Exercises: Approximation and 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 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).

An athlete recorded a time of $10.438$ seconds in a 100-metre sprint. Round their time to hundredths. Was the rounding an overestimate or an underestimate?

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.

Approximate the number $0.666...$ (pure recurring) in the following ways:

1. a) Truncating to hundredths.
2. b) Rounding to hundredths.
3. c) Rounding to tenths.

Look at the following number line where point $P$ is marked. Write down the exact decimal number represented by $P$ and then round this value to the nearest whole number.

If the side of a square measures $6.8$ cm, estimate its area by first rounding the side length to the nearest whole number. Draw a sketch of the square showing the approximate measurement.

You can do it on paper and send a photo with your answer.

State whether the following statements about approximating $4.572$ are true or false:

1. a) Truncating to tenths gives $4.5$.
2. b) Rounding to tenths gives $4.5$.
3. c) Rounding to hundredths gives $4.57$.
4. d) Truncating to units gives $4$.

In a 2nd Year Mathematics class, two students try to approximate decimal numbers to the hundredths, making common mistakes:

• Cian has to round $12.6449$ to the hundredths and reasons as follows: "Since the last digit is a $9$, which is greater than or equal to $5$, it rounds up the preceding digit, which goes from $4$ to $5$; that $5$ in turn rounds up the preceding digit; this way, I apply the change from right to left in a chain until reaching the hundredths and I get $12.65$".

• Aoife has to round $8.3250$ to the hundredths and states: "Since the digit following the hundredths is a $5$, the number stays the same at $8.32$, because before a $5$ you should not alter the previous digit".

1. a) Explain what the mistake in Cian's procedure is (known as "cascade rounding") and indicate what is the only digit that determines the rounding to the hundredths.
2. b) Explain the mistake in Aoife's statement and recall what the international mathematical convention is when the following digit is a $5$.
3. c) Correct both exercises by providing the correct rounding to the hundredths for each number and indicating whether the approximation obtained is an overestimate (by excess) or an underestimate (by defect).

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.

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.

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 $E_a = |V_{real} - V_{aproximado}|$.

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 whole euro before adding, what is the difference between the actual total price and your estimate? Have you overestimated or underestimated?

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

1. a) State which two integers it lies between.
2. b) State which number you would get if you rounded to one decimal place.
3. c) What is the maximum error bound when rounding to one decimal place?

A tailor needs to buy fabric to make 7 cushion covers. Each cover requires exactly $1.428$ metres of fabric.

1. a) Calculate the exact total amount of fabric needed.
2. b) If the tailor decides to buy the fabric by rounding the total obtained to one decimal place, will he have fabric left over or be short? Calculate the exact value of that difference.

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.

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.

A transport company calculates the cost of fuel for a long trip. The price of diesel is $1.387$ €/litre and the lorry consumes $32.4$ litres per 100 km. The trip is 450 km long.

1. a) Calculate the exact total cost of fuel for the trip.
2. b) If the company estimates the cost by rounding the price per litre to hundredths and the consumption to units (litres), what would the estimated budget be?
3. c) Calculate the difference between the actual cost and the estimated cost. Is it a safe estimation for the company (an overestimate) or a risky one (an underestimate)?

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.

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).

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 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.

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?

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?

We want to design a square plate whose exact area is $30$ cm². Since we cannot measure with infinite precision, we approximate the side of the square ($\sqrt{30}$) by rounding to the hundredths.

1. a) What is the value of the approximate side?

1. b) If we make the plate with that approximate side, what will its actual area be?

1. c) Calculate the absolute error made in the area with respect to the original $30$ cm².

Two methods of approximation for the irrational number $\pi$ are being compared.

• Method A uses rounding to four decimal places ($3.1416$).

• Method B uses Archimedes' fraction $\frac{22}{7}$.

Calculate the absolute error of each method given that the value of $\pi$ to seven decimal places is $3.1415926$. Which of the two methods is more precise? Justify your answer by comparing the errors.

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).

Uisce Éireann (Irish Water), the public utility managing water resources in Ireland, publishes in its official bulletin that the water stored in reservoirs in the country amounts to $742.8\text{ hm}^3$ (cubic hectometres), a figure that the computer system rounds to the tenths.

1. a) Write the exact interval $[V_{\text{min}}, V_{\text{max}})$ in which the actual volume of stored water lies before being rounded.
2. b) Knowing that $1\text{ hm}^3 = 1\,000\,000\text{ m}^3$ and that the population of the country consumes an average of $1\,250\,000\text{ m}^3$ of water per day, calculate the difference in days of water supply between the minimum value and the maximum value of that interval.
3. c) Calculate the upper bound of the absolute error and the maximum percentage relative error (rounding the latter to three decimal places) that the published figure of $742.8\text{ hm}^3$ can have with respect to the actual volume.
4. d) Based on the previous results, interpret whether an error from rounding to tenths is negligible or crucial for hydrological planning at the scale of a whole country.

In a pharmaceutical packaging plant in Ireland, a quality audit is carried out on two production lines:

• Line A: packages $20\text{ ml}$ saline solution vials. The filling sensor records an actual sample of $19.4\text{ ml}$ as $20\text{ ml}$.

• Line B: packages $5000\text{ ml}$ hospital disinfectant jugs. The sensor records an actual sample of $4980\text{ ml}$ as $5000\text{ ml}$.

Seán, the trainee auditor, states in his preliminary report:

"Line A is much more accurate and acceptable than Line B, since the absolute error made in Line A is only $0.6\text{ ml}$, whereas in Line B the absolute error is $20\text{ ml}$, that is, more than thirty times greater".

1. a) Identify and explain the serious conceptual mistake in Seán's reasoning regarding the precision of the two processes.
2. b) Calculate the absolute error and the percentage relative error for each of the two measurements with respect to their actual values (round percentages to the hundredths).
3. c) In light of the relative errors, refute Seán's conclusion by demonstrating which of the two lines is actually making a larger error in proportion to the size of what it measures.
4. d) If the quality regulations of the Health Products Regulatory Authority (HPRA) require that the packaging relative error must never exceed $1\,\%$, determine with justification which production line would pass the inspection and which one should be stopped and recalibrated.

Unit challenge. How do decimals affect home energy spending? You are going to audit the cost of electricity using real data from EirGrid.

1. Collection and concepts: Visit the EirGrid website (eirgrid.ie) and note the domestic electricity price (€/kWh) across three time bands of yesterday: peak, standard, and off-peak (for example, $0.18342$). Classify each number according to its type: terminating decimal, pure recurring decimal, or mixed recurring decimal.

2. Generating fraction and spreadsheet: Design a table in a spreadsheet. Assume that an appliance with a power of $1.45\text{ kW}$ runs for $2.3$ hours during peak hours. Multiply these decimal values to find the total consumption. Then, take the average price of the day, express it as a simplified recurring decimal, and determine its generating fraction in your notebook to calculate the exact cost in fractional form.

3. Approximation and report: Round and truncate the calculated amounts to the hundredths (€0.01, cents). Compare both results by calculating the error incurred and write a brief proposal (maximum one page) with recommendations to reduce the household bill based on the analysed time bands.

The following will be assessed:

• Correct identification of the decimal type and rigour when finding the generating fraction.

• Precision in operations with decimals and efficient use of the spreadsheet.

• Correctness in approximation techniques (rounding and truncation) and error analysis.

• Coherent justification of the savings proposal based on EirGrid data.