Exercises: Approximation and errors

The irrational number $\sqrt{10} = 3.16227766...$ appears on your calculator screen.

Copy the following table into your exercise book and complete it by carrying out the indicated approximations for that number:

Type of approximation	To the nearest unit	To the nearest tenth	To the nearest hundredth	To the nearest thousandth
By truncation
By rounding

You are helping to organize a charity run at your school. When calculating the exact distance of a section of the route with the GPS, you get 2.3458 km.

To make the information posters, you need to simplify this number. Write what this distance would be if we approximate by rounding:

1. a) To the units (km).

1. b) To the tenths.

1. c) To the hundredths.

For part c), use the following number line, which shows the section between 2.34 and 2.35 and where the number falls:

A famous streamer has uploaded a video that has exactly 3 894 215 views.

1. a) Write down this quantity rounded to the nearest million.
2. b) Write down the quantity rounded to the nearest hundred thousand.
3. c) How many significant figures have you used in part b?

When measuring the length of your study desk, the actual factory value is 120 cm. You use a tape measure and get a measurement of 119 cm.

1. a) Calculate the absolute error ($E_a$) made in your measurement. Remember that the absolute error is the positive difference between the actual value and the approximate value: $E_a = |V_{real} - V_{aprox}|$.
2. b) If your classmate measures the school corridor, which actually measures 5 000 cm, and gets 4 999 cm, their absolute error is 1 cm, the same as yours. Does this mean both measurements have the same precision? Reason your answer (you can use the concept of relative error to support your answer).

We are going to calculate the relative error ($E_r$) to compare two different weighings. Remember that the formula is $E_r = \frac{E_a}{|V_{real}|}$.

1. Case A: You weigh an apple whose actual weight is 200 g and your scales show 205 g.

2. Case B: You weigh your dog whose actual weight is 20 000 g (20 kg) and the scales show 20 005 g.

Calculate the relative error in both cases (you can express it as a decimal or percentage) and state which measurement was more precise (higher quality).

Look at the following image showing the digital speedometer of an e-bike:

The speedometer only displays integers (no decimals) and reads 24 km/h rounded to the nearest unit.

This means that the actual speed is not exactly 24 km/h, but falls within an interval. Between which two values (lower bound and upper bound) does the actual speed of the bike lie?

You have gone to the supermarket to buy ingredients for dinner. The actual prices of the products are:

• Pasta: 111 pesos

• Tomato sauce: 129 pesos

• Grated cheese: 208.80 pesos

• Soft drink: 75 pesos

1. a) Make a quick estimate of the total cost by rounding each price to the nearest unit before adding.
2. b) Calculate the exact sum of the prices.
3. c) Calculate the difference (absolute error) between your estimate and the actual price. Do you think it was a good estimate to know if you had 600 pesos in your pocket?

We want to calculate the area of a circle whose radius we have measured with a ruler.

The measurement of the radius is $r = 5$ cm.

We know that the area of a circle is $A = \pi \times r^2$.

1. a) If we use the approximation of $\pi \approx 3$, what is the calculated area?
2. b) If we use the approximation of $\pi \approx 3.14$, what is the calculated area?
3. c) Knowing that the "actual" value (using the $\pi$ key on the calculator) is approximately 78.54 cm², calculate the absolute error incurred in part a and in part b. Which approximation of $\pi$ is more appropriate to use in 3rd grade of secondary math?

In science and technology, it is crucial to know how precisely we are working. Round the following numbers to three significant figures:

1. a) The number of views of a viral video: $14\,592\,301$.
2. b) The mass of a dust particle: $0.00040961$ grams.
3. c) The number $\pi$: $3.141592...$
4. d) The capacity of a hard drive in bytes: $1\,024\,576\,000$.

Afterwards, express each result using standard form.

We are going to analyse the error made when using decimals instead of fractions. Consider the fraction $\frac{7}{3}$.

1. a) Write down its decimal form and round it to two decimal places.
2. b) Calculate the absolute error ($E_a$) made with this approximation.
3. c) Calculate the relative error ($E_r$) and express it as a percentage.

Note: Remember that to calculate the error, you must use the exact value (the fraction) as a reference.

Two students carry out different measurements for a science project.

• Lucía measures the length of a football pitch and gets $105$ m, with an absolute error of $0.5$ m.

• Marcos measures the length of a mobile phone screen and gets $15$ cm, with an absolute error of $0.1$ cm.

Which of the two measurements is more precise (has higher quality)? Justify your answer by calculating the relative errors of both measurements.

At many gas stations, the price of fuel is displayed with three decimal places (thousandths of a pesos), although we pay with coins that only go down to cents (hundredths).

The price of diesel today is $1.489$ pesos/liter.

You are going to refuel with exactly $35$ liters.

1. a) Calculate the exact actual price by multiplying the $35$ liters by the price with three decimal places.
2. b) Calculate the price if we first round the price per liter to two decimal places ($1.49$ pesos/L).
3. c) What is the difference in pesos between both methods? Who benefits from rounding first, the customer or the gas station?

We want to calculate the area of a rectangle whose base is $\sqrt{5}$ cm and whose height is $\sqrt{3}$ cm.

1. a) Calculate the exact area using radicals.
2. b) Approximate $\sqrt{5}$ and $\sqrt{3}$ to two decimal places and calculate the approximate area with those values.
3. c) Determine the absolute error made when using the approximate values compared to the exact value from part a.

Digital bathroom scales usually have a precision of $100$ grams ($0.1$ kg). This means that if the scales read $54.2$ kg, the actual weight could be slightly above or slightly below, but rounded to that decimal place.

1. a) Write down the interval of possible actual values for a measured weight of $54.2$ kg.
2. b) What is the absolute error bound for these scales?

Calculate the relative error made in the following approximations and state which of them is the worst approximation (the one with the largest relative error):

1. a) Approximate the fraction $\frac{1}{3}$ as $0.33$.
2. b) Approximate the number $\sqrt{2}$ as $1.41$.
3. c) Approximate a budget of $9\,000\,000$ pesos as $8\,400\,000$ pesos.

On the news, it was announced that "approximately 25 000 people" attended a demonstration.

1. a) If we know that this figure was obtained by rounding to the nearest thousand, what is the maximum and minimum number of people that could have actually attended?
2. b) Represent this interval on the number line.
3. c) What is the maximum absolute error that could be made with that headline?

In a nanotechnology laboratory and an astronomical observatory, two different measurements have been taken using their respective precision instruments:

1. The length of a microscopic chip: $0.00045 \text{ m}$ (with a possible error of $\pm 0.00001 \text{ m}$).

2. The distance between two satellites: $45000 \text{ km}$ (with a possible error of $\pm 100 \text{ km}$).

Without performing complex calculations, one might think that the measurement of the chip is "more refined" because it is a smaller number, but which of the two measurements has greater scientific quality or precision?

Justify your answer by calculating the relative error of both measurements and expressing it as a percentage.

We want to paint a circular landing pad for helicopters (helipad). To calculate the amount of paint needed, we measure the radius and get $r = 12.5 \text{ m}$. We know that the formula for the area is $A = \pi \times r^2$.

1. a) Calculate the "exact" area using the $\pi$ key on your calculator.
2. b) Calculate the approximate area using the rounded value $\pi \approx 3.14$.
3. c) Calculate the absolute error and the relative error committed when using the approximation of $\pi$ compared to the calculator value.
4. d) If a tin of paint covers $5 \text{ m}^2$, does the error committed affect the number of tins we need to buy?

A jewellery company buys gold based on weight. Its digital scale measures with an accuracy of hundredths of a gram ($0.01 \text{ g}$). However, the invoice they issue truncates the weight to a single decimal place (tenths of a gram) to charge for it.

This week they bought 4 batches of gold whose actual weights on the scale were:

• Batch A: $12.48 \text{ g}$

• Batch B: $25.19 \text{ g}$

• Batch C: $8.76 \text{ g}$

• Batch D: $15.95 \text{ g}$

1. a) Calculate the total actual weight by adding the exact amounts from the scale.
2. b) Calculate the total weight that appears on the invoices by adding the amounts truncated to tenths.
3. c) What is the total absolute error committed with this invoicing method? Who loses out, the buyer or the seller?

We have a water tank in the shape of a cuboid (rectangular box). We measured its dimensions using a tape measure that only marks decimeters, so our measurements have an absolute error bound of $0.5 \text{ dm}$ (that is, the actual value could be 0.5 dm greater or smaller).

The measurements obtained are:

• Length: $20 \text{ dm}$

• Width: $10 \text{ dm}$

• Height: $15 \text{ dm}$

Calculate the volume of the tank ($V = \text{largo} \times \text{ancho} \times \text{alto}$) and determine between which two values (interval) the actual volume of the tank lies, considering the error bounds of each measurement.

We are going to approximate the irrational number $\sqrt{10}$.

1. a) Write the approximation of $\sqrt{10}$ by rounding to tenths, hundredths and thousandths.
2. b) Calculate the absolute error committed in each of the three previous cases (using the calculator value as the "actual value").
3. c) How many decimal places do we need to take for the absolute error to be less than $0.0001$? Justify your answer without needing to write down all the digits, using the logic of error bounds.

A digital bicycle speedometer shows the speed with one decimal place. At a given moment, it displays $24.5 \text{ km/h}$.

We know that this device approximates by rounding to the nearest tenth.

1. a) Write the interval of possible actual values in which the exact speed of the bicycle lies. That is, what is the minimum and maximum speed the cyclist could be traveling at for the device to display 24.5?
2. b) If the cyclist maintains that speed for 2 exact hours, what is the maximum difference in kilometers (absolute error bound) between the distance calculated using the value 24.5 and the possible actual distance?

A rectangular plot of land is to be divided into plots to build houses.

The dimensions of the plot are $\frac{200}{3}$ meters long and $\frac{100}{7}$ meters wide.

If we work with fractions, the calculation is exact, but the workers need decimal numbers.

1. a) Calculate the exact area of the plot using fractions.
2. b) Approximate each dimension to two decimal places (hundredths) and calculate the area using these approximate values.
3. c) What is the relative error made when calculating the area with the approximate values compared to the exact value of the fraction?

Calculate step by step the absolute error bound and the relative error bound in the following combined operation, taking into account the order of operations.

Given the numbers $A = 2.86$ and $B = 4.13$, approximations of some unknown real values, but knowing that both have been rounded to the nearest hundredth.

1. a) Calculate the value of $C = \frac{A + B}{2}$.

1. b) Explain: what is the absolute error bound of the result $C$? (Hint: think about the maximum and minimum values that $A$ and $B$ could have had before being rounded).

1. c) Calculate the relative error bound of $C$, by dividing the absolute error bound by the value of $C$.