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: $\frac{3}{4} - \frac{1}{2} \times \left( \frac{4}{5} + \frac{1}{3} \right)$.

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 $\frac{3}{5}$.

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: $\left( \frac{2}{3} \right)^{-2} \times \left( \frac{9}{4} \right)^3 \div \left( \frac{3}{2} \right)^5$.

To make an artisan perfume, $\frac{2}{5}$ of a liter of lavender essence is mixed with $\frac{1}{4}$ of a liter of pure alcohol and the remainder distilled water to fill a bottle of $1.5$ liters. What fraction of the bottle is distilled water?

Solve the following combined operation, expressing the final result as an irreducible fraction: $\frac{\left( 0,\widehat{3} + \frac{1}{2} \right)^{-1} \times 1.1\widehat{6}}{\sqrt{0.25} - \frac{1}{3}}$.

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 $10$ meters. 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 $28\,000$ pesos. In the end, ticket sales generate $17\,000$ pesos 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.