Calculating with Bounds

1. The Core Idea: Error Intervals

Whenever a measurement is rounded, the original, precise value is lost. However, we know it must lie within a specific range, or error interval. The edges of this range are called the upper and lower bounds.

The fundamental rule for finding these bounds is the "Half-Unit Rule".

As you've just read, the process is:

1. Identify the degree of accuracy: e.g., nearest cm (unit = 1), nearest 10 (unit = 10), 1 decimal place (unit = 0.1).
2. Find the tolerance: This is half of the unit of accuracy.
3. Calculate the bounds:
   • Lower Bound (LB): Rounded Value - Tolerance
   • Upper Bound (UB): Rounded Value + Tolerance

The most important thing to remember is that the true value can be equal to the lower bound, but must be less than the upper bound.
For example, if a length $L$ is 12 cm correct to the nearest cm:

• The unit of accuracy is 1 cm.
• The tolerance is $1÷2=0.5$ cm.
• LB = $12-0.5=11.5$ cm.
• UB = $12+0.5=12.5$ cm.
• The error interval is written as: $11.5\le L<12.5$ cm.

2. Calculations with Bounds: The Four Rules

Things get interesting when you have to perform calculations (addition, subtraction, multiplication, or division) with rounded numbers. To find the maximum or minimum possible outcome, you must choose the correct bound for each number in the calculation.

This is a key area where "method selection" is vital. Addition and multiplication are intuitive, but subtraction and division require you to think carefully.

The table below summarises the logic.

Here's a summary of those crucial rules:

3. Bounds in Action: Worked Exam Questions

The best way to master these rules is to see them applied to IGCSE past-paper questions. The following video works through numerous examples covering all four operations, as well as common exam scenarios like unit conversions.

Subtraction and Addition

Let's see how to find the bounds for subtraction and addition.

Now for addition, which is more straightforward.

A crucial exam tip mentioned here is do not round your final answer unless the question specifically tells you to. The exact value is the required answer.

Multiplication and Division

Now let's tackle multiplication (e.g., for area) and division (e.g., for speed).

Test your understanding!

A triangle has an area of 56 cm², correct to the nearest whole number.
Its height is 8 cm, correct to the nearest cm.

The formula for the area of a triangle is $A=\frac{1}{2}bh$.

Calculate the upper bound for the length of the base, $b$.

Show answer

1. Rearrange the formula to make $b$ the subject:
   $A=\frac{bh}{2}⟹2A=bh⟹b=\frac{2A}{h}$

2. Find the bounds for $A$ and $h$:

   • Area $A=56$ cm² (nearest 1 cm²): Tolerance = 0.5.
     So, $55.5\le A<56.5$. UB(A) = 56.5.
   • Height $h=8$ cm (nearest 1 cm): Tolerance = 0.5.
     So, $7.5\le h<8.5$. LB(h) = 7.5.

3. Apply the division rule for the Upper Bound:
   To find the Upper Bound of $b=\frac{2A}{h}$, we need the largest possible numerator divided by the smallest possible denominator.

   • UB($b$) = $\frac{2\times \text{UB}(A)}{\text{LB}(h)}$

4. Calculate:

   • UB($b$) = $\frac{2\times 56.5}{7.5}=\frac{113}{7.5}=\mathrm{15.0666...}$
   • The upper bound for the base is 15.06̇ cm (or an equivalent fraction).

4. Advanced Application: "Suitable Degree of Accuracy"

A common Grade 9 style question asks you to calculate a value "to a suitable degree of accuracy." This has a very specific method.

1. Calculate both the Upper Bound and the Lower Bound for the final value.
2. Write down both full calculator answers.
3. Find the most precise level of rounding (e.g., nearest integer, 1 d.p., 2 s.f.) at which both bounds give the same value.
4. This rounded value is your answer. You must state the bounds you calculated to justify your choice.

Key Takeaways

• Error Interval: All bounds problems start by finding the error interval for each rounded value using the half-unit rule: $\text{LB}\le x<\text{UB}$.
• The Four Rules: The logic for combining bounds is crucial. Addition/multiplication are straightforward (UB with UB, LB with LB), but subtraction/division are "crossed" to find the maximum or minimum.
• Think Logically: For division like $A/B$, ask yourself: "To make the result as big as possible, do I want the top number to be big or small? What about the bottom number?" This will help you select the right bounds.
• Show Your Working: Write down the bounds you are using (e.g., UB(A) / LB(B)) and the substituted values. This is where method marks are earned.
• Do Not Round (Usually): Unless asked for a "suitable degree of accuracy," give the full, exact answer from your calculator.

In our next lesson, we will cover the final topic in this module: estimation by rounding to 1 significant figure. This is a different skill from bounds. Bounds give the exact limits of possibility, while estimation gives a quick approximation to check if a calculator answer is in the right ballpark.