Exercises: Concept

At a dinner with friends, you have ordered a family pizza that comes divided into $8$ equal slices. If you have eaten $3$ slices, identify the terms of the fraction that represents the part of the pizza you have consumed.

1. a) What is the numerator and what does it indicate in this context?
2. b) What is the denominator and what does it indicate in this context?

Your cell phone battery has a total capacity of $4500$ mAh. If the screen shows that you have exactly $\frac{2}{9}$ of the total charge left, how many mAh of battery do you have available at this moment?

Look at the following geometric figures and write the fraction that represents the colored part in each case. Then, indicate which of them is a proper fraction and which is a fraction equal to one.

Classify the following fractions into proper or improper. Remember that to differentiate them you must look at the relationship between the numerator and the denominator:

A video game takes up $\frac{12}{16}$ of the space on a memory card. Your friend tells you that this value is equivalent to $\frac{3}{4}$ of the card. Check if the two values are equivalent fractions using the cross-multiplication method.

Simplify the following fractions until you obtain their irreducible fraction:

1. a) $\frac{15}{20}$
2. b) $\frac{24}{36}$
3. c) $\frac{10}{100}$

After an athletics training session, you have drunk $\frac{7}{3}$ of a gallon of water. Express this amount as a mixed number to find out how many full gallons and what fraction of the next one you have consumed.

Write the opposite and the reciprocal of each of the following fractions. Pay attention to the sign:

1. a) $\frac{2}{5}$
2. b) $-\frac{3}{7}$
3. c) $4$ (remember that an integer can also be written as a fraction)

In a data center, the storage panel is divided into identical modules. Look at the following graphical representation where the colored squares indicate the space occupied by a social network's database.

1. a) Write the fraction that represents the occupied space and the fraction that represents the free space.
2. b) Is it a proper or improper fraction? Justify your answer by comparing the numerator and the denominator.

A well-known "streamer" has $420\,000$ followers on her main platform. When analyzing her statistics, she observes that $\frac{3}{7}$ of her followers connect from the United States, while $\frac{2}{5}$ do so from Latin America.

1. a) Calculate the number of followers who connect from the United States.
2. b) Calculate the number of followers who connect from Latin America.
3. c) What fraction of the total followers do those who do not connect from either of these two regions represent?

To prepare a lilac paint mixture, an artist uses $\frac{4}{10}$ of a gallon of blue color for every gallon of white paint. If the artist wants to keep exactly the same shade but needs to make a larger quantity:

1. a) Find the value of $x$ in the following equality so that the fractions are equivalent: $\frac{4}{10} = \frac{x}{25}$.
2. b) Write three fractions equivalent to $\frac{4}{10}$ by amplification and one by simplification.

In a school basketball championship, the "Tigers" team has scored $48$ baskets out of $72$ attempts, while the "Falcons" have scored $36$ baskets out of $54$ attempts.

1. a) Write the success fraction for each team.
2. b) Reduce both fractions to their irreducible form using the Greatest Common Factor (GCF) method.
3. c) In light of the results, which of the two teams has a better success rate?

Three friends are downloading the same update file for a video game. After 10 minutes, the progress of each one is as follows:

• Marcus: $\frac{5}{6}$ of the total.

• Lucy: $\frac{3}{4}$ of the total.

• Sam: $\frac{11}{12}$ of the total.

1. a) Reduce the three fractions to a common denominator using the least common multiple (LCM).
2. b) Order the friends from least to most progress using the $<$ or $>$ symbols.

Look at the position of the highlighted point on the following number line:

1. a) Write the improper fraction that represents the orange point.
2. b) Express this fraction as a mixed number.
3. c) If we divided each unit into 8 equal parts instead of 4, what numerator would the equivalent fraction to the previous one have?

In a laboratory experiment, the variation in the temperature of a fluid with respect to the room temperature is recorded. The following values in degrees Fahrenheit are obtained:

1. a) Represent these numbers approximately on a number line.
2. b) Order the values from smallest to largest (from coldest to warmest).
3. c) Indicate which of these values has the greatest absolute value.

Classify the following expressions, indicating whether they represent a natural number, an integer (but not a natural number) or a non-integer rational number:

1. a) $\frac{15}{3}$
2. b) $-\frac{12}{4}$
3. c) $\frac{7}{5}$
4. d) $\frac{0}{8}$
5. e) $-\frac{1}{2}$
6. f) $\frac{24}{6}$

In an archaeological excavation, a papyrus has been found describing a distribution of land using unit fractions (those whose numerator is 1). The scribe noted that a farmer's plot is the sum of three different unit fractions and that the total adds up to $\frac{13}{12}$ of the basic unit of measurement.

If we know that the smallest denominator of these fractions is $2$, find the other two denominators so that the equality is correct.

Look at the following geometric figure representing a square solar panel of side $L$. The panel is divided into different sections to maximize its efficiency.

Calculate what fraction of the total area of the square the shaded region A represents. If the total area is $1$ square foot, express the area of region B as an irreducible fraction.

Two friends, Lucy and Jacob, are comparing their progress in an obstacle course. Lucy has completed $\frac{5}{7}$ of the course and Jacob $\frac{6}{7}$. A third runner, Matthew, is located somewhere between the two (not necessarily at the midpoint).

Find a fraction that represents Matthew's position such that its denominator is a prime number greater than $10$. Justify whether there is more than one possibility.

A content creator on a streaming platform starts their live stream with a certain number of viewers.

• During the first hour, they lose $\frac{2}{9}$ of their followers.

• In the second hour, $\frac{1}{4}$ of those remaining leave.

• At the end of the third hour, $45$ new people join, which means they now have $\frac{3}{4}$ of the initial amount of viewers.

Calculate the total fraction of viewers who left relative to the start and determine how many viewers they had when starting the live stream.

Analyze the following property of fractions: "If we have a positive proper fraction $\frac{n}{d}$ (where $n < d$) and we add the same natural number $k$ to both the numerator and the denominator, we obtain a new fraction $\frac{n+k}{d+k}$".

Prove, using the fraction $\frac{2}{5}$ as an example and adding $k=3$, whether the new fraction is greater than, less than or equal to the original one. Would this conclusion hold if the original fraction were improper ($\frac{5}{2}$)?

A graphic designer is creating a logo based on a regular hexagon. They have divided the hexagon by joining its opposite vertices to form six equilateral triangles. Subsequently, they have divided one of those triangles into four smaller triangles by joining the midpoints of its sides.

If we shade one of those tiny triangles (as shown in orange), what fraction of the total area of the hexagon does it represent? Express the result as an irreducible fraction.

On the number line, point A corresponds to the fraction $\frac{1}{6}$ and point B corresponds to $\frac{1}{2}$.

If we divide the line segment AB into three equal parts, what fractions correspond to the division points? Also place the origin ($0$) on the line and explain what graphical distance (in inches or grid units) there would be between $0$ and point A if the distance between A and B is $12$ in.

Find an irreducible fraction $\frac{a}{b}$ that satisfies the following three conditions simultaneously:

1. It is greater than $\frac{3}{5}$ but less than $\frac{2}{3}$.

2. The sum of its numerator and its denominator is a multiple of $11$.

3. The denominator $b$ is an even number less than $20$.

Justify the steps followed to find it.