Exercises: Rational numbers

Classify the following fractions according to the type of decimal expression they generate (terminating, pure repeating or mixed repeating) without performing the division: $\frac{7}{20}$, $\frac{5}{33}$, $\frac{11}{9}$ and $\frac{3}{14}$.

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

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

Evaluate the following operation with fractions 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 pound of flour to make a cake. If you have $2,5$ lb of flour, how many whole cakes can you make and what fraction of a pound will you have left over?

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

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

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

Evaluate the following operation by first converting the decimal numbers to their 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 harvest in the morning and $\frac{2}{5}$ of the remainder in the afternoon. If he still has $480$ lb left in the warehouse, how many pounds of tomatoes did he harvest in total?

When measuring the length of a piece of wood of exactly $\frac{17}{3}$ in, a carpenter records a value of $5,6$ in. 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 $\frac{x}{42}$ generates a terminating decimal number, knowing that $x$ is a natural number less than $25$. Justify your answer using the prime factorization of the denominator.

In a basketball game, a player scores $5$ out of every $8$ free throws she takes. If she has missed $24$ free throws throughout the season, how many has she taken in total and what is her success percentage expressed as a decimal number?

Simplify the following expression using the laws of exponents and 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 a handmade perfume, $\frac{2}{5}$ of a gallon of lavender essence is mixed with $\frac{1}{4}$ of a gallon of pure alcohol and the rest is distilled water to fill a $1,5$-gallon bottle. What fraction of the bottle does the distilled water represent?

Evaluate 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 repeating decimal number $0,\widehat{9}$ is exactly equal to the natural number $1$.

A gear system is made up of three cogwheels. The first wheel turns $\frac{15}{4}$ of a revolution per minute. The second wheel turns $\frac{5}{2}$ of a revolution for every revolution the first one makes, and the third wheel turns $\frac{4}{3}$ of a revolution for every revolution the second one makes. How many complete revolutions will the third wheel make in $8$ minutes?

An irrigation water tank has two filling 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 be filled in $1$ hour? How long will it take to fill completely?

A special rubber ball is dropped from an initial height of $10$ feet. 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 number.

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 partners invest in a technology company. The first contributes $\frac{1}{4}$ of the total capital, the second contributes $\frac{2}{5}$ and the third contributes the remaining $\$14\,000$. After one year, the business generates a profit of $\$8\,500$. Knowing that the profits are distributed proportionally to the invested capital, what fraction of the profits corresponds to the second partner and what exact amount of money will they receive?

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