Exercises: Rational numbers

Classify the following fractions according to the type of decimal expansion they produce (terminating, pure recurring or mixed recurring) without carrying out the division: $\frac{7}{20}$, $\frac{5}{33}$, $\frac{11}{9}$ and $\frac{3}{14}$.

Calculate the irreducible generating fraction of the following decimal numbers: $3,45$ and $2,\widehat{7}$.

Classify the following numbers, indicating the smallest number set to which they belong ($\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{I}$): $-5$; $\frac{8}{2}$; $1,2\widehat{3}$; $\sqrt{7}$.

Work out the following fraction operation and simplify the result: $\frac{3}{4} - \frac{1}{2} \cdot \left( \frac{4}{5} + \frac{1}{3} \right)$.

Order the following rational numbers from smallest to largest: $\frac{5}{6}$, $0,83$, $0,\widehat{8}$, $\frac{7}{8}$.

A baking recipe requires $\frac{3}{4}$ of a kilogram of flour to make a cake. If you have $2,5$ kg of flour, how many whole cakes can you make and what fraction of a kilogram will you have left over?

Represent the rational numbers $\frac{5}{3}$ and $-\frac{7}{4}$ accurately on the number line.

Find two distinct rational numbers strictly between $\frac{2}{5}$ and $\frac{3}{5}$.

To classify the decimal number generated by the fraction $\frac{21}{140}$ without performing the division, Lucía decomposes the denominator into prime factors: $140 = 2^2 \cdot 5 \cdot 7$. Observing that the factors $2$ and $5$ appear alongside $7$, she firmly states: "Since the factor $7$ appears in the denominator accompanying $2$ and $5$, the fraction necessarily generates a mixed recurring decimal". Explain in detail the mistake Lucía made, indicate which essential preliminary step she omitted, and correctly classify the decimal expression of that fraction by calculating its exact value.

The goal is to construct irreducible fractions whose denominator $d$ is a natural number strictly between $20$ and $30$ ($20 < d < 30$):

1. a) Write an irreducible fraction that satisfies that condition and whose decimal expression is terminating.
2. b) Write another irreducible fraction that satisfies the condition but whose decimal expression is pure recurring.
3. c) Write a third irreducible fraction that satisfies the condition and whose decimal expression is mixed recurring.

In each of the three cases, justify your answer using the prime factor decomposition of the denominator you chose.

Calculate the irreducible generating fraction of the decimal number $4,1\widehat{36}$.

Work out the following operation by first converting the decimal numbers into generating fractions, and simplify the result: $1,\widehat{5} + 0,2\widehat{3} - \frac{5}{6}$.

A farmer sells $\frac{1}{3}$ of his tomato crop in the morning and $\frac{2}{5}$ of the remainder in the afternoon. If he still has $480$ kg left in the storehouse, how many kilograms of tomatoes did he harvest in total?

When measuring the length of a piece of wood that is exactly $\frac{17}{3}$ cm, a carpenter records a value of $5,6$ cm. Calculate the absolute error and the relative error made in the measurement, expressing the results as irreducible fractions.

Determine the value of $x$ so that the fraction $\frac{x}{42}$ produces a terminating decimal, knowing that $x$ is a natural number less than $25$. Justify your answer using the prime factorisation of the denominator.

In a basketball match, a player scores $5$ out of every $8$ free throws she takes. If she missed $24$ free throws over the entire season, how many did she take in total, and what is her success rate expressed as a decimal?

Simplify the following expression using index laws, leaving the result as an irreducible fraction: $\left( \frac{2}{3} \right)^{-2} \cdot \left( \frac{9}{4} \right)^3 \div \left( \frac{3}{2} \right)^5$.

To make an artisan perfume, $\frac{2}{5}$ of a litre of lavender essence is mixed with $\frac{1}{4}$ of a litre of pure alcohol and the remainder distilled water to fill a bottle of $1,5$ litres. What fraction of the bottle is distilled water?

Before precisely operating on the expression $E = 2,\widehat{6} \cdot 0,75 - 0,1\widehat{6}$:

1. a) Make an initial estimate by rounding each decimal number to the tenths and mentally calculate the approximate value to predict between which two consecutive integers the result will lie.
2. b) Determine the irreducible generating fraction of each of the three terms and calculate the exact value of $E$, expressing the final result as an irreducible fraction.
3. c) Obtain the decimal expression of the exact result from part b, compare it with your initial estimate, and argue whether the obtained result is reasonable or if it indicates a calculation error.

On the C-5 line of Cercanías Madrid, a transit time study is being conducted. A train usually takes $12,1\widehat{6}$ minutes to cover the route between two stations, while the intercity bus line running the same route takes $18,45$ minutes.

1. a) Model the train's time by expressing it as an irreducible fraction of an hour (remember that $1\text{ hour} = 60\text{ minutes}$).
2. b) During a technical maintenance day, the train suffers a breakdown that increases its travel time by one third compared to its usual time. Model this new situation by calculating the train's total time in minutes as an irreducible fraction and classify the type of decimal number it represents.
3. c) With that time increase, is the train still faster than the bus? Justify your answer by formally comparing both time fractions in minutes by finding a common denominator.

Solve the following combined operation, expressing the final result as an irreducible fraction: $\frac{\left( 0,\widehat{3} + \frac{1}{2} \right)^{-1} \cdot 1,1\widehat{6}}{\sqrt{0,25} - \frac{1}{3}}$.

Prove mathematically, using the algorithm to obtain the generating fraction, that the pure recurring decimal $0,\widehat{9}$ is exactly equal to the natural number $1$.

A gear system consists of three cogs. The first cog makes $\frac{15}{4}$ of a turn per minute. The second cog makes $\frac{5}{2}$ of a turn for every turn of the first cog, and the third cog makes $\frac{4}{3}$ of a turn for every turn of the second cog. How many full turns will the third cog make in $8$ minutes?

An irrigation water tank has two inlet taps and a drain. The first tap fills it in $4$ hours, the second tap fills it in $6$ hours, and the drain empties it completely in $8$ hours. If all three are opened at the same time when the tank is empty, what fraction of the tank will fill in $1$ hour? How long will it take to fill completely?

A special rubber ball is dropped from an initial height of $10$ metres. On each bounce, it reaches a height equal to $\frac{3}{4}$ of the height of the previous bounce. Calculate the height it will reach on the fourth bounce and express it as a terminating decimal.

Find a positive rational number of the form $\frac{a}{b}$ such that adding its multiplicative inverse gives a result of $\frac{25}{12}$.

Three friends organise an esports tournament in their city. The first contributes $\frac{1}{4}$ of the total budget, the second contributes $\frac{2}{5}$ and the third contributes the remaining $1\,400$ €. In the end, ticket sales generate $850$ € in revenue. Given that the income is distributed proportionally to the money contributed, what fraction of the revenue belongs to the second friend and what exact amount of money will they receive?

Determine all integer values of $n$ for which the algebraic fraction $\frac{3n + 15}{n + 2}$ represents an integer.

You are asked to insert exactly three distinct rational numbers strictly between $\frac{5}{11}$ and $\frac{6}{11}$. Two solving strategies are proposed:

• Method A (Successive midpoints): Calculate the midpoint $m_1 = \frac{a+b}{2}$ between the given endpoints and, subsequently, the midpoints between each endpoint and $m_1$.

• Method B (Direct amplification): Multiply the numerator and denominator of both fractions by a suitable integer to open a sufficient interval between consecutive integer numerators.

1. a) Apply both methods to find, using each of them, three inserted rational numbers.
2. b) Compare the computational cost of both procedures and argue reasonably which of the two methods you would choose if, instead of three numbers, you were asked to insert $50$ rational numbers between $\frac{5}{11}$ and $\frac{6}{11}$.
3. c) Relying on the mathematical property that justifies Method A, prove why in the set of rational numbers $\mathbb{Q}$ it lacks mathematical rigor to speak of the "rational number immediately following $\frac{5}{11}$".

A mixed recurring decimal with a non-zero non-recurring part has the form $0,a\widehat{b}$, where $a$ and $b$ represent distinct decimal digits ($a \in \{1, 2, \ldots, 9\}$ and $b \in \{0, 1, \ldots, 9\}$). When simplifying its generating fraction as much as possible, an irreducible fraction $\frac{p}{30}$ is obtained. Knowing that the difference between the repetend (recurring part) and the non-recurring part is $b - a = 3$:

1. a) Set up the algebraic expression for the generating fraction in terms of $a$ and $b$, and substitute the condition $b = a + 3$ to simplify it in terms of the digit $a$.
2. b) Analyze all possible values of the digit $a$ that make the irreducible denominator exactly $30$, uniquely determining the digits $a$ and $b$ and the irreducible fraction $\frac{p}{30}$.
3. c) Justify from the prime factors of $30$ why the result is a mixed recurring decimal. What type of decimal number would have been obtained if the pair of digits $(a, b) = (6, 9)$ had been chosen? Justify your answer by relating it to the property of repetend $9$.