Exercises: Rational numbers (fractions)

Simplify the following fraction to its simplest form: $\frac{1260}{1500}$.

Work out the following combined calculation with fractions and simplify the result: $\frac{3}{4} - \frac{1}{2} \cdot \left( \frac{5}{6} + \frac{1}{3} \right)$.

Express the purely recurring decimal number $3,\overline{45}$ as an irreducible fraction.

Without performing the division, determine which of the following fractions will result in a terminating decimal and which in a recurring decimal, justifying your answer by factorising the denominators: $\frac{13}{40}$, $\frac{5}{21}$, $\frac{27}{250}$.

Look at the following number line. Identify which rational numbers (as irreducible fractions) correspond to the points marked with the letters A, B and C.

Marta makes the following statement in class: "To find out what type of decimal the fraction $\frac{21}{750}$ generates, I directly factor its denominator: since $750 = 2 \cdot 3 \cdot 5^3$, the factors $2$ and $5$ appear, but since the factor $3$ is also present, the fraction definitely generates a mixed recurring decimal". Without performing manual long division: a) Identify the conceptual mistake Marta made when analyzing the fraction. b) Explain in detail why her deduction failed and show the correct steps to classify the decimal number generated by that fraction.

The Ministry of Transport and Sustainable Mobility of Spain funded a transport pass campaign for young people with a fixed budget allocation. During the first month of implementation, $\frac{3}{10}$ of the total budget was spent. In the second month, exactly half of the money remaining after the first month was spent. After these two months, the campaign ended with a remaining surplus of $63\,000$ euros. a) Model the situation by calculating step by step the irreducible fraction of the total budget represented by the leftover money. b) Determine the total amount in euros initially allocated to the campaign. c) Explain which of the two months recorded a higher spending on passes.

An employee uses $\frac{1}{4}$ of his monthly salary to pay the rent for his home and $\frac{2}{5}$ of the remainder for food and utility expenses. If after making these payments he has $810$ euros left in his account, what is his total monthly salary?

A reservoir used for irrigation was filled to $\frac{7}{8}$ of its total capacity. After a week of severe drought, an amount of water equivalent to $\frac{1}{5}$ of the total capacity of the reservoir evaporates and is consumed. Later, heavy rainfall adds an amount of water equal to $\frac{1}{4}$ of the water remaining at that moment. If the total capacity of the reservoir is $1200$ cubic hectometres, what fraction of the total capacity of the reservoir is now full, and how many cubic hectometres does it contain?

A rubber ball special for physics experiments is dropped from an initial height of $10$ metres. After hitting the ground, on each bounce it reaches a height equal to $\frac{3}{4}$ of the height reached on the immediately preceding bounce. What height does the ball reach after the third bounce? Express the final result first as an irreducible fraction and then as a decimal number.

We are looking for a positive proper irreducible fraction $\frac{a}{b}$ that satisfies all three of the following conditions at the same time: 1) Its decimal value lies strictly between $\frac{1}{4}$ and $\frac{1}{3}$. 2) Its denominator $b$ is a two-digit number less than $20$. 3) When converted to a decimal number, it results in a mixed recurring decimal whose non-periodic part (anteperiod) has a single digit. a) Reason what prime factors the denominator $b$ must necessarily have and why they cannot cancel out with the numerator $a$. b) Find a fraction that meets all the conditions and verify step by step that it satisfies the requirements.

In an agricultural region in southern Spain, a community irrigation pond was $\frac{4}{5}$ full of its total capacity at the beginning of summer. During a heatwave recorded by AEMET, farmers consumed $\frac{3}{8}$ of the water the pond contained for drip irrigation tasks. Days later, a storm struck, contributing a volume of water equivalent to $\frac{1}{10}$ of the pond's total capacity. a) Before computing exact fractions, estimate with justification whether after these two events the pond will be above or below half of its capacity, explaining your prediction. b) Calculate accurately the irreducible fraction of the total capacity contained in the pond after the storm. c) Compare your result with your previous estimate and assess whether the obtained value is consistent.

A clockwork mechanism consists of a system of three gears A, B and C connected in series (A drives B, and B drives C). Gear A has $12$ teeth, gear B has $18$ teeth and gear C has $30$ teeth. Knowing that the number of turns a gear makes is inversely proportional to its number of teeth, if gear A turns exactly $\frac{15}{4}$ of a turn, what fraction of a turn will gear C make?

In the architectural design of a mosaic, an initial square of area $1 \text{ m}^2$ is drawn. Inside it, a second square is inscribed by joining the midpoints of the sides of the first. This geometric process is repeated to draw a third square inscribed in the second, and a fourth square inscribed in the third. What fraction of the area of the first square represents the area contained exclusively between the third and fourth squares?

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

To find out whether the expression $E = \left(\frac{7}{12} - \frac{1}{8}\right) \cdot \frac{6}{25}$ generates a terminating, pure recurring, or mixed recurring decimal without performing decimal division, two students choose different strategies: Lucía decides to first perform the subtraction inside the parentheses by calculating the traditional least common multiple, then multiply straight across, simplify by dividing by the g.c.d. at the end, and factor the resulting denominator. Marcos, on the other hand, chooses to factor all the terms into prime factors from the beginning and simplify common factors before carrying out any multiplication of large numbers. a) Solve the exercise in detail following each of the two methods and classify the resulting decimal. b) Evaluate the advantages and disadvantages of both approaches and justify which of the two methods is more efficient if you were working with fractions whose terms had three- or four-digit numbers.

According to studies by the National Statistics Institute (INE) on household budgets, an average household distributes its monthly income as follows: it sets aside $\frac{3}{8}$ of the salary to pay for home rent; of the remaining money, it allocates $\frac{2}{5}$ to food and utilities; with what is still left, it puts aside $\frac{1}{3}$ for unexpected events and emergencies; finally, it dedicates $\frac{1}{4}$ of the remaining final money to savings plans, leaving a free balance of $315$ euros for leisure. a) Set up the necessary chained operations to determine what irreducible fraction of the total salary those $315$ euros represent. b) Calculate the family's total monthly salary and the exact amount in euros spent per month on rent. c) A family member states: "If we add up the fractions spent $\frac{3}{8} + \frac{2}{5} + \frac{1}{3} + \frac{1}{4}$, we get more than $1$, so the family would have spent more money than it earns". Rigorously explain the mathematical error in this reasoning.