Upper and Lower Bounds [H]
Upper and Lower Bounds
At Higher tier, limits of accuracy are applied to compound expressions involving multiple operations — ratios, nested fractions, formulae with several measured quantities. The strategy is always the same: identify the bounds for each measured quantity, then combine them using the rules that produce the maximum or minimum result for the expression. Compound expressions require careful thought about which combination of bounds maximises or minimises the overall value.
Reviewing the Combining Rules
For measured values \( a \) and \( b \) with lower bounds LB and upper bounds UB:
| Expression | Maximum | Minimum |
|---|---|---|
| \( a + b \) | UB(\(a\)) + UB(\(b\)) | LB(\(a\)) + LB(\(b\)) |
| \( a - b \) | UB(\(a\)) \( - \) LB(\(b\)) | LB(\(a\)) \( - \) UB(\(b\)) |
| \( a \times b \) | UB(\(a\)) \( \times \) UB(\(b\)) | LB(\(a\)) \( \times \) LB(\(b\)) |
| \( a \div b \) | UB(\(a\)) \( \div \) LB(\(b\)) | LB(\(a\)) \( \div \) UB(\(b\)) |
Compound Expressions
For an expression like \( \dfrac{a + b}{c} \), work outward: first find the maximum and minimum of the numerator \( (a + b) \), then combine with the bounds of the denominator \( c \). Think of it as: to maximise a fraction, maximise the numerator and minimise the denominator simultaneously.
\[ \text{Maximum of } \frac{a+b}{c} = \frac{\text{UB}(a) + \text{UB}(b)}{\text{LB}(c)} \] \[ \text{Minimum of } \frac{a+b}{c} = \frac{\text{LB}(a) + \text{LB}(b)}{\text{UB}(c)} \]Worked Examples
\( x = 6.4 \) (1 dp), \( y = 2.7 \) (1 dp). Find max/min of \( x + y \) and \( x - y \).
Limits: \( 6.35 \leq x < 6.45 \) and \( 2.65 \leq y < 2.75 \).
\[ \text{Max}(x+y) = 6.45 + 2.75 = 9.20 \] \[ \text{Min}(x+y) = 6.35 + 2.65 = 9.00 \] \[ \text{Max}(x-y) = 6.45 - 2.65 = 3.80 \] \[ \text{Min}(x-y) = 6.35 - 2.75 = 3.60 \]\( p = 3.8 \) (1 dp), \( q = 1.5 \) (1 dp). Find max/min of \( pq \) and \( p/q \) to 3 sf.
Limits: \( 3.75 \leq p < 3.85 \) and \( 1.45 \leq q < 1.55 \).
\[ \text{Max}(pq) = 3.85 \times 1.55 = 5.9675 \approx 5.97 ext{ (3 sf)} \] \[ \text{Min}(pq) = 3.75 \times 1.45 = 5.4375 \approx 5.44 ext{ (3 sf)} \] \[ \text{Max}(p/q) = \frac{3.85}{1.45} = 2.6552\ldots \approx 2.66 ext{ (3 sf)} \] \[ \text{Min}(p/q) = \frac{3.75}{1.55} = 2.4194\ldots \approx 2.42 ext{ (3 sf)} \]Find the max and min of \( \dfrac{a+b}{c} \) where \( a = 7.4 \), \( b = 2.1 \), \( c = 3.2 \), all to 1 dp.
Bounds: \( a \): [7.35, 7.45), \( b \): [2.05, 2.15), \( c \): [3.15, 3.25).
Maximum: maximise numerator, minimise denominator.
\[ \frac{7.45 + 2.15}{3.15} = \frac{9.60}{3.15} = 3.0476\ldots \approx 3.05 \text{ (3 sf)} \]Minimum: minimise numerator, maximise denominator.
\[ \frac{7.35 + 2.05}{3.25} = \frac{9.40}{3.25} = 2.8923\ldots \approx 2.89 \text{ (3 sf)} \]Key Takeaways
- For compound expressions, treat each sub-expression in turn and select bounds that push the overall result in the desired direction.
- To maximise a fraction: maximise the numerator and minimise the denominator simultaneously.
- The subtraction rule applies inside compound numerators: max of \( (a-b) \) uses UB(\(a\)) \( - \) LB(\(b\)), not UB(\(a\)) \( - \) UB(\(b\)).
- Always state bounds of each measured quantity explicitly before combining — this demonstrates the method clearly and reduces errors.
- Round bounds to an appropriate number of significant figures at the end, not during intermediate steps.