Rational Numbers. Concept

At a dinner with friends, you ordered a family pizza divided into $8$ equal slices. If you ate $3$ slices, identify the parts of the fraction representing the fraction of the pizza you ate.

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

Look at the following geometric shapes and write down the fraction representing the shaded part in each case. Then, state which of them is a proper fraction and which is a fraction equal to one.

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 right now?

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

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

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

Classify the following expressions, stating 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) $\frac{24}{6}$

After track practice, you drank $\frac{7}{3}$ of a liter of water. Express this amount as a mixed number to find out how many full liters and what fraction of the next one you consumed.

Simplify the following fractions to their simplest form:

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

Find the unknown value $x$ in each of the following equalities so that the fractions are equivalent:

1. a) $\frac{x}{12} = \frac{20}{48}$

1. b) $\frac{5}{9} = \frac{x}{45}$

1. c) $\frac{14}{x} = \frac{2}{7}$

1. d) $\frac{27}{81} = \frac{1}{x}$

To prepare a lilac paint mixture, an artist uses $\frac{4}{10}$ of a liter of blue paint for every liter of white paint. If the artist wants to keep exactly the same shade but needs to produce 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 down three equivalent fractions to $\frac{4}{10}$ by multiplying (amplification) and one by simplifying.

In a school basketball tournament, the "Tigers" team scored $48$ baskets out of $72$ attempts, while the "Hawks" scored $36$ baskets out of $54$ attempts.

1. a) Write down the success fraction for each team.
2. b) Simplify both fractions to their simplest form using the Highest Common Factor (HCF) method.
3. c) Looking at the results, which of the two teams has a better success rate?

Write down 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)

Compare the following pairs of rational numbers using the symbols $<$ (less than), $>$ (greater than), or $=$ (equal to):

1. a) $\dfrac{4}{7} \square \dfrac{6}{7}$

1. b) $\dfrac{3}{2} \square \dfrac{3}{5}$

1. c) $\dfrac{1}{3} \square \dfrac{4}{12}$

Order the following rational numbers from smallest to largest. To do this, first convert them to a common denominator using the lowest common multiple:

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

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

At a high-mountain weather station, the following temperature variations with respect to the seasonal average have been recorded, in degrees Celsius:

• Monday: $-\frac{9}{4}\ ^\circ\text{C}$

• Tuesday: $-2.1\overset{\frown}{6}\ ^\circ\text{C}$ (the 6 repeats: $-2.1666\ldots$)

• Wednesday: $-|-2.2|\ ^\circ\text{C}$

• Thursday: $-\frac{22}{10}\ ^\circ\text{C}$

Represent these four values on a number line like the one in the figure (copy it into your notebook) and order them in increasing order (from least to greatest) using the symbols $<$ and $=$ when necessary. Justify your answer by converting all the numbers to their irreducible fraction form.

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

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

Analyse 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 get a new fraction $\frac{n+k}{d+k}$".

Show, 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. Would this conclusion still hold if the original fraction were improper ($\frac{5}{2}$)?

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 denominator is a multiple of $13$.

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

Justify the steps you took to find it.

Three friends are filling a soccer sticker album. Alex has stuck in $\frac{1}{3}$ of the total stickers, Beatriz has stuck in $\frac{2}{5}$ of the total, and Carlos has the remaining stickers.

1. a) What fraction of the album has Carlos completed?
2. b) If the album has a total of $300$ stickers, how many has each person stuck in?
3. c) Which of the three friends has stuck in the most stickers?

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

Calculate what fraction of the square's total area is represented by the shaded region A. If the total area is $1$ square meter, express the area of region B as an irreducible fraction.

A graphic designer is creating a logo based on a regular hexagon. He has divided the hexagon by joining its opposite vertices to form six equilateral triangles. Subsequently, he has 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.

Design a square mosaic for a swimming pool. The mosaic is divided into three areas of different colors:

• Zone A occupies $\frac{2}{5}$ of the total.

• Zone B occupies $\frac{3}{7}$ of what remains after setting aside Zone A.

• Zone C occupies the remainder of the surface.

If the total surface area of the swimming pool is $140 \text{ m}^2$, calculate what fraction of the total Zone C represents and how many square meters correspond to it. Express the result for the surface area of Zone C as an exact decimal number.

In an archaeological excavation, a papyrus describing land distribution using unit fractions (those with a numerator of 1) has been found. The scribe noted that a farmer's plot is the sum of three distinct 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 holds true.