Exercises: Approximation and rounding

In a video game shop, the price of a poster is €$7.82$. If you only have one-euro coins and you want to know, approximately, how many whole euros it costs:

On the number line in the figure, the two long marks are two consecutive whole numbers (the ones marked with “?”) and each small mark is one tenth.

1. a) Between which two whole numbers does the price lie?
2. b) Which of the two numbers is it closer to? Round to the nearest unit.

To carry out an experiment in Science class, we need to measure exactly $4.576$ grams of salt. Complete the following table by applying the two requested approximation methods:

Approximation to hundredths	Result
Truncation
Rounding

A 100-metre sprinter has recorded a time of $12.45$ seconds on the stopwatch. If we want to express this time with a single decimal digit:

1. a) Round to the nearest tenth.
2. b) Was it an upward or a downward approximation? Explain why you looked at the hundredths digit.

On the number line, the runner's time is marked with a point between $12.4$ and $12.5$, and each small mark represents $0.01$ s.

Look at the following decimal numbers and round them to one decimal place (tenths):

1. a) $3.14$
2. b) $9.68$
3. c) $0.45$
4. d) $1.03$

The download speed of an internet connection is $92.578$ Mbps. The technician tells us that, for simplicity, we can truncate the number to two decimal places (hundredths).

1. a) What value will we obtain?
2. b) Draw a small number line and plot the original value and the truncated value.

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

In the supermarket, fruit scales usually round the weight. If we buy a bag of oranges that weighs $2.341$ kg, state what weight would be displayed if the scale:

1. a) Rounds to the units.
2. b) Rounds to the hundredths.

Approximate the number $\pi \approx 3.141592...$ in the following ways:

1. a) By rounding to three decimal places (thousandths).
2. b) By truncation to three decimal places (thousandths).
3. c) In which case is the result larger?

Imagine you are going to pay with a $20$ € note for a purchase that costs $14.23$ €. The cashier has no small cent coins and suggests rounding the price to the nearest tenth of a euro (that is, to a multiple of $10$ cents, with a single decimal digit).

1. a) What would the new rounded price be?
2. b) How much money would the cashier give you back after the rounding?

Cian has to approximate the number $6.378$ to the nearest hundredth in a class exercise. In his notebook, he writes: "$6.378 \approx 6.37$ because the thousandths digit $8$ does not reach ten, so it does not change the preceding digit."

1. a) Explain what conceptual error Cian makes and what approximation method he actually applied without realizing it.
2. b) Perform the correct rounding to the nearest hundredth and explain, using the distance between decimal numbers, why your result is closer to the original number than the value Cian wrote.

In a local stationery shop, Sara wants to buy a notebook that costs $2.85$ €, a pack of markers for $4.15$ €, and a folder for $1.90$ €. She has a $5$ € coin and two $2$ € coins in her purse.

1. a) Before calculating the exact price, estimate the total expenditure by rounding the price of each item to the nearest whole euro. According to your previous estimate, will she have enough money to pay for the purchase?
2. b) Calculate the exact cost of the purchase by adding the three prices.
3. c) Check whether your estimate was accurate by calculating the difference between the exact price and the estimated price. Do you think the estimate was useful for making a quick decision before going to the checkout? Justify your answer.

Copy and complete the following table in your exercise book, finding the given approximations for the decimal number $n = 14.3728$.

Approximation to...	Truncation	Rounding
Tenths
Hundredths
Thousandths

Look at the following number line where a decimal number $x$ is shown under a magnifying glass.

Knowing that the orange dot represents the number $x = 3.774$:

1. a) Between which two hundredths does the number $x$ lie?
2. b) If you had to approximate $x$ to tenths by rounding, which value would you choose? Justify your answer by looking at the position of the dot on the line relative to the midpoint ($3.75$).

In an app store for video games, the price of a new texture pack is $\text{€}\ 12.957$.

1. a) If the store decides to truncate the price to two decimal places to display it in the catalogue, what price will appear?
2. b) If the payment system rounds to two decimal places (euro cents) when taking the final payment, how much will the user actually pay?
3. c) What is the difference in price between the truncated price and the rounded price?

Three athletes took part in a 60-metre sprint at school. The times recorded by the stopwatch were as follows:

• Aoife: $8.429$ seconds.

• Cian: $8.421$ seconds.

• Darragh: $8.425$ seconds.

1. a) If the results are published on the school website rounded to two decimal places (hundredths), what time will appear for each person?
2. b) Looking at the rounded results, does it seem that they took the same time? Is that true according to the original times?

A mathematics teacher calculates the average mark for an exam and gets $6.4782$.

1. a) Approximate the mark to one decimal place by truncation.
2. b) Approximate the mark to one decimal place by rounding.
3. c) Which of the two approximations is more favourable to the student? Why do you think school report cards usually use rounding instead of truncation?

We want to put a protective fence around a small rectangular garden. When measuring with a digital tape measure, we find that the sides measure $4.56$ metres and $3.27$ metres.

1. a) Calculate the perimetre of the garden.
2. b) If the garden owner buys the fence from a DIY store that only sells in whole metres and always rounds up (rounding up to the nearest unit) to make sure there is enough material, how many metres of fencing will he buy?

Think and answer this decimal number riddle:

"I am a number with three decimal places. If you truncate me to two decimal places (hundredths), I am $7.42$. However, if you round me to two decimal places, I am $7.43$."

Write three different numbers that satisfy these conditions. What is the smallest possible number it could be? And the largest?

A pair of precision scales shows that a bag of apples weighs $2.345$ kg.

1. a) Express the weight in grams (remember that $1$ kg = $1,000$ g).
2. b) If the shopkeeper tells us that the weight is approximately $2.3$ kg, what type of approximation has he used?
3. c) Calculate the error made in the shopkeeper's approximation (the difference between the actual weight and the approximate weight). Express your answer in kilograms and in grams.

The athletics club of a town in County Kilkenny is going to paint a continuous white line around its training track. The total length they must paint measures exactly $345\text{ m}$. According to the manufacturer's specifications, $1\text{ litre}$ of special paint can cover $25\text{ m}$ of line. In the local shop, the paint is sold exclusively in sealed $1\text{ litre}$ tins (fractions of a tin are not sold).

1. a) Perform the division with two decimal places to find out how many exact litres of paint are theoretically needed.
2. b) If you apply the standard mathematical rounding rule to the nearest whole unit on that decimal result, how many tins would you buy? If you buy that amount, could you paint the entire track or would you run out of paint? Check it by calculating the metres you could paint with those tins.
3. c) Justify what type of intentional approximation must be applied in this material purchasing problem and how many tins must finally be acquired to guarantee the complete job.

We are looking for a mysterious decimal number $N$ that has three decimal places. To discover it, we have four clues:

1. Its integer part is the number $7$.

2. If we truncate $N$ to the tenths, we obtain $7.4$.

3. If we round $N$ to the hundredths, we obtain $7.46$.

4. Its thousandths digit is an odd number greater than $3$.

1. a) Find all possible decimal numbers that satisfy these four conditions simultaneously.
2. b) Take the smallest of the numbers you found and truncate it to the hundredths. Does this truncation coincide with the rounding to the hundredths from clue 3? Explain why this coincidence occurs or does not occur.

We are investigating a "mystery" decimal number, which we will call $N$, which has three decimal places. We know the following clues about it:

1. Its integer part is the smallest two-digit prime number.

2. If we round it to two decimal places (hundredths), we get $11.38$.

3. If we truncate it to one decimal place (tenths), we get $11.3$.

4. The sum of its three decimal digits is $19$.

Which number is it? Reason your answer by ruling out the options that do not meet the conditions.

At a budget petrol station, the price per litre of Unleaded 95 petrol is $1.419$ €. A group of friends fills up the tank of their van for a trip with $52.6$ litres.

The petrol station's till is programmed to calculate the total and then round the final result to the nearest euro cent (two decimal places). However, one of the friends insists that they should first truncate the price per litre to the nearest cent and then multiply by the number of litres.

Calculate the difference in price between both methods. Which of the two benefits the customer and which benefits the petrol station?

In a biology laboratory, the growth of a bacterium is being measured. Its current length is $0.03478$ millimetres.

• Scientist A decides to use an approximation by rounding to three decimal places (thousandths).

• Scientist B prefers to use an approximation by truncation to four decimal places (ten-thousandths).

State which of the two values is larger and calculate the exact difference between both approximations. Which of the two do you think is "closer" to the actual value of the bacterium?

Find a decimal number that satisfies these three "boundary" conditions simultaneously:

• It is greater than $7.45$ and less than $7.46$.

• When rounded to thousandths, the result is $7.458$.

• The sum of its four decimal digits is equal to $25$.

Is this number unique or are there several possibilities? Justify your answer.

A Year 7 class is designing a part for a 3D printer. The software gives a measurement of $12.666...$ (recurring) millimetres for the thickness of a fitting. The printer has three precision modes:

• Eco Mode: Truncates to tenths.

• Standard Mode: Rounds to hundredths.

• Pro Mode: Rounds to thousandths.

Calculate the value that the printer would use in each mode. If the slot where the part must fit measures exactly $12.67$ mm, in which modes will the part fit and in which will it not? (Assume that to fit, the part must be less than or equal to the slot size).

Look at the following number line where two approximations of the number $X$ are shown.

Knowing that the truncation to hundredths is $3.14$ and the rounding to hundredths is $3.15$, state whether the following statements are true or false, justifying each one:

• a) The thousandths digit of $X$ must be greater than or equal to $5$.

• b) The number $X$ could be $3.1449$.

• c) If we add $0.001$ to the number $X$, its truncation to hundredths will still be $3.14$.

A modern chef needs to weigh three ingredients using high-precision scales that measure to thousandths of a gram:

• Salt: $2.341$ g

• Pepper: $1.256$ g

• Saffron: $0.897$ g

The chef wants to find the total weight of the spice mixture. Compare these two procedures:

1. Procedure 1: Add the three exact weights and then round the final result to tenths.

2. Procedure 2: Round each ingredient to tenths and then add them.

Calculate both results. Are they equal? Explain why rounding individual values can lead to a larger error than rounding at the end of the calculation.

Create a real-life problem (shopping, sports, science...) where both of these conditions are met simultaneously:

1. It is necessary to use a number with at least four decimal places.

2. The difference between truncating that number to hundredths and rounding it to hundredths is exactly $0.01$.

Write the full question statement and explain how it would be solved using the rounding and truncation rules.

A small agricultural cooperative in Wexford is preparing to ship four fruit orders to a supermarket. The exact weight recorded by the precision scale for each order is:

• Order 1: $142.38\text{ kg}$

• Order 2: $85.36\text{ kg}$

• Order 3: $64.39\text{ kg}$

• Order 4: $107.37\text{ kg}$

To organize the transport delivery note, two workers propose approximating the weights to the nearest tenth of a kilogram:

• Aoife's method: Round each individual order to the tenths first and then add the four rounded values.

• Niamh's method: Add the four exact weights first and only round the final result to the tenths.

1. a) Calculate step by step the total result obtained with Aoife's method and the one obtained with Niamh's method.
2. b) Calculate the exact sum without approximating and determine the error made by each method (the positive difference between the method's result and the exact value).
3. c) In light of the results, which of the two methods is more reliable? Justify why rounding at each intermediate step usually accumulates a larger error than rounding only at the end.

At a Met Éireann weather station, digital sensors measure temperature with several decimal places, but on the public screen of the outdoor display, the temperature rounded to the tenths is shown. On a hot day, the screen shows exactly $38.4\text{ }^\circ\text{C}$.

1. a) Between which two possible values is the actual temperature $T$ recorded by the sensor located for that value to be displayed? Write your answer in the form of an inequality like $A \le T < B$, using two decimal places for $A$ and for $B$.
2. b) What is the greatest possible difference (the maximum error) that can exist between the temperature we see on the screen ($38.4\text{ }^\circ\text{C}$) and the actual temperature from the sensor?
3. c) If the outdoor display were programmed to truncate to the tenths instead of rounding, what would the new interval $C \le T < D$ for the actual temperature be? Reasonably explain why rounding minimizes the maximum error compared to truncation and what fundamental role the digit $5$ plays in that advantage.

Topic challenge. How much does the energy you use at home cost? Carry out a mini energy audit in a spreadsheet following these steps:

1. Data collection: Access the official EirGrid website (eirgrid.ie) or your electricity supplier's website (e.g. Electric Ireland, Bord Gáis Energy) and write down yesterday's small consumer price ($\text{€}/kWh$) at 3 different hours: early morning, midday, and night.

2. Approximation and calculation: Official prices have up to five decimal places (for example, $0.14823$ €). Round each one to the nearest hundredth (cents). Then, calculate the cost of running a $2$ kW oven for $1.25$ hours during each time slot by multiplying power, time, and the rounded price.

3. Splitting and generating fraction: Add the three costs together and divide the total amount among $3$ household members. Identify what type of decimal number results (terminating, pure recurring, or mixed recurring) and find its irreducible generating fraction to express the exact share without rounding errors.

4. Proposal: Prepare a brief, reasoned proposal for your family with a comparative table, recommending the most economical time to cook and calculating the weekly savings.

Assessment criteria:

• Retrieval of real data from EirGrid or an electricity supplier and accuracy in rounding to the nearest hundredth.

• Correctness in combined operations with decimal numbers.

• Identification of the type of decimal and exact calculation of the generating fraction.

• Clear and coherent economic justification in the final proposal.