Bounds and Error Intervals [H]

This lesson covers bounds and error intervals for AQA GCSE Mathematics Higher Tier. When a measurement is rounded, the true value could lie anywhere within a range. Understanding this range — defined by upper and lower bounds — is essential for questions on accuracy and for using bounds in calculations.

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What Are Bounds?

When a value has been rounded or truncated, the actual value lies within a range. The limits of this range are called bounds.

• The lower bound (LB) is the smallest value that would round to the given number.
• The upper bound (UB) is the smallest value that would round up to the next number.

Key Principle

If a measurement is given as x, rounded to a certain degree of accuracy, then the true value v satisfies:

Lower Bound <= v < Upper Bound

Note: the lower bound is included (<=) but the upper bound is excluded (<).

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Finding Bounds for Rounded Values

The bounds depend on the degree of accuracy used in the rounding.

The Rule

Lower bound = rounded value - half the degree of accuracy

Upper bound = rounded value + half the degree of accuracy

Worked Examples

Rounded Value	Degree of Accuracy	Half the Accuracy	Lower Bound	Upper Bound
7 cm	Nearest cm (1 cm)	0.5 cm	6.5 cm	7.5 cm
35 kg	Nearest kg (1 kg)	0.5 kg	34.5 kg	35.5 kg
4.7 m	1 d.p. (0.1 m)	0.05 m	4.65 m	4.75 m
150 g	Nearest 10 g	5 g	145 g	155 g
3,200	Nearest 100	50	3,150	3,250
5.83 s	2 d.p. (0.01 s)	0.005 s	5.825 s	5.835 s

Exam Tip: The degree of accuracy is determined by the smallest place value used. For example, 4.7 is given to 1 decimal place, so the degree of accuracy is 0.1, and you add/subtract half of that (0.05).

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Error Intervals

An error interval states the range of possible values using inequalities.

Worked Example

A length is measured as 12.4 cm, correct to 1 decimal place. Write the error interval.

• Lower bound = 12.4 - 0.05 = 12.35
• Upper bound = 12.4 + 0.05 = 12.45

Error interval: 12.35 <= length < 12.45

Worked Example: Nearest Integer

The population of a town is given as 8,400 to the nearest 100. Write the error interval.

• Lower bound = 8,400 - 50 = 8,350
• Upper bound = 8,400 + 50 = 8,450

Error interval: 8,350 <= population < 8,450

Exam Tip: Always use <= for the lower bound and < (strict inequality) for the upper bound. This is because a value exactly at the lower bound would round to the given value, but a value at the upper bound would round up.

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Truncation Bounds

When a number has been truncated (not rounded), the bounds are different.

For a number truncated to n decimal places:

• Lower bound = the truncated value
• Upper bound = the truncated value + the smallest unit at that decimal place

Worked Example

A number is truncated to 6.4 (1 decimal place). Write the error interval.

• Lower bound = 6.4 (the truncated value itself)
• Upper bound = 6.4 + 0.1 = 6.5

Error interval: 6.4 <= number < 6.5

Compare with rounding: if 6.4 was rounded to 1 d.p., the error interval would be 6.35 <= number < 6.45.

graph TD
    A["Value Given"] --> B{"Rounded or Truncated?"}
    B -->|Rounded| C["LB = value - half accuracy"]
    B -->|Rounded| D["UB = value + half accuracy"]
    B -->|Truncated| E["LB = truncated value"]
    B -->|Truncated| F["UB = truncated value + unit"]
    C --> G["LB <= true value < UB"]
    D --> G
    E --> G
    F --> G

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Using Bounds in Calculations

When performing calculations with rounded values, you need to find the maximum and minimum possible results.

Rules for Calculations with Bounds

Operation	For Maximum Result	For Minimum Result
Addition (A + B)	UB(A) + UB(B)	LB(A) + LB(B)
Subtraction (A - B)	UB(A) - LB(B)	LB(A) - UB(B)
Multiplication (A x B)	UB(A) x UB(B)	LB(A) x LB(B)
Division (A / B)	UB(A) / LB(B)	LB(A) / UB(B)

Key Insight for Subtraction and Division

To make a result as large as possible:

• For subtraction: use the largest possible first number and the smallest possible second number.
• For division: use the largest possible numerator and the smallest possible denominator.

To make a result as small as possible:

• Do the opposite.

Exam Tip: The most common mistake is using the wrong combination of bounds. Think about what makes the answer bigger or smaller. For division, a bigger numerator with a smaller denominator gives the biggest result.

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Worked Example: Bounds in Calculations

A rectangle has a length of 8.3 cm (to 1 d.p.) and a width of 5.7 cm (to 1 d.p.). Calculate the upper and lower bounds of the area.

Step 1: Find the bounds of each measurement.

Measurement	Lower Bound	Upper Bound
Length	8.25 cm	8.35 cm
Width	5.65 cm	5.75 cm

Step 2: Calculate the bounds of the area (length x width).

• Maximum area = UB(length) x UB(width) = 8.35 x 5.75 = 48.0125 cm squared
• Minimum area = LB(length) x LB(width) = 8.25 x 5.65 = 46.6125 cm squared

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Worked Example: Speed Calculation

A car travels 240 km (to the nearest 10 km) in 3.5 hours (to the nearest 0.1 hours). Calculate the upper and lower bounds of the speed.

Step 1: Find the bounds.

Measurement	Lower Bound	Upper Bound
Distance	235 km	245 km
Time	3.45 hours	3.55 hours

Step 2: Speed = Distance / Time

• Maximum speed = UB(distance) / LB(time) = 245 / 3.45 = 71.0 km/h (3 s.f.)
• Minimum speed = LB(distance) / UB(time) = 235 / 3.55 = 66.2 km/h (3 s.f.)

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Suitable Degree of Accuracy

A common exam question asks: "To what degree of accuracy can the answer be given?"

Method

1. Calculate the upper and lower bounds of the answer.
2. Round both bounds to increasing degrees of accuracy.
3. The answer can be given to the degree of accuracy where both bounds round to the same value.

Worked Example

Using the rectangle example above (area between 46.6125 and 48.0125 cm squared):