🎬 See the interval around a rounded value
A rounded value represents every original value inside its error interval.
🎯 By the end of this lesson...
You should be able to:
- Distinguish between exact and measured values.
- Explain why measured values contain uncertainty.
- Identify the unit to which a value was rounded.
- Calculate the lower bound of a rounded value.
- Calculate the upper bound of a rounded value.
- Write error intervals using inequality symbols.
- Find bounds for whole numbers and decimal measurements.
- Apply bounds to practical measurement problems.
🤔 Think About This...
Is the bag exactly 2 kilograms?
A bag of sugar is labelled:
Does the bag have a mass of exactly 2.000000 kg?
Its actual mass could instead be:
- 1.998 kg;
- 2.001 kg;
- 2.003 kg.
The value shown on the label may have been rounded.
1️⃣ Exact Values and Measured Values
Exact values
Exact values are normally obtained by counting.
There is no rounding uncertainty.
Measured values
Measured values are obtained using instruments.
The displayed value may have been rounded.
💡 Did You Know?
Even highly accurate scientific instruments have limits.
Scientists and engineers therefore record the range within which the actual value is expected to lie.
2️⃣ What Is a Bound?
A bound describes the smallest or largest possible actual value represented by a rounded measurement.
Lower bound
The smallest possible value that would round to the stated measurement.
Upper bound
The value at which the measurement would begin rounding to the next number.
Example 1: A length is 15 cm to the nearest centimetre
Values from 14.5 cm up to, but not including, 15.5 cm round to 15 cm.
Upper bound = 15.5 cm
3️⃣ The Half-Unit Rule
To find bounds, first identify the unit used for rounding.
Then find half of that unit.
| Rounded to the nearest... | Rounding unit | Half of the unit |
|---|---|---|
| 10 | 10 | 5 |
| Whole number | 1 | 0.5 |
| One decimal place | 0.1 | 0.05 |
| Two decimal places | 0.01 | 0.005 |
| Three decimal places | 0.001 | 0.0005 |
Lower bound
Upper bound
🧠 StudyNest Method: Finding Bounds
Step 2: Find half of that rounding unit.
Step 3: Subtract the half-unit from the rounded value to find the lower bound.
Step 4: Add the half-unit to the rounded value to find the upper bound.
Step 5: Write the error interval using:
lower bound ≤ actual value < upper bound
4️⃣ Bounds to the Nearest Whole Number
Example 2: A mass is 40 kg to the nearest kilogram
The rounding unit is 1 kg.
Lower bound
Upper bound
Upper bound = 40.5 kg
Example 3: A distance is 126 m to the nearest metre
5️⃣ Bounds to One Decimal Place
Example 4: A distance is 3.6 km to one decimal place
One decimal place means the number was rounded to the nearest 0.1.
Example 5: A temperature is 28.4°C to one decimal place
6️⃣ Bounds to Two Decimal Places
Example 6: A height is 1.72 m to two decimal places
Two decimal places means the number was rounded to the nearest 0.01 m.
7️⃣ Bounds to the Nearest Ten
Example 7: A crowd is estimated as 480 people to the nearest ten
The rounding unit is 10.
8️⃣ Writing Error Intervals
An error interval shows the complete range of possible values.
Example 8: A length is 20 cm to the nearest centimetre
9️⃣ Why Is the Upper Bound Not Included?
Consider a value recorded as 20 to the nearest whole number.
20.49
This rounds down to 20.
Included20.50
This rounds up to 21.
Not included🔍 Inequality Symbols in Error Intervals
Less than or equal to
The lower bound is included.
Less than
The upper bound is not included.
🔟 Finding the Rounded Value from an Error Interval
Example 9
The midpoint of the interval is:
The interval has a width of 1, so the measurement was rounded to the nearest whole number.
🌍 Real-Life Example: Carpentry
Example 10
A carpenter records the length of a wooden plank as 2.4 m, correct to the nearest 0.1 m.
🌍 Real-Life Example: Road Distance
Example 11
A road distance is recorded as 4.8 km, correct to the nearest 0.1 km.
🌍 Real-Life Example: Package Mass
Example 12
The mass of a parcel is recorded as 6.35 kg, correct to the nearest 0.01 kg.
🧠 The StudyNest Bounds Strategy
Step 2: Write the rounding unit.
Step 3: Divide the rounding unit by 2.
Step 4: Subtract that amount for the lower bound.
Step 5: Add that amount for the upper bound.
Step 6: Write the interval as:
lower bound ≤ value < upper bound
Step 7: Include the correct measurement units.
🎯 Exam Tips
⚠ Common Mistakes
Adding and subtracting the full rounding unit instead of half of it.
Assuming every whole number has bounds 0.5 away without checking whether it was rounded to the nearest ten, hundred or another unit.
Including the upper bound using ≤.
Using 0.5 instead of 0.05 for a value rounded to one decimal place.
Confusing the lower and upper bounds.
Forgetting measurement units in the final answer.
Treating a counted quantity as though it were a rounded measurement.
🎮 Your Turn!
🟢 Foundation Questions
-
State whether each value is exact or measured:
a) 18 learners
b) 1.6 metres - Find the lower and upper bounds of 15 kg, correct to the nearest kilogram.
- Find the lower and upper bounds of 28 cm, correct to the nearest centimetre.
- Write the error interval for 42 m, correct to the nearest metre.
- Find the bounds of 350, correct to the nearest ten.
- Find the bounds of 7200, correct to the nearest hundred.
🔵 Developing Questions
- Find the lower and upper bounds of 5.8 kg, correct to one decimal place.
- Find the bounds of 16.42 cm, correct to two decimal places.
- Write the error interval for 7.3 litres, correct to one decimal place.
- Write the error interval for 2.68 metres, correct to two decimal places.
- A temperature is recorded as 31.6°C, correct to one decimal place. Find its error interval.
- A package has a recorded mass of 4.25 kg, correct to the nearest 0.01 kg. Find its bounds.
🔴 Challenge Questions
- A road is recorded as 4.8 km long, correct to the nearest 0.1 km. Find its lower bound, upper bound and error interval.
- The length of a field is recorded as 120 m, correct to the nearest 10 m. Write its error interval.
-
The error interval for a mass is:
17.5 ≤ m < 18.5
State the recorded mass and the degree of accuracy. -
The error interval for a length is:
6.25 ≤ x < 6.35
State the recorded length and the unit to which it was rounded. -
A student writes the following interval for 12.4 cm, correct to one
decimal place:
12.35 ≤ x ≤ 12.45
Explain the mistake and write the correct interval. -
A package is labelled 3 kg, correct to the nearest kilogram. Could
its actual mass be:
a) 2.51 kg?
b) 3.49 kg?
c) 3.50 kg?
✅ Show Answers
-
a) Exact
b) Measured -
Lower bound = 14.5 kg
Upper bound = 15.5 kg -
Lower bound = 27.5 cm
Upper bound = 28.5 cm - 41.5 ≤ x < 42.5
-
Rounded to nearest 10:
Half-unit = 5
345 ≤ x < 355 -
Rounded to nearest 100:
Half-unit = 50
7150 ≤ x < 7250 -
Half of 0.1 = 0.05
Lower bound = 5.75 kg
Upper bound = 5.85 kg -
Half of 0.01 = 0.005
Lower bound = 16.415 cm
Upper bound = 16.425 cm - 7.25 ≤ x < 7.35
- 2.675 ≤ x < 2.685
- 31.55°C ≤ t < 31.65°C
- 4.245 kg ≤ m < 4.255 kg
-
Lower bound = 4.75 km
Upper bound = 4.85 km
4.75 ≤ d < 4.85 -
Half of 10 m = 5 m
115 m ≤ length < 125 m -
Recorded mass = 18
Rounded to the nearest whole unit. -
The midpoint is:
(6.25 + 6.35) ÷ 2 = 6.30
The interval width is 0.10, so it was rounded to the nearest 0.1.
Recorded length = 6.3 -
The upper bound must not be included.
Correct interval:
12.35 ≤ x < 12.45 -
The valid range is:
2.5 ≤ m < 3.5
a) Yes
b) Yes
c) No
🧠 Remember This
Measured values may have been rounded.
Find half of the rounding unit.
Lower bound = rounded value − half-unit.
Upper bound = rounded value + half-unit.
Write error intervals as:
Lower bound ≤ actual value < upper bound.
The upper bound is not included.
🎯 Before Moving On...
- ✅ Can I distinguish exact and measured values?
- ✅ Can I identify the rounding unit?
- ✅ Can I find half of the rounding unit?
- ✅ Can I calculate a lower bound?
- ✅ Can I calculate an upper bound?
- ✅ Can I write an error interval?
- ✅ Do I know why the upper bound is excluded?
- ✅ Can I solve a practical bounds problem?
🎉 Well Done!
You can now calculate bounds and describe the complete range represented by a rounded measurement.
Identify the rounding unit → Find half → Subtract for the lower bound → Add for the upper bound → Write the interval
📈 Your Progress
Lesson 17