Exercises: Operations with rational numbers

Work out the following combined operation with fractions, respecting the order of operations, and express the result as an irreducible fraction:

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

A baking recipe requires $\frac{3}{4}$ of a kilogram of flour to make a whole cake. If we only have enough ingredients to make half the recipe, how much flour will we need? Express the result as an irreducible fraction.

Solve the following complex fraction, simplifying the numerator and the denominator before performing the main division:

$ \frac{1 + \frac{1}{2}}{1 - \frac{1}{3}} $

Find the value of the unknown $x$ that makes the following equation true:

$ \frac{2}{3} \cdot x - \frac{1}{4} = \frac{5}{12} $

Marcos wants to calculate the cost per hour of electricity consumption and comes across the combined operation $\frac{3}{5} + \frac{1}{5} \cdot \frac{2}{3}$. To solve it, he states: "Since the first two fractions have the same denominator, I add first and get $\frac{3}{5} + \frac{1}{5} = \frac{4}{5}$; then I multiply by $\frac{2}{3}$ and the result is $\frac{8}{15}$". a) Reasonably explain the conceptual error that Marcos makes and which rule of the order of operations he has violated. b) Calculate the correct result, expressing it as an irreducible fraction.

According to a mobility survey by the Instituto Nacional de Estadística (INE) in a Spanish metropolitan area, $\frac{5}{12}$ of workers commute to their jobs by public transport and $\frac{3}{8}$ do so by private motor vehicle. The rest of the working active population commutes on foot or by bicycle. a) Set up the operation using fractions and determine the irreducible fraction of the total number of workers who use active and sustainable means of transport (on foot or by bike). b) If $24\,000$ working people live in that metropolitan area, calculate how many of them commute on foot or by bicycle.

A cyclist completes $\frac{1}{3}$ of a stage in the first hour. In the second hour, he covers $\frac{2}{5}$ of the remaining distance. If after these two hours he still has 30 km left to reach the finish line, what is the total length of the stage?

A technology company allocates $\frac{3}{8}$ of its annual budget to salaries, $\frac{1}{5}$ to marketing campaigns and $\frac{1}{4}$ of the remaining budget to research and development (R&D). What fraction of the total budget remains available for other operating expenses?

A metal alloy used in aerospace engineering consists of $\frac{3}{7}$ copper, $\frac{2}{5}$ zinc, and the rest is tin. If the total mass of a part made from this alloy is 140 kg, how many exact kilograms of tin does it contain?

One tap can fill a swimming pool in 4 hours, while a second tap can fill it in 6 hours. If both taps are opened at the same time, what fraction of the pool will be filled after 1 hour? How much total time (in hours and minutes) will they take to fill the pool completely?

A reservoir in a river basin in southern Spain has a maximum capacity of $450\text{ hm}^3$, and at the beginning of autumn it is filled to $\frac{2}{5}$ of its total capacity. During a rainstorm, it collects a volume equivalent to $\frac{1}{3}$ of the total capacity of the reservoir, but at the same time, for safety reasons, an amount equal to $\frac{1}{10}$ of the water stored at the beginning must be released. a) Before calculating the exact volume in $\text{hm}^3$, make a reasoned estimate by comparing the fractions: will the reservoir exceed half of its total capacity after the storm? Justify your answer. b) Calculate the exact irreducible fraction of the total capacity that remains stored and find the final volume in $\text{hm}^3$ to check the validity of your estimate.

To simplify the complex fraction $\frac{\frac{3}{4} - \frac{1}{6}}{\frac{5}{8}}$, two 4th-year ESO students propose different strategies: Sara prefers to first perform the subtraction in the numerator by finding the LCM of the denominators $4$ and $6$, and then divide by $\frac{5}{8}$ by multiplying by the reciprocal fraction. Sergio, on the other hand, proposes multiplying the overall numerator and overall denominator by $24$ (the LCM of $4$, $6$, and $8$) to eliminate all intermediate fractions at once in the first step. a) Work out both methods step by step in writing and verify that they lead to the same irreducible fraction. b) Analyze and argue which of the two methods you consider safer and more efficient to avoid fraction errors in this type of expression.

Work out the exact sum of the following series of fractions. (Hint: Observe the pattern and split each fraction into the difference of two simpler fractions, for example $\frac{1}{n \cdot (n+1)} = \frac{1}{n} - \frac{1}{n+1}$):

$ \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + \dots + \frac{1}{9 \cdot 10} $

A town council's Youth Department distributes its annual budget as follows: it allocates $\frac{1}{4}$ of the total to the maintenance of sports facilities; of the remaining money, it allocates $\frac{2}{5}$ to vocational training courses and workshops; and the final amount left unassigned, which amounts to exactly $45\,000$ €, is reserved for a transport scholarship fund. a) Model the situation by setting up a single linear equation where the unknown $x$ is the initial total budget, expressing the fractions in terms of $x$. b) Solve the equation to find out the department's total budget. c) Check the result by calculating the monetary allocations for each of the three purposes and verifying that their sum equals the total.

On a day of high renewable energy production in Spain, the electrical system operator (Red Eléctrica) records that wind power covers $\frac{3}{8}$ of the national demand, solar photovoltaic generates an amount equivalent to $\frac{1}{3}$ of the wind power contribution, and hydroelectric power provides $\frac{1}{4}$ of the remaining demand that still needed to be covered. Non-renewable power plants cover the remaining $240\text{ GWh}$ necessary to supply the total consumption. a) Determine, using fraction operations, what irreducible fraction of the total daily demand was represented by non-renewable production. b) Calculate the total electricity demand for the day in $\text{GWh}$ and how many $\text{GWh}$ were specifically generated by solar energy. c) If, looking ahead to the following year, the solar contribution were to increase by half as much again (that is, by a factor of $\frac{3}{2}$) compared to its current share, and the other renewables maintained identical fractions, demonstrate through fractional calculation whether or not the sum of all renewable energies would reach $\frac{3}{4}$ of the country's total demand.