant figures.
  5. Estimate:

    โˆš63.7 ร— 19.8
  6. Estimate:

    (402 ร— 0.049) รท 1.98
  7. A shop sells 198 notebooks at US$2.97 each. Estimate the total value of the notebooks.
  8. A bus travels 452 km using 39.6 litres of fuel. Estimate the distance travelled per litre.
  9. A learner calculates 79.6 ร— 3.98 and obtains 31.6808. Use estimation to explain why this answer is incorrect.
  10. Explain the difference between rounding a number to 2 decimal places and rounding it to 2 significant figures.
โœ… Show Answers
  1. 374 โ‰ˆ 370
  2. 685 โ‰ˆ 690
  3. 4638 โ‰ˆ 4600
  4. 9752 โ‰ˆ 10 000
  5. 45 381 โ‰ˆ 50 000
  6. 7.386 โ‰ˆ 7.4
  7. 12.745 โ‰ˆ 12.75
  8. 9.0048 โ‰ˆ 9.005
  9. 0.00408 has 3 significant figures.
  10. 12.030 has 5 significant figures.
  11. 5837 โ‰ˆ 5800
  12. 76 492 โ‰ˆ 76 500
  13. 0.006873 โ‰ˆ 0.0069
  14. 3.9951 โ‰ˆ 4.00

    The zeros show 3 significant figures.
  15. 62.1 + 38.7 + 19.6

    โ‰ˆ 60 + 40 + 20

    = 120
  16. 39.4 ร— 5.2

    โ‰ˆ 40 ร— 5

    = 200
  17. 804 รท 19.7

    โ‰ˆ 800 รท 20

    = 40
  18. (29.8 ร— 6.12) รท 0.301

    โ‰ˆ (30 ร— 6) รท 0.3

    = 180 รท 0.3

    = 600
  19. 0.00056789 โ‰ˆ 0.000568
  20. 987 654 โ‰ˆ 990 000
  21. 48.675923 โ‰ˆ 48.676
  22. 0.008046 โ‰ˆ 0.00805 kg
  23. โˆš63.7 ร— 19.8

    โ‰ˆ โˆš64 ร— 20

    = 8 ร— 20

    = 160
  24. (402 ร— 0.049) รท 1.98

    โ‰ˆ (400 ร— 0.05) รท 2

    = 20 รท 2

    = 10
  25. 198 ร— 2.97

    โ‰ˆ 200 ร— 3

    = US$600
  26. 452 รท 39.6

    โ‰ˆ 450 รท 40

    โ‰ˆ 11.25 km per litre

    A rougher estimate of about 11 km/L is also reasonable.
  27. 79.6 ร— 3.98

    โ‰ˆ 80 ร— 4

    = 320

    Therefore, 31.6808 is far too small. The decimal point was probably entered or placed incorrectly.
  28. Decimal places count digits after the decimal point.

    Significant figures begin at the first non-zero digit, regardless of the decimal pointโ€™s position.

๐Ÿง  Remember This

Approximation replaces a value with a nearby simpler value.

Estimation uses approximate values to predict an answer.

For place value and decimal places, locate the requested position and inspect the next digit.

For significant figures, begin at the first non-zero digit.

Digits 0โ€“4 round down, while digits 5โ€“9 round up.

Use โ‰ˆ for approximate values.

Estimation helps you check whether an exact answer is sensible.

๐ŸŽฏ Before Moving On...

๐ŸŽ‰ Well Done!

You can now round values accurately and use estimation to make quick, sensible mathematical decisions.

Remember:

Identify the required accuracy โ†’ Locate the rounding digit โ†’ Check the next digit โ†’ Round correctly โ†’ Test whether the answer is sensible

๐Ÿ“ˆ Your Progress

Lesson 16 of 19

Real Numbers Progress 84%