Approximation and rounding

You are checking the prices on an online shop and find an accessory for your console that costs $24.9632$ €. Approximate this price by truncation to hundredths (that is, to euro cents).

In a Design and Technology class, you have measured the thickness of a part using digital callipers, obtaining the value $3.749$ mm. Approximate this value to tenths in two different ways:

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

In the decimal place value 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) The tenths.
2. b) The hundredths.
3. c) The thousandths.
4. d) The units.

You go to a stationery shop with a $\text{€}\ 2$ coin and a $\text{€}\ 10$ banknote (a total of $\text{€}\ 12$). You need to buy a compass that costs $\text{€}\ 4.85$, a drawing pad for $\text{€}\ 3.15$, and a set square pair for $\text{€}\ 3.90$.

1. a) Without doing calculations on paper or using a calculator, mentally estimate the total cost by rounding the price of each item to the nearest unit (whole euros). According to your initial estimate, do you have enough money to pay?
2. b) Now calculate the exact total cost of the purchase and determine the absolute error you made in your estimate.
3. c) Was your estimate an overestimate or an underestimate? Reason whether in everyday life, when shopping with a limited budget, it is more prudent to always round prices up (by excess) instead of using the standard rule of rounding to the nearest unit.

Consider the mixed recurring decimal $n = 4.12\widehat{6}$, that is, $n = 4.12666\ldots$ (the 6 repeats indefinitely).

1. a) Obtain its generating fraction simplified as much as possible.

1. b) Approximate the number $n$ by rounding to the thousandths.

1. c) Calculate the absolute value of the difference between the actual value (using the generating fraction) and the approximation from part b). Which error concept does this result correspond to?

Consider the irrational number $\pi \approx 3.14159265...$. In the figure you can see the value of each decimal place.

1. a) Truncate to the ten-thousandths.

1. b) Round to the ten-thousandths.

1. c) Compare both results and explain why, in this particular case, rounding and truncation do not coincide.

In a metal carpentry factory, aluminium profiles for windows are cut with an automatic machine. The operator can configure the machine using two different methods of approximation to the nearest whole millimetre:

• Method 1 (Truncation): cuts the profile simply by dropping the tenths of a millimetre.

• Method 2 (Rounding): cuts the profile by applying the mathematical rule of rounding to the nearest millimetre unit.

In an order, three pieces must be cut whose exact design measurements are: Piece A: $120.3\text{ mm}$; Piece B: $120.5\text{ mm}$; Piece C: $120.9\text{ mm}$.

1. a) Determine the cutting measurement that each method will produce for the three pieces and calculate the absolute error made in each of the six cuts.
2. b) What is the maximum absolute error that can be made with Method 1 for any decimal measurement? And with Method 2?
3. c) Compare both methods and justify which of the two is preferable to use in the factory if the goal is to minimise the maximum error in the cuts. Also reason whether either of the methods introduces a systematic bias (always cutting below what is required).

Complete the following table in your notebook for the number $7.4628$. For each case, indicate whether the approximation obtained by rounding is rounded down or rounded up.

An unknown positive decimal number $x$ has exactly three significant decimal places (its expression is of the form $a,bcd$). Find out what number it is knowing that it simultaneously meets the following four conditions:

1. Its integer part is $6$.

2. Its truncation to the tenths is $6.3$.

3. Its rounding to the hundredths is $6.40$.

4. The sum of its three decimal digits ($b + c + d$) is exactly equal to $20$.

1. a) From conditions 2 and 3, deduce with reasoning which digits must necessarily occupy the tenths ($b$) and hundredths ($c$) places. Justify why the hundredths digit cannot be a $4$.
2. b) Determine all the possible digits for the thousandths ($d$) that make the rounding to the hundredths equal $6.40$.
3. c) Use condition 4 to find the final value of the number $x$ and calculate the absolute error made if this number is replaced by its rounding to the hundredths.

A precision sensor at a weather station records the relative humidity of the air. For data storage reasons, the computer system automatically rounds the value to the hundredths. If the screen shows a value of $64.38\,\%$, determine the exact interval (minimum value and maximum value) in which the actual humidity captured by the sensor lies before being rounded.

The figure shows the number line with the value on the screen and the two values with hundredths next to it.

The official time for a 100-metre sprint is measured to thousandths of a second. A runner took $10.437$ seconds.

1. a) If the news only reports the time rounded to the nearest hundredth of a second, what time will they state?
2. b) If another runner took $10.432$ seconds, what time would they be assigned using the same rounding criterion?
3. c) Would it be fair to use this rounding if both runners were competing in the same heat? Reason your answer based on precision.

The space probe Voyager 1 is approximately $2.348 \times 10^{10}$ kilometres from Earth. An astronomer decides to approximate this distance by truncating the number to the nearest thousand million (that is, keeping only two significant figures: $2.3 \times 10^{10}$ km).

In the figure, both values, the exact one and the approximate one, are marked on a number line.

Calculate the absolute error and the relative error made with this approximation. Express the relative error as a percentage rounded to the nearest thousandth.

In an electronics component factory, a rectangular part must measure $12.456$ mm long and $8.321$ mm wide. The machine operator rounds both measurements to one decimal place before making the cut.

1. Calculate the area of the part with the original measurements (unapproximated).

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

3. Determine the absolute error made in the area calculation due to these prior roundings.

A farmer has measured the area of his rectangular plot of land and obtained a value of $124.3756$ $m^2$.

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

An influencer is analysing their follower growth. 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 analytics system only displays whole numbers 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 displaying the truncated figure compared to the actual 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 $15$ km at a speed of $35$ km/h:

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

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

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