Exercises: Decimal numbers

Classify the following decimal numbers, indicating whether they are terminating, purely recurring, mixed recurring or irrational:

• $A = 3,14159$

• $B = 2,\widehat{7}$

• $C = 0,1\widehat{23}$

• $D = 1,010010001\dots$

Without performing the division, determine the type of decimal expansion (terminating, purely recurring or mixed recurring) corresponding to each of the following fractions by analysing the prime factors of their denominators:

• $\frac{7}{20}$

• $\frac{5}{21}$

• $\frac{11}{6}$

Find the irreducible generating fraction of the following purely recurring decimal numbers:

• $3,\widehat{45}$

• $0,\widehat{7}$

• $12,\widehat{3}$

Determine the irreducible generating fraction of the following mixed recurring decimal numbers:

• $0,2\widehat{71}$

• $1,0\widehat{5}$

• $3,12\widehat{4}$

Marcos has to classify the type of decimal number generated by the fraction $\frac{27}{150}$ without carrying out the division. In his notebook, he writes:

"I factorise the denominator into prime factors: $150 = 2 \cdot 3 \cdot 5^2$. Since $2$ and $5$ appear, but the prime factor $3$ is also present, the fraction must necessarily generate a mixed recurring decimal".

However, when entering $\frac{27}{150}$ into the calculator, the screen shows $0,18$, which is a terminating decimal.

1. a) Find the mistake that Marcos made and explain what essential theoretical step he forgot to take before factorising the denominator.
2. b) Correct Marcos's procedure with the fraction $\frac{27}{150}$ and formally prove why it turns out to be a terminating decimal.
3. c) Apply the correct procedure to determine, without carrying out the division, what type of decimal number is generated by the fraction $\frac{21}{280}$.

Three students who share a flat in Granada decide to buy a joint quarterly city bus pass online. The pass costs exactly $68\text{ \euro}$ in total, and they agree to pay one third of the amount each.

1. a) Before calculating with precision, make a mental estimate: between which two whole numbers of euros must the individual contribution lie? Give a first reasoned approximation including cents.
2. b) Express each person's exact contribution as an irreducible fraction and find its decimal expansion. What type of decimal is it? Justify your answer by analysing the prime factors of the denominator.
3. c) If each student makes a Bizum transfer of $22,66\text{ \euro}$, is the total price of the pass covered? How much is left over or how much is missing? Evaluate whether your initial estimate was accurate and reflect on the problem that arises when trying to pay a recurring amount with real money (coins and cents).

Work out the following combined calculation, first converting the decimal numbers to their generating fractions. Express the final result as an irreducible fraction and as a decimal number:

$1,\widehat{3} + 0,\widehat{5} - 0,1\widehat{6}$

Determine the value or values that the digit $x$ can take (from $0$ to $9$) so that the fraction $\frac{x}{15}$ yields a terminating decimal. Reason your answer based on the prime factorisation of the denominator.

In a home in Seville on the regulated electricity tariff (PVPC), the breakdown of one month's consumption reflects the following costs according to the time band: off-peak period $12,\widehat{4}\text{ \euro}$, mid-peak period $21,\widehat{4}\text{ \euro}$, and peak period $33,\widehat{4}\text{ \euro}$. In addition, they have a social discount that subtracts $5,\widehat{1}\text{ \euro}$ from the bill.

To calculate the total net cost before taxes, the family proposes two calculation methods:

• Method 1: Round each of the four amounts to the nearest cent ($12,44\text{ \euro}$, $21,44\text{ \euro}$, etc.) and operate directly with the rounded decimals.

• Method 2: Obtain the irreducible generating fraction of each term, operate with fractions, and round only the final result to the nearest cent.

1. a) Apply Method 1 and calculate the resulting net cost.
2. b) Apply Method 2 by calculating the generating fractions and operating with them.
3. c) Compare both numerical results. Do they match to the cent, or does a discrepancy occur? Justify which of the two methods is correct and argue why intermediate amounts should never be rounded in banking or billing computer systems.

We want to find a positive, irreducible proper fraction $\frac{a}{b}$ (with $a < b$) that simultaneously satisfies the following conditions:

1. Its decimal expansion is a mixed recurring number of the form $0,1\widehat{y}$ (a non-recurring part of one digit equal to $1$ and a recurring period of a single unknown digit $y$).

2. The denominator $b$ is a two-digit natural number strictly less than $16$.

1. a) Write the theoretical formula of the generating fraction for a number of the form $0,1\widehat{y}$ without simplifying. From its denominator, deduce what prime factors $b$ can have so that the number is a mixed recurring decimal, and why $b$ must necessarily be a divisor of $90$.
2. b) Check the two-digit numbers less than $16$ and identify the only possible value for the denominator $b$.
3. c) With the value of $b$ found, determine what digit $y$ must be so that the numerator $a$ is an integer and the fraction is irreducible. Check the final result by dividing $a$ by $b$.

Find the digit in the $250$th decimal place in the decimal expansion of the fraction $\frac{3}{13}$. Explain the steps followed to find the recurring sequence and calculate the position.

Two vehicles are travelling in a straight line. Vehicle A has a constant speed of $33,\widehat{3}$ m/s, while vehicle B travels at $120$ km/h. If both maintain their speed for exactly $1,5$ hours, calculate the exact difference in metres between the distances travelled by each. Work with fractions to avoid rounding errors.

Observe the diagram illustrating the process of nested intervals to locate an irrational number on the number line. Given that the process continues infinitely, determine which famous irrational number the intervals are approaching and write an interval of width $0,001$ that contains this number.

According to the Hotel Occupancy Survey of the National Statistics Institute (INE), during a holiday weekend in a coastal town with $1\,300$ hotel beds offered, $400$ overnight stays by foreign tourists were recorded. The proportion of foreign occupancy relative to the total capacity was therefore $\frac{400}{1\,300} = \frac{4}{13}$.

1. a) Without carrying out the division, justify in a reasoned manner what type of decimal number the irreducible fraction $\frac{4}{13}$ represents by analysing the prime factorisation of its denominator.
2. b) Carry out the division $4 : 13$ until obtaining the complete period. Express the result using recurring notation (with an overbar or arc) and state the length of this period.
3. c) The municipality's statistics department uses a digital validation code based on the decimals of this proportion: the code corresponds to the digit occupying the $2\,025$th decimal place after the decimal point. Find this digit in a reasoned manner, explaining clearly the meaning of the quotient and the remainder of the integer division you use to find it.

A train on Renfe Cercanías line C-1 completes the $35\text{ km}$ journey between two stations in exactly $33\text{ minutes}$.

1. a) Calculate the average speed of the train in kilometres per hour ($\text{km/h}$) expressed as an irreducible fraction $\left(\text{remember that } 33\text{ minutes} = \frac{33}{60}\text{ hours}\right)$.

1. b) Classify the type of decimal generated by this fraction by analysing the prime factors of the denominator, and write its exact value using repeating decimal notation.

1. c) At the railway traffic control centre, two options are evaluated for the software estimating distances travelled on long journeys:

• Option A: The software stores and calculates speed rounded to the hundredths ($63,64\text{ km/h}$).

• Option B: The software stores and calculates speed using the exact irreducible fraction obtained in section a).

Calculate the total theoretical distance that each option would record after $165\text{ hours}$ of cumulative operation of that unit. What discrepancy or error in kilometres does Option A show compared to Option B? Explain the importance of working with generating fractions in control software as opposed to decimal approximations.

Unit Challenge. Become a climate analyst by investigating how number sets appear in our country's weather.

1. Data collection. Go to the AEMET website and record data for 3 different municipalities: maximum and minimum temperatures of a cold day (make sure to include at least one below zero) and the count of rainy days in the last month (natural number, $\mathbb{N}$).

2. Spreadsheet and operations. Build a table and calculate:

• The temperature range of each location by subtracting temperatures (operations in $\mathbb{Z}$ and decimals).

• The irreducible fraction representing rainy days over the total days of the month (rational number, $\mathbb{Q}$) and obtain its terminating or repeating decimal expression.

• The mean temperature of the minimums by operating with fractions and rounding to the tenths place.

3. Reasoned report. In a brief document (one page at most), present your tables and classify each result in $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, or $\mathbb{R}$, justifying why they belong to each set. Conclude by arguing which municipality has more extreme conditions.

Assessment criteria:

• Rigorous collection of real data from AEMET and tidy use of the spreadsheet.

• Accuracy in operations with integers, fractions, and decimal expressions using a decimal comma (e.g. $-2,4$ or $15,6$).

• Correct identification and classification of values into the sets $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, and $\mathbb{R}$.

• Coherence and argumentative clarity in the report's conclusions.