Rounding and Approximation Error

In the decimal number system, each digit has a value according to its position. Given the number $58,347$, state which digit occupies each of the following positions:

1. a) Tenths.
2. b) Hundredths.
3. c) Thousandths.
4. d) Ones.

Look at the following number line where the point $P$ has been marked. Write the exact decimal number that represents $P$ and then round this value to the nearest ones place (to the nearest integer).

In Technology class you have measured the thickness of a part with a digital caliper, obtaining the value $3,749$ inches. Approximate this value to the nearest tenth in two different ways:

1. a) By truncation.
2. b) By rounding.

State whether the following statements about the approximation of $4,572$ are true or false:

1. a) Truncation to the nearest tenth is $4,5$.
2. b) Rounding to the nearest tenth is $4,5$.
3. c) Rounding to the nearest hundredth is $4,57$.
4. d) Truncation to the nearest ones place is $4$.

Consider the irrational number $\pi \approx 3,14159265...$.

1. a) Perform a truncation to ten-thousandths.
2. b) Perform a rounding to ten-thousandths.
3. c) Compare both results and explain why in this particular case rounding and truncation do not match.

You are checking the prices of an online shop and you find an accessory for your console that costs $\$24,9632$. Approximate this price by truncation to the nearest hundredth (i.e., to cents).

An athlete has recorded a time of $10,438$ seconds in a 100-meter sprint. Round their time to the nearest hundredth. Was it an overestimate or an underestimate?

Copy and complete the following table in your notebook for the number $7,4628$. For each case, indicate whether the approximation obtained is an underestimate or an overestimate.

The official time of a 100-meter sprint is measured in thousandths of a second. A runner has taken $10,437$ seconds.

1. a) If the news only reports the time rounded to the nearest hundredth, what time will they say?
2. b) If another runner has taken $10,432$ seconds, what time would they be assigned following the same rounding criteria?
3. c) Would it be fair to use this rounding if both runners competed in the same heat? Justify your answer based on accuracy.

A precision sensor in a weather station records the relative humidity of the air. For data storage reasons, the computer system automatically rounds the value to two decimal places. If a value of $64,38\,\%$ appears on the screen, determine the exact interval (minimum value and maximum value) in which the real humidity captured by the sensor lies before being rounded.

Approximate the number $0,666...$ (pure repeating decimal) in the following ways:

1. a) Truncating to the nearest hundredth.
2. b) Rounding to the nearest hundredth.
3. c) Rounding to the nearest tenth.

Represent the pure repeating decimal $2,\widehat{6}$ on a number line. To do this:

1. a) State which two integers it lies between.
2. b) State what number you would obtain if you round to the nearest tenth.
3. c) What is the maximum error bound we make when rounding to the nearest tenth?

Consider the mixed repeating number $n = 4,12\widehat{6}$.

1. a) Obtain its fully simplified generating fraction.
2. b) Approximate the number $n$ by rounding to three decimal places.
3. c) Calculate the difference between the real value (using the generating fraction) and the approximation from section b. What error concept does this result correspond to?

If the side of a square measures $6,8$ inches, estimate its area by first rounding the measurement of the side to the nearest integer. Draw a sketch of the square indicating the approximate measurement.

In a computer shop, the prices of three components are $\$15,99$, $\$32,45$, and $\$12,80$.

If you decide to estimate the total cost by rounding to the nearest dollar before adding, what is the difference between the actual total price and your estimate? Have you made an overestimate or an underestimate?

A trucking company calculates the fuel cost for a long journey. The price of diesel is $\$1,387$/gallon and the truck consumes $32,4$ gallons per 100 miles. The journey is 450 miles.

1. a) Calculate the exact total cost of fuel for the journey.
2. b) If the company estimates the cost by rounding the price per gallon to the nearest hundredth and the consumption to the nearest unit (gallons), what would the estimated budget be?
3. c) Calculate the difference between the actual and estimated cost. Is it a safe estimate for the company (overestimate) or a risky one (underestimate)?

A farmer has measured the area of his rectangular plot and obtained a value of $124,3756$ $m^2$.

1. a) Calculate the value of the area by rounding to the nearest hundredth.
2. b) Calculate the absolute error made when performing this approximation. Remember that the absolute error is $E_a = |V_{real} - V_{aproximado}|$.

In an electronic components factory, a rectangular piece must measure $12,456$ inches long and $8,321$ inches wide. The machine operator rounds both measurements to one decimal place before making the cut.

1. Calculate the area of the piece with the original measurements (without approximating).

2. Calculate the area of the piece with the rounded measurements.

3. Determine the absolute error made in the calculation of the area due to those previous roundings.

We want to design a square plate whose exact area is $30$ sq in. As we cannot measure with infinite precision, we approximate the side of the square ($\sqrt{30}$) by rounding to the nearest hundredth.

1. a) What is the value of the approximate side?
2. b) If we manufacture the plate with this approximate side, what will its actual area be?
3. c) Calculate the absolute error made in the area compared to the original $30$ sq in.

The Voyager 1 space probe is approximately $2,348 \cdot 10^{10}$ miles from Earth. An astronomer decides to approximate this distance by truncating the number to the billions (that is, leaving only two significant figures in the integer part of the standard form: $2,3 \cdot 10^{10}$ mi).

Calculate the absolute error and the relative error made with this approximation. Express the relative error as a percentage rounded to three decimal places.

An influencer is analyzing the growth of their followers. On Monday they had $1.240.500$ followers. On Tuesday, their audience grew by $0,125\,\%$.

1. Calculate the exact number of new followers.

2. If the statistics system only shows integers and uses truncation, what total number of followers will the influencer see on Tuesday?

3. What is the relative error (as a percentage) made by the platform when showing the truncated data compared to the real growth?

A courier company uses drones to deliver parcels. The navigation software estimates the arrival time by dividing the distance by the average speed.

For a journey of $15$ miles at a speed of $35$ mph:

1. Calculate the exact time in hours (as a fraction and as a repeating decimal).

2. The software provides an estimate by rounding the time to the nearest whole minute.

3. Calculate the relative error made by the software when informing the customer in minutes, comparing it with the real time (express both times in the same unit).