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Learning Resources

Approximations and Errors

Interactive curriculum lessons, worked examples, and geometric problem-solving techniques designed to help Kenyan students master CBC, KPSEA, KCSE, and IGCSE mathematics.

Grade 09 Pathway: N/A

First Principles

Objective: Round numbers to given decimal places or significant figures and calculate absolute, relative, and percentage errors.

Concrete scenario: You are at a local duka in Machakos buying three items priced at KSh 47.85, KSh 32.15, and KSh 18.40. You quickly estimate the total by rounding each price to the nearest ten shillings: \(50 + 30 + 20 = 100\) KSh. The cashier's till, however, calculates the exact sum as KSh 98.40. The difference of KSh 1.60 is the error introduced by rounding. Approximations speed up mental calculation, but knowing the bounds of our error ensures we do not run short of cash!

Geometric insight: Picture a standard metre ruler. When a length is rounded to the nearest centimetre (\(0.01\text{ m}\)), it is snapped to the nearest tick mark. The true measurement could lie anywhere from half a division below to half a division above that tick. Therefore, the maximum possible absolute error is always half of the smallest measuring unit (\(\frac{1}{2} \times 10^{-n}\)).

Algebraic rules:

  • Absolute Error (\(E_{abs}\)): \(E_{abs} = |x_{true} - x_{approx}|\) or \(|\text{measured} - \text{actual}|\).
  • Relative Error (\(E_{rel}\)): \(E_{rel} = \frac{E_{abs}}{|x_{true}|}\). This gives error as a proportion of the true value.
  • Percentage Error (\(E_{\%}\)): \(E_{\%} = E_{rel} \times 100\%\).

Key Formulas

\[E_{abs} = |x_{true} - x_{approx}|\]

Absolute Error: The magnitude of the numerical difference between the true value and its approximation. It always has the same units as the measured quantity and is strictly non-negative.

\[E_{rel} = \frac{E_{abs}}{|x_{true}|} = \frac{|x_{true} - x_{approx}|}{|x_{true}|}\]

Relative Error: Dimensionless ratio that measures how significant an error is relative to the total quantity being measured.

\[E_{\%} = E_{rel} \times 100\% = \frac{|x_{true} - x_{approx}|}{|x_{true}|} \times 100\%\]

Percentage Error: The relative error expressed as a percentage.

\[\text{Maximum Absolute Error} = \frac{1}{2} \times (\text{Unit of precision})\]

Tolerance / Maximum Error: For a measurement rounded to \(n\) decimal places, unit of precision is \(10^{-n}\), giving a maximum error of \(\pm 0.5 \times 10^{-n}\).

\[\text{Lower Bound} = x - E_{abs}, \quad \text{Upper Bound} = x + E_{abs}\]

Measurement Limits: The true value \(X\) lies in the interval \(x - E_{abs} \le X < x + E_{abs}\).

Worked Examples

Example 1 (Easy): A sack of rice has an actual mass of \(48.7\text{ kg}\). A trader estimates it as \(50.0\text{ kg}\). Find the absolute error and the percentage error.
  1. Calculate Absolute Error: \[E_{abs} = |x_{true} - x_{approx}| = |48.7 - 50.0| = 1.3\text{ kg}\]
  2. Calculate Relative Error: \[E_{rel} = \frac{E_{abs}}{x_{true}} = \frac{1.3}{48.7} \approx 0.026694\]
  3. Calculate Percentage Error: \[E_{\%} = 0.026694 \times 100\% \approx 2.67\%\]
Example 2 (Medium): A builder measures the length of a classroom floor as \(12.4\text{ m}\) to the nearest \(0.1\text{ m}\). Find the range of possible true values and the percentage error in this measurement.
  1. Find the maximum absolute error: \[\text{Unit of precision} = 0.1\text{ m} \implies E_{abs} = \frac{0.1}{2} = 0.05\text{ m}\]
  2. Determine the limits (bounds): \[\text{Lower Bound} = 12.4 - 0.05 = 12.35\text{ m}\] \[\text{Upper Bound} = 12.4 + 0.05 = 12.45\text{ m}\] Thus, \(12.35\text{ m} \le \text{Length} < 12.45\text{ m}\).
  3. Calculate maximum percentage error: \[E_{\%} = \frac{0.05}{12.4} \times 100\% \approx 0.403\%\]
Example 3 (Hard): A rectangular plot of land in Eldoret is measured as \(20\text{ m}\) by \(10\text{ m}\), with each dimension rounded to the nearest metre. Calculate the maximum possible percentage error in the calculated area.
  1. State limits for each dimension: \[\text{Length } L: 19.5\text{ m} \le L < 20.5\text{ m}\] \[\text{Width } W: 9.5\text{ m} \le W < 10.5\text{ m}\]
  2. Nominal Area: \[A_{nominal} = 20 \times 10 = 200\text{ m}^2\]
  3. Find maximum and minimum areas: \[A_{max} = 20.5 \times 10.5 = 215.25\text{ m}^2\] \[A_{min} = 19.5 \times 9.5 = 185.25\text{ m}^2\]
  4. Determine the maximum difference from nominal: \[|A_{max} - A_{nominal}| = |215.25 - 200| = 15.25\text{ m}^2\] \[|A_{min} - A_{nominal}| = |185.25 - 200| = 14.75\text{ m}^2\] Maximum absolute error in area is \(15.25\text{ m}^2\).
  5. Calculate maximum percentage error: \[E_{\%} = \frac{15.25}{200} \times 100\% = 7.625\%\]

Common Mistakes

Mistake Dividing by the estimated/approximate value instead of the true value when computing relative error: \(E_{rel} = \frac{|x_{true} - x_{approx}|}{x_{approx}}\).
Correction Relative error must always be compared against the actual/true value: \(E_{rel} = \frac{|x_{true} - x_{approx}|}{x_{true}}\).
Why it feels right Often, the estimated value is the number we just calculated, so students instinctively place it in the denominator.
Mistake Assuming the absolute error for a measurement rounded to the nearest whole unit is \(1\) rather than \(0.5\).
Correction A number rounds to the nearest whole number if it is within \(\pm 0.5\) of that number.
Why it feels right The smallest division on the ruler is 1 unit, so it feels like the uncertainty is 1 full unit.
Mistake Rounding numbers at every step of a multi-step calculation.
Correction Maintain full precision throughout all working steps and round only the final calculated answer.
Why it feels right Rounding early makes intermediate numbers shorter and easier to write, but it compounds rounding errors drastically.

Real World

Construction & Carpentry: When cutting timber for roofing in Nairobi, a carpenter measures lengths to the nearest millimetre. If a \(4\text{ m}\) beam has an absolute error of \(\pm 5\text{ mm}\), accumulating this over 20 trusses can cause roof misalignments of up to \(10\text{ cm}\).
Agricultural Trade: Coffee and tea factories in Kericho weigh produce in tonnes. A small relative error of \(0.5\%\) on a \(50\text{-tonne}\) delivery represents \(250\text{ kg}\) of coffee, worth tens of thousands of Kenyan Shillings.
Medicine & Pharmacology: Drug dosages in hospitals are measured in milligrams (mg) or micrograms (\(\mu\text{g}\)). A \(5\%\) error in a high-potency pediatric antibiotic can be toxic, which is why digital precision balances with errors under \(0.001\text{ g}\) are required.
Mobile Data & Billing: Telecommunication companies round data usage to the nearest kilobyte or megabyte. Understanding accumulated rounding error ensures billing transparency for millions of M-Pesa and data bundle subscribers.

Practice

Jackson is measuring the length of a furrow in his shamba using a tape marked in metres with a smallest division of 1 cm (0.01 m). He records the length as 12.37 m. What is the maximum absolute error of this measurement in metres? (Type only the number, e.g., 0.005)
Review the concepts above.
Jackson measures one side of his shamba using a measuring tape and records the length as 12.347 m. What is this length expressed in centimetres, rounded to the nearest centimetre? (Type only the number, e.g., 1235)
Review the concepts above.
Jackson uses a measuring rope that is exactly 7.2 metres long to measure the perimeter of his family garden. He counts exactly 15 full lengths of the rope. What is the perimeter of the garden in metres? (Type only the number, e.g., 108)
Review the concepts above.
A shopkeeper in Nairobi approximates the weight of a sack of maize as 50 kg, but the actual weight is 48.7 kg. What is the relative error of this approximation, rounded to three decimal places? (Type only the number, e.g., 0.027)
Review the concepts above.
A farmer uses a steel measuring tape that has stretched uniformly so that each nominal metre is actually 1.02 m. He measures the side of a wheat field as 120 m using this tape. What is the actual length of the side in metres? (Type only the number, e.g., 122.4)
Review the concepts above.
A rectangular garden is measured as 23.6 m by 15.4 m, where each length is rounded to the nearest 0.1 m. What is the maximum possible absolute error in the calculated area, rounded to one decimal place? (Type only the number, e.g., 2.0)
Review the concepts above.