Exercises: Absolute and relative error

Calculate the absolute error and the relative error made when approximating the number $3,456$ by $3,46$. Express the relative error as a percentage.

Round the number $x = 12,7891$ to two decimal places and calculate the absolute error made in this approximation.

When measuring the length of a corridor, an approximate value of $150$ cm was obtained with a maximum absolute error of $2$ cm. Write down the interval in which the actual value of the length definitely lies.

Two measurements were taken in a technology class: in the first, an absolute error of $0,5$ cm was made when measuring a strip of wood of $20$ cm, and in the second, an absolute error of $2$ cm when measuring a plank of $100$ cm. Which of the two measurements is more precise? Justify your answer by calculating the relative errors.

Given that the relative error of a weighing in the laboratory is $5\ \%$ and that the true value of the mass is $250$ g, determine the absolute error made.

Approximate the number $\frac{2}{3}$ to two decimal places, first by truncation and then by rounding. Calculate the absolute error in each case and state which approximation is closer to the true value.

The true value of the acceleration due to gravity on Earth is $g = 9,81$ m/s$^2$. If in a physics problem you use the approximate value of $10$ m/s$^2$ to simplify calculations, find the absolute error and the percentage relative error you are introducing.

Determine the absolute and relative error made when using the fraction $\frac{22}{7}$ as an approximation of the irrational number $\pi$. (Use the value of $\pi$ provided by your calculator and express the final results rounded to four decimal places).

In a school debate about the cost of public transport in Sevilla, it is stated that the official price of a single urban bus ticket is $1,40\ €$. A student claims in their speech that the ticket costs "around $1,50\ €$". To assess the precision of this estimate, their classmate David performs the following calculations on the board:

1. a) Identify and explain in detail the two conceptual errors David made: one regarding the units of the absolute error and the other regarding the value placed in the denominator of the relative error.
2. b) Correct David's calculations by finding the correct value of $E_a$ (with its units) and $E_r$ (as a percentage, rounded to two decimal places).

At the Parque del Retiro weather station (Madrid), run by the Agencia Estatal de Meteorología (AEMET), a high-precision digital rain gauge records an accumulated precipitation of $14,6\text{ mm}$ after a spring storm. At the same time, in a nearby urban community garden, an analog-reading rain gauge is used that rounds to the nearest millimetre ($p = 1\text{ mm}$), marking $15\text{ mm}$.

1. a) Calculate the absolute error (stating its units) and the percentage relative error (to two decimal places) of the urban garden's reading compared to the AEMET reference value.
2. b) According to the property of ideal rounding to a unit $p$, what is the theoretical maximum bound that the absolute error could have due solely to rounding in that analog instrument?
3. c) Check whether the absolute error obtained in part a) satisfies this bound and explain what would happen if the garden had recorded $16\text{ mm}$.

A space telescope measures the distance to a distant star, obtaining $4,25 \cdot 10^8$ km. Given that the value considered exact by the astronomical community after multiple calibrations is $4,23 \cdot 10^8$ km, calculate the absolute error and the percentage relative error of the telescope's measurement.

On a greengrocer's counter, the price of one kilogram of cherries is $14,99$ €. If a customer buys $0,350$ kg, the scale calculates the exact price by multiplying both quantities and then rounds the result to the nearest cent in order to charge them. Calculate the absolute error and the percentage relative error made in the final price to be paid.

The actual size of a bacterium observed under a microscope is $2,5 \cdot 10^{-6}$ m. When noting down the data in their laboratory notebook, a student makes a mistake in the exponent and writes $2,5 \cdot 10^{-5}$ m. Calculate the absolute error and the relative error made by the student. In light of the relative error, is this a negligible error or a serious mistake?

In a 100-metre sprint, the official timer has a precision of one hundredth of a second (that is, the absolute error bound is $0,01$ s). If the time displayed on the scoreboard for the winning athlete is $9,81$ s, calculate the relative error bound affecting this measurement and express it as a percentage.

The radius of a circular metal part has been measured using callipers, obtaining an approximate value of $r = 15$ mm, given that the instrument has a maximum margin of error of $\pm 0,5$ mm. Calculate the interval containing the actual area of the part. Then, using the measured value to find an approximate area, determine the maximum absolute error bound that could be made for this area.

To prepare a solution in a chemistry laboratory, exactly $250$ ml of distilled water is required. When using a measuring cylinder, a student adds $248$ ml. On the other hand, the recipe requires $5$ g of salt, but when weighing it on the balance, $5,1$ g is added. Calculate the relative error in each of the two measurements and determine which of the two processes was more precise.

The capacity of an external hard drive is advertised on the box as $500$ GB (where $1$ commercial GB is equivalent to $10^9$ bytes). However, the computer's operating system counts gigabytes in binary base ($1$ computer GB is equivalent to $2^{30}$ bytes), so the user expected $500$ computer GB. Calculate the absolute error and the percentage relative error made when taking the manufacturer's advertised capacity as an approximation of $500$ computer GB (exact value), expressing both values in bytes.

If approximating an unknown $x$ by the value $3,14$ leads to a relative error of approximately $0,00050696...$ (assuming that the true value is $\pi$), verify mathematically whether this approximation is an underestimate or an overestimate, and check that the relative error formula holds with these data.

At a school track and field meet, a runner completes the $60\text{ m}$ sprint. The electronic photo-finish system records an official time of $7,84\text{ s}$, while a teacher in the stands manually times it at $8,10\text{ s}$.

1. a) Before doing any written calculations, make a mental estimate: will the relative error of the manual stopwatch exceed $5\ %$ or will it be lower? Explain which simple, rounded values you mentally compare the difference in times ($0,26\text{ s}$) and the official time ($7,84\text{ s}$) with to answer.
2. b) Calculate the exact absolute error (with its unit) and the percentage relative error (rounded to two decimal places).
3. c) Check whether your initial estimate was correct and reflect: even though the absolute error is barely a few tenths of a second, why do you think this relative error is unacceptable in high-level competitions?

At an agricultural cooperative in El Bierzo (León), bags of chestnuts are packed with an advertised nominal net weight of $500\text{ g}$. Quality control regulations automatically reject any package if the percentage relative error with respect to the advertised nominal weight exceeds $1,6\ %$.

1. a) Set up the corresponding inequality and determine the interval of actual weights $[x_{\min}, x_{\max}]$ that a bag must have to be accepted by the sorting machine.
2. b) During an inspection, a defective package containing $488\text{ g}$ is found. Calculate the absolute error and the percentage relative error it has with respect to the nominal value of $500\text{ g}$, and indicate whether the machine will reject it.
3. c) If another defective bag is rejected because it was overfilled, having an exact relative error of $2,4\ %$, set up the inverse equation and calculate how many grams of chestnuts that bag actually contains.

The true value of an unknown physical quantity is $x$ and its approximate value, measured by a sensor, is $y$. Given that the relative error made in the measurement is exactly $4\ \%$ and the absolute error is $0,2$ units, set up the corresponding equations and determine the two possible true values of $x$ and their respective approximations $y$.

A surveyor measures the side of a square plot of land, obtaining $L = 40$ m. They know that the relative error of their measuring instrument is $2\ \%$ (that is, $E_r(L) = 0,02$). Show mathematically that the relative error bound made when calculating the area of the land from this measurement is approximately $4\ \%$.

Reason mathematically and discuss under what strict conditions the absolute error of an approximation coincides exactly with its relative error. Provide a concrete numerical example giving the true value and the approximate value that illustrates this peculiar situation.

A mechanical component factory specifies that the ideal length of a bolt is $50$ mm. For a batch of bolts to pass automated quality control, the relative error of the length of each part cannot exceed $0,5\ \%$. Determine, in interval notation, the range of acceptable actual lengths for a bolt to be considered suitable for sale.

Suppose we approximate the irrational number $\sqrt{2}$ by taking only its first $n$ exact decimal places using truncation. Find a mathematical expression bounding the absolute error in terms of $n$. From this, work out the minimum number of decimal places $n$ we must take if we want to guarantee that the relative error of our approximation is strictly less than $10^{-4}$.

In a density experiment, the mass of an object is measured, obtaining $m = 120$ g with an absolute error bound of $2$ g, and its volume $V = 30$ cm$^3$ with an absolute error bound of $1$ cm$^3$. Calculate the maximum and minimum possible actual values of the density ($d = \frac{m}{V}$). Using the central approximate value ($d_{aprox} = \frac{120}{30}$), what is the absolute error bound for the density?

A student has calculated the relative error of a laboratory measurement and obtained the irreducible fraction $\frac{1}{15}$. Given that they also know that the absolute error was exactly $3$ units and that the approximation was an overestimate, reconstruct the problem to determine the exact true value of the quantity and the approximate value used in the calculation.

In the code of an old computer program, the fraction $A = \frac{22}{7}$ is used as a fixed approximation for $\pi$ in calculating the volume of spheres. A client orders the design of a spherical tank of radius $R = 10$ m and contractually requires that the absolute error of the volume calculated by the software be less than $1$ m$^3$. Will the program meet the client's requirement? Justify your answer by calculating the absolute error of the volume (use the value of $\pi$ from a scientific calculator to determine the volume considered to be actual).

For a solar panel project on a house in Toledo, an installer measures a rectangular surface using a laser distance meter that rounds to the nearest centimetre ($p = 1\text{ cm} = 0,01\text{ m}$). The measurements obtained are $1,60\text{ m}$ in length and $1,00\text{ m}$ in width. To calculate the area and bound the error that propagates to the result, two working methods are proposed:

• Method A (Direct estimate): Calculates the area as $1,60 \cdot 1,00 = 1,60\text{ m}^2$ and states that, since the absolute error bound for each side is $0,005\text{ m}$, the absolute error of the area will not exceed the sum of both bounds ($0,005 + 0,005 = 0,01\text{ m}^2$).

• Method B (Bounding by extremes): Determines the interval of possible values for each side according to the rounding bound ($E_a \le p/2$), calculates the extreme values of the area ($Área_{\min}$ and $Área_{\max}$), and obtains the rigorous bound of the absolute error from the largest difference with respect to $1,60\text{ m}^2$.

1. a) Fully develop Method B: state the intervals for the length and width, calculate $Área_{\min}$ and $Área_{\max}$, and deduce the maximum bound of the absolute error of the area.
2. b) Compare the results of both methods and justify why Method A is incorrect for operations such as multiplication, proving which of the two approaches is mathematically rigorous.

The Dirección General de Tráfico (DGT) uses mobile speed cameras on dual carriageways with a generic limit of $120\text{ km/h}$. Official metrological regulations establish that, for speeds above $100\text{ km/h}$, the maximum error margin permitted for a mobile speed camera is a relative error of $7\ %$ with respect to the vehicle's actual speed ($E_r \le 0,07$). To avoid issuing improper fines due to this technical inaccuracy, the DGT only fines if the minimum possible actual speed of the vehicle definitely exceeds $120\text{ km/h}$.

1. a) A mobile speed camera records a reading of $131\text{ km/h}$ for a vehicle. Knowing that $E_r = |v_{\text{real}} - 131| / v_{\text{real}} \le 0,07$, set up and solve the inequality to deduce the minimum actual speed at which this vehicle could have been travelling.
2. b) With the result obtained, provide a reasoned justification as to whether the driver of that vehicle will receive a fine or benefit from the metrological margin of doubt.
3. c) Model the general situation: set up the equation that relates the speed limit ($120\text{ km/h}$) to the $7\ %$ margin, and determine from which exact speed reading on the mobile speed camera (rounded up to the next integer of $\text{km/h}$) the DGT has the absolute mathematical guarantee to fine a driver on that stretch of road.

Unit Challenge. Is the paper you use mathematically perfect? Investigate the presence of irrational numbers in everyday design by solving this challenge.

1. Data collection: Measure with a ruler the length and width (in cm) of an A4 sheet of paper you have at home. Calculate the decimal ratio by dividing $\frac{\text{length}}{\text{width}}$.

2. The real line: The industrial standard states that this theoretical ratio must be exactly the irrational number $\sqrt{2}$. Using GeoGebra or on paper, plot $\sqrt{2}$ on the real line using the Pythagorean theorem ($1^2 + 1^2$) and place your measured value on it to compare them.

3. Approximations and errors: Open a spreadsheet. Truncate and round $\sqrt{2}$ to $2$ and $4$ decimal places. Then, calculate the absolute error ($E_a = |\text{measurement} - \sqrt{2}|$) and the relative percentage error ($E_r = \frac{E_a}{\sqrt{2}} \cdot 100\,\%$) of your sheet.

4. Final product: Hand in a brief report (one page) with the error table, the number line plot, and a conclusion: is the manufacturer's margin of error when cutting the paper negligible?

Assessment criteria:

• Rigorous measurement and correct calculation of approximations and errors ($E_a$ and $E_r$).

• Exact geometric representation of $\sqrt{2}$ and correct ordering on the real line.

• Appropriate use of digital tools (GeoGebra or spreadsheet).

• Clear writing, justification of the result, and correct use of the decimal comma.