Rounding Estimation And Bounds

Learn it by playing

Answer these questions to earn energy, then fish and explore. No account needed.

For teachers: ready-to-use lesson slides, revision notes, diagrams for Rounding Estimation And Bounds (Maths [CIE], Core) — use them in your lesson, or run the topic as a live class game.

Notes

Rounding to a Given Place Value

Identify the digit in the required place value, then count in that unit to find the two possible rounded values (one below, one above).
Circle the digit immediately to the right of the required place; if it is **5 or more**, round up; if **less than 5**, round down.
For decimal places, ensure the answer has exactly the specified number of decimal places (e.g., 2.40 for 2 d.p., not 2.4).
When rounding to a place value (e.g., nearest 100), fill any following places before the decimal point with zeros (e.g., 1567.45 → 1600).

Rounding to Significant Figures

The **first significant figure** is the leftmost non-zero digit (e.g., 0.006207 → first s.f. is 6).
Count digits from the first s.f. to locate the required s.f.; zeros between non-zero digits are significant.
Round using the same rules: circle the next digit; $if \ge 5$ round up, else round down.
For large numbers, fill with zeros up to the decimal point (e.g., 34 568 to 2 s.f. → 35 000).
For decimals, insert zeros between decimal point and first s.f. as needed (e.g., 0.003 435 to 3 s.f. → 0.003 44).
Unless stated otherwise, give final answers to **3 significant figures**; in money use 2 d.p.; in angles use 1 d.p.

Estimation

Estimate by rounding each number to **1 significant figure** (or a convenient value) before calculating.
Avoid rounding small numbers to zero, especially in denominators (division by zero is undefined).
For addition/multiplication: rounding both numbers up gives an **overestimate**; rounding both down gives an **underestimate**.
For subtraction/division: rounding the first number up and the second down gives an overestimate; the opposite gives an underestimate.
Estimation is useful for checking if an exact answer is reasonable.

Upper & Lower Bounds

Bounds describe the range a rounded number can lie between: **lower bound (LB)** $\le x <$ **upper bound (UB)**.
To find bounds: divide the degree of accuracy by 2, then add to get UB and subtract to get LB.
Example: 3.6 km (1 d.p.) → degree $= 0.1$ , half $= 0.05$ → $LB = 3.55, UB = 3.65$ .
The error interval is written as $LB \le x < UB (e.g., 3.55 \le l < 3.65)$ .
Always identify the degree of accuracy correctly (e.g., nearest metre, 2 s.f., etc.).

Rounding to 1 Decimal Place

Significant Figures: First Non-Zero Digit

Estimation by Rounding to 1 s.f.

Upper and Lower Bounds on a Number Line

Practice questions

Free preview — 8 of 32 questions. Sign up to see them all.

1.Write 0.3728 correct to 1 decimal place.
Easy
- A0.4
- B0.37
- C0.3
- D0.38
2.The height, h metres, of a tower is 128 m, correct to the nearest metre. Complete the statement about the value of $h: ___ \le h < ___$
Easy
- A $127.5 \le h < 128.5$
- B $127 \le h < 129$
- C $127.5 \le h \le 128.5$
- D $128 \le h < 129$
3.Write 867 correct to the nearest ten.
Easy
- A870
- B860
- C900
- D800
4.The length, l cm, of a pencil is 18 cm, correct to the nearest centimetre. Complete the statement about the value of $l: ___ \le l < ___$
Easy
- A $17.5 \le l < 18.5$
- B $17 \le l < 19$
- C $17.5 \le l \le 18.5$
- D $18 \le l < 19$
5.Write 849.481 correct to 1 decimal place.
Easy
- A849.5
- B849.4
- C849.48
- D850.0
6.Write 97.4236 correct to 3 decimal places.
Easy
- A97.424
- B97.423
- C97.420
- D97.400
7.Write 3.72194 correct to 3 decimal places.
Easy
- A3.722
- B3.721
- C3.720
- D3.700
8.Write 8379 correct to the nearest hundred.
Easy
- A8400
- B8300
- C8000
- D9000

Summary

## Rounding, Estimation & Bounds ### Rounding to a Given Place Value - Identify the digit in the required place value. - Count in the units you are rounding to (e.g., nearest 100 → count in 100s). - Circle the digit to the right: if ≥5 round up, else round down. - Put zeros in following place values before the decimal point. - For decimal places, give exactly the number of decimal places asked (e.g., 2.40 not 2.4). ### Rounding to Significant Figures - First significant figure: first non-zero digit from left. - Count to the required significant figure; use normal rounding rules. - For large numbers, fill with zeros up to the decimal point. - For decimals, fill zeros between decimal point and first significant figure. - Unless stated, round final answers to 3 significant figures. ### Estimation - Round each number to 1 significant figure (avoid rounding to 0). - Perform the calculation with rounded numbers. - Overestimate: numerator rounded up and/or denominator rounded down (for division). - Underestimate: numerator rounded down and/or denominator rounded up. ### Upper & Lower Bounds - Bounds: LB ≤ x < UB. - Degree of accuracy (e.g., nearest 100, 1 d.p., 2 s.f.). - Half the degree of accuracy: add to get UB, subtract to get LB. - Error interval: LB ≤ x < UB. - Examples: - 128 m (nearest m): 127.5 ≤ h < 128.5 - 18.3 cm (nearest mm): 18.25 ≤ l < 18.35 - 115 g (nearest 5 g): 112.5 ≤ m < 117.5 - 1.5 L (nearest 0.1 L): 1.45 ≤ w < 1.55

Key terms

Rounding: Approximating a number to a given place value or significant figure by looking at the next digit (≥5 round up, else round down).
Significant figures: The digits in a number that carry meaning, starting from the first non-zero digit from the left.
Estimation: Finding an approximate value by rounding numbers (usually to 1 significant figure) and calculating.
Upper bound (UB): The largest value that a rounded number can be, but not including (x < UB).
Lower bound (LB): The smallest value that a rounded number can be, including (LB ≤ x).
Error interval: The range between the lower and upper bounds, written as LB ≤ x < UB.

Slides

Rounding to a Given Place Value

Identify the digit in the required place value.
Count in the units you are rounding to (e.g., 100s, 0.01s).
Circle the digit to the right; if 5 or more, round up; otherwise round down.

Round 849.481 to 1 decimal place.

849.5
849.4
849.48
850.0

Rounding to Decimal Places

When rounding to a given number of decimal places, count the digits after the decimal point.
Circle the digit to the right of the required decimal place.
If that digit is 5 or more, round up; if less than 5, round down.
Always give the answer with the exact number of decimal places requested (e.g., 2.40 not 2.4).

Write 97.4236 correct to 3 decimal places: ____

Rounding to Significant Figures

The first significant figure is the first non-zero digit from the left.
Count along to the right to find the required significant figure.
Use normal rounding rules: circle the next digit; if 5 or more, round up.
Fill with zeros to maintain place value for large numbers (e.g., 345 256 → 345 000 to 3 s.f.).

Write 0.047883 correct to 2 significant figures.

0.048
0.047
0.0479
0.05

Estimation

Estimation helps check if your answer is reasonable.
Round each number to 1 significant figure (or a convenient value) before calculating.
Avoid rounding small numbers to zero, especially in denominators.

Estimate $37.8 \times 13.2 / (28.5 + 22.1)$ by rounding each number to 1 significant figure: ____

Upper & Lower Bounds

Bounds are the smallest and largest values a rounded number can take.
Lower bound (LB) is included, upper bound (UB) is not: $LB \le x < UB$ .
To find bounds, divide the degree of accuracy by 2, then add/subtract from the given value.

The height h of a tower is 128 m, correct to the nearest metre. Which is the correct error interval?

$127 \le h < 129$
$127.5 \le h < 128.5$
$128 \le h < 129$
$127 \le h \le 129$

Bounds for Different Accuracies

For rounding to a given number of decimal places, the degree of accuracy is 0.1, 0.01, etc.
For rounding to the nearest 10, 100, etc., the degree of accuracy is 10, 100, etc.
Always divide the accuracy by 2 to find the half-range.

The length l of a pencil is 18 cm, correct to the nearest centimetre. Complete: $____ \le l < ____$

Overestimates and Underestimates

If you round numbers up in addition or multiplication, you get an overestimate.
If you round numbers down in addition or multiplication, you get an underestimate.
For subtraction and division, rounding up the first number or down the second gives an overestimate, and vice versa.

Which of these are true?

Rounding 17.3 to 20 and 3.81 to 4 in a numerator gives an overestimate.
Rounding 11.5 to 10 in a denominator gives an overestimate.
Rounding both numbers up in a division always gives an overestimate.

Estimation with Significant Figures

In exam questions, you are often asked to round each number to 1 significant figure.
Show your working: write the rounded numbers, then the calculation.
Estimation can be used to check if your final answer is plausible.

Estimate $71 \times 32.4 / 4.82$ by rounding each number to 1 significant figure.

Bounds for Measurements

When a measurement is given correct to a certain unit, the bounds are half that unit above and below.
For example, 76.3 m correct to 1 d.p. means accuracy 0.1 m, half is 0.05 m, so $LB = 76.25 m, UB = 76.35 m$ .
Always write the error interval as $LB \le x < UB$ .

The height h of a tower is 76.3 m, correct to 1 decimal place. Complete: $____ \le h < ____$

Summary

Rounding: identify the place value, look at the next digit, round up if 5 or more.
Estimation: round to 1 significant figure and calculate.
Bounds: divide accuracy by 2, add/subtract to find LB and UB.

The mass m of a horse is 429 kg, correct to the nearest kilogram. Which is correct?

$428.5 \le m < 429.5$
$428 \le m < 430$
$429 \le m < 430$
$428.5 \le m \le 429.5$

Flashcards

Round 0.3728 to 1 decimal place.

0.4

The height h of a tower is 128 m, correct to the nearest metre. Complete: ... $\le h <$ ...

$127.5 \le h < 128.5$

Round 867 to the nearest ten.

870

The length l of a pencil is 18 cm, correct to the nearest centimetre. Complete: ... $\le l <$ ...

$17.5 \le l < 18.5$

Round 849.481 to 1 decimal place.

849.5

Round 97.4236 to 3 decimal places.

97.424

Round 3.72194 to 3 decimal places.

3.722

Round 8379 to the nearest hundred.

8400

Round 6789 to the nearest 100.

6800

Round 6789 to 3 significant figures.

6790

The mass m of a horse is 429 kg, correct to the nearest kilogram. Complete: ... $\le m <$ ...

$428.5 \le m < 429.5$

Round 8473 to the nearest ten.

8470

Round 16.086 to 2 decimal places.

16.09

Round 3.058 to 3 significant figures.

3.06

Genaro measures his desk as 120 cm, correct to the nearest centimetre. Complete: ... $\le l <$ ...

$119.5 \le l < 120.5$

Round 1850 to 2 significant figures.

1900

By rounding each number to 1 significant figure, estimate $(37.8 \times 13.2) / (28.5 + 22.1)$ .

$(40 \times 10) / (30 + 20) = 400 / 50 = 8$

The height h of a tower is 76.3 m, correct to 1 decimal place. Complete: ... $\le h <$ ...

$76.25 \le h < 76.35$

By rounding each number to 1 significant figure, estimate $(71 \times 32.4) / 4.82$ .

$(70 \times 30) / 5 = 2100 / 5 = 420$

Calculate $3\sqrt{40}$ correct to 4 decimal places.

3.4199

Write 0.047883 correct to 2 significant figures.

0.048

Write 6.8167 correct to 3 significant figures.

6.82

The amount of fuel t is 135 litres, correct to the nearest litre. Complete: ... $\le t <$ ...

$134.5 \le t < 135.5$

The length l of a line is 18.3 cm, correct to the nearest millimetre. Complete: ... $\le l <$ ...

$18.25 \le l < 18.35$

The mass m of a banana is 115 g, correct to the nearest 5 g. Complete: ... $\le m <$ ...

$112.5 \le m < 117.5$

Unlock all 32 questions, slides, flashcards & more

Create a free account to see every question, the slides, flashcards and revision notes for this topic.

Past papers

Past-paper practice for this topic is coming soon.

🗂️ Coming soon