Choose endpoint combinations that genuinely bound a positive expression.
Extended · Syllabus E1.10
Notes and key methods
For positive lengths, a product is smallest at both lower bounds and has its upper bound at both upper bounds. For a positive quotient, the lower bound uses lower numerator divided by upper denominator; reverse these choices for the upper bound.
Check positivity before applying these rules. An upper bound need not be an attainable maximum when the original rounding interval excludes its upper endpoint. Give enough digits to preserve the bound rather than rounding it inward.
Check the conditions
Use the stated units and precision.
Show the method as well as the final answer.
Three worked examples
Example 1 · Bounds in calculations
A rectangle has recorded length 13 cm and width 8 cm. Each was independently rounded to the nearest \(1\) cm. Both true dimensions are positive.
Find a lower bound for the rectangle area in \(\displaystyle \mathrm{cm}^{2}\).
Find an upper bound for its area in \(\displaystyle \mathrm{cm}^{2}\).
Find a lower bound for its perimeter in cm.
Find an upper bound for its length-to-width ratio.
Show the reasoning
(a)
Use the correct lower/upper endpoint combination for the stated increasing or quotient expression.
\(\displaystyle \frac{375}{4}\)
Answer: \(\displaystyle \frac{375}{4}\).
(b)
Use the correct lower/upper endpoint combination for the stated increasing or quotient expression.
\(\displaystyle \frac{459}{4}\)
Answer: \(\displaystyle \frac{459}{4}\).
(c)
Use the correct lower/upper endpoint combination for the stated increasing or quotient expression.
\(40\)
Answer: \(40\).
(d)
Use the correct lower/upper endpoint combination for the stated increasing or quotient expression.
\(\displaystyle \frac{9}{5}\)
Answer: \(\displaystyle \frac{9}{5}\).
Example 2 · Bounds in calculations
A rectangle has recorded length 14 cm and width 9 cm. Each was independently rounded to the nearest \(\displaystyle \frac{1}{10}\) cm. Both true dimensions are positive.
Find a lower bound for the rectangle area in \(\displaystyle \mathrm{cm}^{2}\).
Find an upper bound for its area in \(\displaystyle \mathrm{cm}^{2}\).
Find a lower bound for its perimeter in cm.
Find an upper bound for its length-to-width ratio.
Show the reasoning
(a)
Use the correct lower/upper endpoint combination for the stated increasing or quotient expression.
\(\displaystyle \frac{49941}{400}\)
Answer: \(\displaystyle \frac{49941}{400}\).
(b)
Use the correct lower/upper endpoint combination for the stated increasing or quotient expression.
\(\displaystyle \frac{50861}{400}\)
Answer: \(\displaystyle \frac{50861}{400}\).
(c)
Use the correct lower/upper endpoint combination for the stated increasing or quotient expression.
\(\displaystyle \frac{229}{5}\)
Answer: \(\displaystyle \frac{229}{5}\).
(d)
Use the correct lower/upper endpoint combination for the stated increasing or quotient expression.
\(\displaystyle \frac{281}{179}\)
Answer: \(\displaystyle \frac{281}{179}\).
Example 3 · Bounds in calculations
A rectangle has recorded length 15 cm and width 5 cm. Each was independently rounded to the nearest \(1\) cm. Both true dimensions are positive.
Find a lower bound for the rectangle area in \(\displaystyle \mathrm{cm}^{2}\).
Find an upper bound for its area in \(\displaystyle \mathrm{cm}^{2}\).
Find a lower bound for its perimeter in cm.
Find an upper bound for its length-to-width ratio.
Show the reasoning
(a)
Use the correct lower/upper endpoint combination for the stated increasing or quotient expression.
\(\displaystyle \frac{261}{4}\)
Answer: \(\displaystyle \frac{261}{4}\).
(b)
Use the correct lower/upper endpoint combination for the stated increasing or quotient expression.
\(\displaystyle \frac{341}{4}\)
Answer: \(\displaystyle \frac{341}{4}\).
(c)
Use the correct lower/upper endpoint combination for the stated increasing or quotient expression.
\(38\)
Answer: \(38\).
(d)
Use the correct lower/upper endpoint combination for the stated increasing or quotient expression.
\(\displaystyle \frac{31}{9}\)
Answer: \(\displaystyle \frac{31}{9}\).
Common mistakes
Keep intermediate values exact until the final step.