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:

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:

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?

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?

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?

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.

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).

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.

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.

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?

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?

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?

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?