Exercises: Operations with rational numbers

In a survey carried out in your class about favorite social networks, $\frac{12}{30}$ of the students chose a short video platform. Find the irreducible fraction to know what part of the total this group represents.

Observe the following figure representing a chocolate cake divided into equal portions. Write down the fraction that represents the missing part and calculate the opposite fraction and the reciprocal fraction of that amount.

A teenager dedicates $\frac{1}{3}$ of his free time in the afternoon to playing sports and $\frac{2}{5}$ to studying math.

1. a) Calculate what fraction of his total free time these two activities occupy.
2. b) What fraction of time does he have left to play on his console?

Your cell phone battery has a total capacity of $4500$ mAh. If at this moment the screen indicates that it has exactly $\frac{4}{9}$ of its capacity left:

1. a) How many mAh does it have available?
2. b) If the power saving mode is activated when $\frac{1}{5}$ of the total is left, how many mAh does that limit represent?

To prepare a healthy smoothie, you have mixed $\frac{3}{4}$ of a gallon of oat milk with $\frac{1}{2}$ gallon of orange juice. When serving it, $\frac{1}{8}$ of a gallon is spilled. Perform the operation step by step to find out how much mixture is left in the pitcher.

On a shelf in a comic book store, $\frac{2}{3}$ of the copies are superhero comics. Of those superhero comics, half ($\frac{1}{2}$) belong to a famous American publisher. Calculate the fraction of the total shelf that the comics from that specific publisher represent by performing a multiplication of fractions.

We want to distribute $\frac{3}{2}$ of a pound of jelly beans into small bags. If we put $\frac{1}{10}$ of a pound in each bag:

1. a) How many bags will we be able to fill completely?
2. b) Perform the operation using division of fractions: $\frac{3}{2} : \frac{1}{10}$

Solve the following operation respecting the order of operations:

Solve the following expression, simplifying the result until you obtain the irreducible fraction:

Calculate the result of the following expression, simplifying the result until you obtain the irreducible fraction:

Three friends decide to buy a present for a classmate who is moving to another school. The first contributes $\frac{2}{5}$ of the total price, the second contributes $\frac{1}{3}$ of the remainder and the third completes the payment with $\$16$.

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

An architect is designing a play area in a rectangular park. He has decided to divide the land as shown in the following diagram:

If the "Grass" area occupies exactly half of the land 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 feet does the "Sand" area measure?

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

A content creator has recorded a video for his channel. During editing, he cuts $\frac{1}{6}$ of the total footage due to sound errors. Then, he removes $\frac{2}{5}$ of what was left because the pace was too slow. In the end, the edited video has a duration of $18$ minutes.

1. a) What fraction of the original footage does the final video of $18$ minutes represent?
2. b) How long was the original recording before starting to edit?

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

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

1. a) Perform all operations working exclusively with fractions.
2. b) How many gallons of mixture are left 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 expression, first transforming the decimal numbers into their respective generating fractions and simplifying the intermediate steps:

A young influencer is organizing her monthly budget derived from her collaborations. She allocates $\frac{2}{5}$ of her income to the rent of her studio and $\frac{1}{4}$ of the remainder to video editing. If after these expenses she still has $\frac{5}{9}$ of what was left to save and she still has $\$480$ available 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 content is alcohol. $12$ gallons of the mixture are extracted and replaced with $12$ gallons of pure alcohol. If the mixture now contains $\frac{2}{5}$ of alcohol, how many gallons of liquid are there in total in the tank?

Observe 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 area represents and calculate how many cm$^2$ correspond to it.

In an esports competition, the total prize is distributed as follows: the winning team takes $\frac{4}{7}$ of the total, the runner-up takes $\frac{2}{3}$ of the remainder, and the third place takes the rest. If the difference between the second and third prizes is $\$3.500$, calculate the total value of the prize 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, with 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 spend on the return journey? Express the result also in minutes and seconds.

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 together minus $\$18.000$.

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