Exercises: Approximation and errors

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

Copy the following table into your notebook and complete it by making the indicated approximations for that number:

Type of approximation	To the nearest unit	To the tenths	To the hundredths	To the thousandths
By truncating
By rounding

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

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

1. a) To the nearest unit (miles).
2. b) To the tenths.
3. c) To the hundredths.

Use the following number line to help you visualize where the number falls:

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

1. a) Write this amount rounded to the nearest million.
2. b) Write the amount 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 true factory value is 120 inches. You use a tape measure and get a measurement of 119 inches.

1. a) Calculate the absolute error ($E_a$) made in your measurement. Remember that the absolute error is the positive difference between the true value and the approximate value: $E_a = |V_{real} - V_{aprox}|$.
2. b) If your classmate measures the school hallway, which is actually 5,000 inches long, and gets 4,999 inches, his absolute error is 1 inch, the same as yours. Does this mean that both measurements have the same precision? Explain your answer (you can rely on the concept of relative error).

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 true weight is 200 g and your scale shows 205 g.

2. Case B: You weigh your dog whose true weight is 20,000 g (20 kg) and the scale shows 20,005 g.

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

Look at the following image representing the digital speedometer of an electric bicycle:

The speedometer only shows whole numbers (without decimals) and reads 24 mph by rounding to the nearest unit.

This means that the true speed is not exactly 24 mph, but lies within an interval. Between which two values (lower bound and upper bound) is the true speed of the bicycle?

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

• Pasta: $\$1.85$

• Tomato sauce: $\$2.15$

• Grated cheese: $\$3.48$

• Soft drink: $\$1.25$

1. a) Make a quick estimate of the total cost by rounding each price to the nearest dollar before adding.
2. b) Calculate the exact sum of the prices.
3. c) Calculate the difference (absolute error) between your estimate and the true price. Do you think it was a good estimate to know if you had $\$10$ 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 $r = 5$ in.

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

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 "true" value (using the $\pi$ key on the calculator) is approximately 78.54 sq in., calculate the absolute error made in part a and in part b. Which approximation of $\pi$ is more appropriate to use in 8th grade math?

In the scientific and technological field, it is crucial to know with what precision 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: $1\,024\,576\,000$.

Then, express each result using scientific notation.

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

1. a) Write its decimal expression and round it to the hundredths.
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 for the calculation of the error, you must use the exact value (the fraction) as a reference.

Two students make different measurements for a science project.

• Lucy measures the length of a soccer field and gets $105$ ft, with an absolute error of $0,5$ ft.

• Mark measures the length of a cell phone screen and gets $15$ in, with an absolute error of $0,1$ in.

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

In many gas stations, the price of fuel is shown with three decimal places (thousandths of a dollar), although we pay with coins that only go up to cents (hundredths).

The price of diesel today is $\$1,489$/gallon.

You are going to fill up with exactly $35$ gallons.

1. a) Calculate the exact true price by multiplying the $35$ gallons by the price with three decimal places.
2. b) Calculate the price if we first round the price per gallon to two decimal places ($\$1,49$/gal).
3. c) What is the difference in dollars between both methods? Who benefits from the prior rounding, the customer or the gas station?

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

1. a) Calculate the exact area using radicals.
2. b) Approximate $\sqrt{5}$ and $\sqrt{3}$ to the hundredths and calculate the approximate area with those values.
3. c) Determine the absolute error made when using the approximate values with respect 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 scale reads $54,2$ kg, the true weight could be slightly above or slightly below, but rounded to that decimal.

1. a) Write the interval of possible true values for a measured weight of $54,2$ kg.
2. b) What is the absolute error bound of this scale?

Calculate the relative error made in the following approximations and state which of them is the worst approximation (the one with the highest 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 $\$150.000$ as $\$140.000$.

In the news, they 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 who 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 this headline?

In a nanotechnology laboratory and an astronomical observatory, two different measurements have been taken with 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 chip measurement is "finer" because it is the smaller number, but which of the two measurements has higher 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 helicopter landing pad (helipad) which has a circular shape. To calculate the amount of paint needed, we measure the radius and obtain $r = 12,5 \text{ m}$. We know that the formula for the area is $A = \pi \cdot r^2$.

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 made when using the approximation of $\pi$ compared to the calculator value.
4. d) If a can of paint covers $5 \text{ m}^2$, does the error made affect the number of cans we must buy?

A jewelry company buys gold based on weight. Their digital scale measures with a precision 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) for charging.

This week they have 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 actual total 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 the tenths.
3. c) What is the total absolute error made with this billing method? Who loses out, the buyer or the seller?

We have a water tank in the shape of a rectangular prism. We have measured its dimensions with 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 larger 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.

Let's approximate the irrational number $\sqrt{10}$.

1. a) Write the approximation of $\sqrt{10}$ by rounding to the nearest tenth, hundredth, and thousandth.
2. b) Calculate the absolute error made 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 so that the absolute error is less than $0,0001$? Justify your answer without needing to write all the digits, using the logic of error bounds.

A digital bicycle speedometer shows the speed to one decimal place. At a given moment, it shows $24,5 \text{ mph}$.

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 for the device to show 24.5?
2. b) If the cyclist maintains this speed for exactly 2 hours, what is the maximum difference in miles (absolute error bound) between the distance calculated with the value 24.5 and the possible actual distance?

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

The dimensions of the plot are $\frac{200}{3}$ feet long and $\frac{100}{7}$ feet 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 with 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 and relative error 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 actual values, but knowing that both have been rounded to two decimal places (hundredths).

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

Reason: What is the absolute error bound of the result $C$? (Hint: Think about the maximum and minimum value that A and B could have before being rounded).