Exercises: Operations with rational numbers

In a survey conducted in your class about their favourite social network, $\frac{12}{30}$ of the students chose a short-video platform. Find the irreducible fraction to work out what fraction of the total this group represents.

Look at the following figure, which represents a chocolate cake divided into equal slices. Write down the fraction representing the missing part, and calculate the opposite fraction and the reciprocal fraction of that amount.

A teenager spends $\frac{1}{3}$ of their afternoon free time doing sport and $\frac{2}{5}$ studying maths.

1. a) Calculate what fraction of their total free time these two activities take up.
2. b) What fraction of time do they have left to play on the games console?

Your mobile phone battery has a total capacity of $4500$ mAh. If the screen currently shows that exactly $\frac{4}{9}$ of its capacity remains:

1. a) How many mAh are available?
2. b) If battery saver mode is activated when $\frac{1}{5}$ of the total remains, how many mAh does that limit represent?

To make a healthy smoothie, you have mixed $\frac{3}{4}$ of a litre of oat milk with $\frac{1}{2}$ litres of orange juice. When serving it, $\frac{1}{8}$ of a litre is spilled. Work through the operation step by step to find out how much of the mixture remains in the jug.

On a shelf in a comic shop, $\frac{2}{3}$ of the titles are about superheroes. Of those superhero comics, half ($\frac{1}{2}$) belong to a famous American publisher. Calculate the fraction of the whole shelf represented by the comics of that specific publisher by carrying out a fraction multiplication.

We want to share $\frac{3}{2}$ of a kilogram of gummy sweets into small bags. If we put $\frac{1}{10}$ of a kilogram into each bag:

1. a) How many bags can we fill completely?
2. b) Carry out the operation using division of fractions: $\frac{3}{2} : \frac{1}{10}$

Solve the following mixed operation respecting the order of operations:

Elena is solving a homework assignment and states the following: “To calculate $-\frac{3}{4} : \frac{2}{5}$, since dividing means sharing, I divide the numerator by the numerator and the denominator by the denominator: $\frac{-3 : 2}{4 : 5}$. Since it does not divide evenly, I then simply change the sign of the second fraction and multiply straight across: $-\frac{3}{4} \cdot \left(-\frac{2}{5}\right) = \frac{6}{20} = \frac{3}{10}$”.

1. a) Identify the two conceptual errors Elena makes in her reasoning and explain the fundamental difference between the “reciprocal fraction” and the “opposite fraction”.
2. b) Solve the operation correctly step by step, leaving the final result as an irreducible fraction.

Before performing long calculations with different denominators, it is very useful to estimate the size of the result to anticipate the answer and detect possible errors. You have the following operation: $E = \left( 1 - \frac{1}{6} \right) \cdot \left( \frac{1}{2} + \frac{5}{11} \right)$.

1. a) Without performing the exact calculation, reasonably estimate the result by answering these guiding questions:

• Is the factor $\left( 1 - \frac{1}{6} \right)$ greater than or less than $1$? Is it close to $1$?

• Knowing that $\frac{5}{10} = \frac{1}{2}$, is the fraction $\frac{5}{11}$ greater than or less than $\frac{1}{2}$? Will the sum in the second pair of parentheses be more or less than $1$?

• Based on the above, deduce with reasoning whether the final product $E$ will be slightly less than $1$, exactly $1$, or noticeably greater than $1$.

1. b) Now perform the exact calculation of $E$ by resolving the parentheses and multiplying the fractions. Express the result as an irreducible fraction.
2. c) Find the decimal expression of your exact result (to two decimal places) and check whether your initial estimate was accurate.

Solve the following combined expression, simplifying the result to its simplest form:

Calculate the result of the following combined operation, simplifying the result to its simplest form:

Three friends decide to buy a gift for a classmate who is changing schools. The first contributes $\frac{2}{5}$ of the total price, the second contributes $\frac{1}{3}$ of what is left, and the third completes the payment with $16$ €.

1. a) What fraction of the total does the amount contributed by the second friend represent?
2. b) What is the total price of the gift?
3. c) How much money did each person put in?

An architect is designing a play area in a rectangular park. They have decided to divide the ground as shown in the following diagram:

If the "Grass" area occupies exactly half of the ground and the "Swings" area is $\frac{3}{10}$ of the total park:

1. a) What fraction of the total does the "Sand" area represent?
2. b) If the park has a total area of $1200$ $m^2$, how many square metres is the "Sand" area?

Find the value of the following expression taking into account the order of operations and the sign rules:

A content creator recorded a video for their channel. During editing, they cut $\frac{1}{6}$ of the total footage due to sound errors. Then, they removed $\frac{2}{5}$ of what was left because the pacing was too slow. In the end, the edited video is $18$ minutes long.

1. a) What fraction of the original footage does the final $18$-minute video represent?
2. b) How long was the original recording before editing began?

Calculate the value of the following mixed operation with negative powers and simplify the result:

In a chemistry laboratory, a container holds $2,4$ litres of an acid solution. For an experiment, $\frac{2}{3}$ of the contents is first extracted. Then, $0,5$ litres of distilled water are added to dilute the mixture. Finally, $\frac{3}{5}$ of the resulting mixture is used.

1. a) Carry out all operations working exclusively with fractions.
2. b) How many litres of mixture remain in the container at the end? Express the result both as a fraction and as a decimal number.

To prepare a paired quiz in Math class, you must pose a numerical puzzle with the following incomplete equality:

1. a) Applying the properties of inverse operations and respecting the order of operations, isolate step by step and calculate which irreducible fraction must be placed in the box $\Box$.
2. b) Explain, using the sign rules for multiplication, why the content of the parentheses necessarily had to be negative, and justify why the fraction you are looking for $\Box$ must be greater than $\frac{2}{5}$.
3. c) Check your solution by substituting the obtained fraction into the original expression and solving it step by step.

At the end of spring, a reservoir in the Guadalquivir river basin is filled to $\frac{4}{5}$ of its total capacity. During the summer, due to evaporation and agricultural irrigation, $\frac{3}{8}$ of the water the reservoir contained is consumed. In September, thanks to the first autumn storms, an amount of water equivalent to $\frac{1}{10}$ of the reservoir's total capacity flows in.

1. a) What fraction of the total capacity of the reservoir does the water consumed during the summer represent?
2. b) Write a single combined operation with fractions that expresses the fraction of the total capacity that the reservoir contains after the September rains, and calculate it, simplifying to an irreducible fraction.
3. c) If the total maximum capacity of the reservoir is $160\text{ hm}^3$, how many cubic hectometres of water are actually stored at the end of September?

Calculate the exact value of the following combined expression, first converting the decimals into their respective generating fractions and simplifying the intermediate steps:

A young influencer is organising her monthly budget from sponsorships. She allocates $\frac{2}{5}$ of her income to renting her studio and $\frac{1}{4}$ of the remainder to video editing. If after these expenses she still sets aside $\frac{5}{9}$ of what was left for savings and still has $480$ € left for personal expenses, what is her total monthly income?

In a chemistry laboratory, a tank contains a mixture of alcohol and water. Initially, $\frac{2}{7}$ of the contents is alcohol. $12$ litres of the mixture are removed and replaced with $12$ litres of pure alcohol. If the mixture now contains $\frac{2}{5}$ alcohol, how many litres of liquid are in total in the tank?

Look at the design of this modern tile for a cultural centre. The outer square has an area of $1$ m$^2$. Different regions have been shaded following a geometric pattern. Determine what fraction of the total the orange shaded region represents and calculate how many cm$^2$ correspond to it.

In an eSports competition, the total prize money is distributed as follows: the winning team gets $\frac{4}{7}$ of the total, the runner-up $\frac{2}{3}$ of what is left, and the third-placed team the remainder. If the difference between the prize for second and third place is $3.500$ €, calculate the total value of the prize money and how much money each of the three teams receives.

Find the value of $x$ in the following equality of powers with rational numbers, applying the necessary properties to simplify the expression:

A surveillance drone makes a round trip between two bases. On the outward journey, it flies with a tailwind at a constant speed. On the return journey, into a headwind, its speed is reduced to $\frac{3}{5}$ of the outward speed. If the total flight time was $1$ hour and $20$ minutes, what fraction of the total time did it take for the return journey? Express the result in minutes and seconds as well.

Three partners invest in a technology business. The first contributes $12.000$ €, the second contributes $\frac{2}{5}$ of the total, and the third contributes the same amount as the other two combined minus $18.000$ €.

1. a) What is the total investment in the business?
2. b) What fraction of the total does each partner contribute? (Express the fractions in their simplest form).

To calculate the exact value of the expression $M = \frac{3}{4} \cdot \left( \frac{8}{9} - \frac{16}{21} \right)$, Sofía and David debate which is the smartest approach:

• Sofía's method (parentheses first): she finds the LCM of $9$ and $21$, solves the subtraction inside the parentheses, and then multiplies by $\frac{3}{4}$, simplifying the resulting fraction.

• David's method (distributive property): he applies the distributive property calculating $\frac{3}{4} \cdot \frac{8}{9} - \frac{3}{4} \cdot \frac{16}{21}$, simplifying each product before carrying out the subtraction.

1. a) Solve the expression in detail following the method proposed by Sofía until you reach the irreducible fraction.
2. b) Solve the expression following David's method, clearly indicating the cross-simplifications prior to each multiplication.
3. c) Analyze the size of the denominators and numerators you worked with in each approach. Justify which of the two methods is more efficient and carries a lower risk of calculation error in this particular case.

In a demanding mountain stage of the Vuelta a España:

• On the first section, $\frac{1}{12}$ of the cyclists who started the race drop out.

• On the second section, during the climb up a special category mountain pass, $\frac{2}{11}$ of the riders still in the race drop out.

• On the third section, due to a severe hailstorm on the descent, $\frac{1}{3}$ of the cyclists who cleared the mountain pass abandon.

• At the start of the final climb to the finish line, only $60$ riders remain competing in the stage.

1. a) Express the portion of the peloton still in the race section by section using a chained product of fractions, and simplify by cross-cancelling factors to prove that the $60$ riders represent exactly $\frac{1}{2}$ of the starting peloton.
2. b) Deduce how many cyclists started the stage.
3. c) During the TV broadcast, a commentator asserts: “Since $\frac{1}{12}$ dropped out at the start, $\frac{2}{11}$ on the mountain pass, and $\frac{1}{3}$ on the descent, we add $\frac{1}{12} + \frac{2}{11} + \frac{1}{3} = \frac{79}{132}$, which means almost $60\,\%$ of the peloton has abandoned”. Justify with mathematical precision the commentator's serious conceptual error and state the actual fraction of the starting peloton that abandoned the race.

Unit challenge. How does the weather influence our environment? Become a weather analyst by examining real data from your province.

1. Data collection: Access the official AEMET website and record the maximum and minimum temperatures (signed integers or decimals) and precipitation for the past $7$ days.

2. Integers and fractions: Calculate the daily thermal oscillation ($T_{\text{max}} - T_{\text{min}}$) applying operations with integers and decimals. Express the number of rainy days as a fraction of the weekly total, find its irreducible fraction, and find an equivalent fraction projected onto a month of $28$ days.

3. Calculation and conclusions: Enter the data into a spreadsheet to find the average temperature of the week and estimate the economic savings in irrigation knowing that each litre collected represents $0,18$ € (precipitation in l/m² gives the litres collected per square metre, so express the savings in €/m²). Write a brief report reasoning your results.

The following will be assessed:

• Correct application of signs and subtraction of integers/decimals in the thermal oscillation.

• Rigorous determination of the irreducible fraction and its equivalent.

• Precision in combined operations with decimals in the spreadsheet.

• Clarity and mathematical foundation of the final report.