Operations with Rational Numbers

In a survey conducted in your class about their favorite 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 liter of oat milk with $\frac{1}{2}$ liters of orange juice. When serving it, $\frac{1}{8}$ of a liter is spilled. Work through the operation step by step to find out how much of the mixture remains in the jug.

A teenager spends $\frac{1}{3}$ of their afternoon free time playing sports and $\frac{2}{5}$ studying math.

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 video game console?

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

Your cell 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?

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.

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 $2.4$ liters of an acid solution. For an experiment, $\frac{2}{3}$ of the contents is first extracted. Then, $0.5$ liters 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 liters 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:

Solve the following mixed operation respecting the order of operations:

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:

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?

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 $960$ pesos.

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 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 $210\,000$ pesos, calculate the total value of the prize money and how much money each of the three teams receives.

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 meters is the "Sand" area?

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 $28\,800$ pesos 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. $12$ liters of the mixture are removed and replaced with $12$ liters of pure alcohol. If the mixture now contains $\frac{2}{5}$ alcohol, how many liters of liquid are in total in the tank?

Look at the design of this modern tile for a cultural center. 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.

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 $720\,000$ pesos, the second contributes $\frac{2}{5}$ of the total, and the third contributes the same amount as the other two combined minus $1\,080\,000$ pesos.

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).