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.

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.

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?

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: $\frac{3}{2} : \frac{1}{10}$

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.

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.

Solve the following mixed operation respecting the order of operations:

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

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

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

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

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

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.

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

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?

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?

In a demanding mountain stage of the Rás (Tour of Ireland):

• 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.

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 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.

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 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.

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?

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.