Exercises: 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?

Your mobile 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?

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.

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:

Simplify the following fractions to their simplest form:

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 litre of water. Express this amount as a mixed number to find out how many whole litres and what fraction of the next one you have drunk.

In the figure, each litre is divided into 3 equal parts (thirds) and is shown on a number line. You can use it to help you.

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

In a Mathematics exam on rational numbers, Cian had to answer a question about the number $-\frac{3}{5}$. On his paper, he wrote:

"The reciprocal of $-\frac{3}{5}$ is $\frac{5}{3}$, because when inverting a fraction its sign always changes and becomes positive."

1. a) Explain which two different theoretical concepts Cian is confusing and why his mathematical statement is incorrect.
2. b) Clearly write the opposite and the reciprocal of $-\frac{3}{5}$, justifying each one based on its definition.

In a mountain stage of the Rás Tailteann, the organizers place a special sprint banner at the kilometer mark that corresponds exactly to $\frac{47}{6}$ of a reference climb.

1. a) Before doing any exact calculations, estimate with reasoning between which two consecutive whole kilometers this banner is located and which of the two you think it will be closer to.
2. b) Convert the fraction $\frac{47}{6}$ into a mixed number (integer plus proper fraction) and check if your estimate was correct. Explain how the remaining proper fraction confirms which whole number the kilometer mark is closer to.

In a data centre, the storage panel is divided into identical modules. Look at the following diagram where the coloured 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.

To prepare a lilac paint mixture, an artist uses $\frac{4}{10}$ of a litre of blue paint for every litre 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?

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.

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) $\frac{24}{6}$

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

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

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

1. a) What fraction of the album has Wei 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?

To simplify the fraction $\frac{420}{1260}$ to its irreducible form, two classmates suggest different strategies:

• Aoife suggests: "I'm going to successively divide the numerator and denominator using simple divisibility rules (by 2, by 3, by 5...) until I can't do it anymore."

• Cian suggests: "I prefer to break down both numbers into prime factors, calculate the $\text{gcd}(420, 1260)$, and do a single division."

1. a) Apply both methods step by step in your notebook to obtain the irreducible fraction.
2. b) Critically evaluate both procedures: what advantages and disadvantages does each method have in this case with three- and four-digit numbers? Why does calculating the $\text{gcd}$ mathematically guarantee that the resulting fraction cannot be simplified any further?

A study on urban sustainability in two Irish municipalities analyzes bicycle use. In municipality A, 180 out of every 450 people surveyed use a bicycle daily. In municipality B, 252 out of every 588 people surveyed claim to use it daily.

1. a) Model the proportion of daily cyclists in each municipality using an irreducible fraction.
2. b) Without calculating the decimal expression of the fractions, compare both proportions from their irreducible forms and indicate with reasoning in which municipality there is a greater habit of bicycle use.
3. c) Justify which number sets ($\mathbb{N}$, $\mathbb{Z}$, or only $\mathbb{Q}$) the values of these proportions belong to, and what would theoretically have to happen in the survey for the resulting fraction to belong to the set $\mathbb{N}$.

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.

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 metre, express the area of region B as an irreducible fraction.

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

Find a fraction that represents Cian'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 ($\frac{5}{2}$)?

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.

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.

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.

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

• 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 metres correspond to it. Express the result for the surface area of Zone C as an exact decimal number.

Find a positive, irreducible fraction $\frac{a}{b}$ that simultaneously satisfies these three conditions:

1. Its expression as a mixed number has 3 as its whole-number part (that is, $\frac{a}{b} = 3 + \frac{r}{b}$, with $0 < r < b$).

2. The reciprocal of the remaining proper fraction $\frac{r}{b}$ is an exact integer.

3. The product of the numerator and the denominator, $a \times b$, is strictly greater than 50 and less than 80.

Deduce step by step which value or values $a$ and $b$ can take, reasoning why the condition that the initial fraction is irreducible forces $r = 1$, and ruling out with mathematical arguments any options that do not satisfy all the constraints.

During a classroom discussion, Aoife makes the following statement about fraction simplification and equivalence:

"If I have any proper, irreducible fraction $\frac{a}{b}$ (with $a < b$) and I add the same natural number $k$ ($k > 0$) to both the numerator and the denominator, I get an equivalent fraction $\frac{a+k}{b+k}$, because I have added the exact same amount to both parts."

1. a) Provide a simple numerical counterexample to prove that the new fraction obtained by Aoife is not equivalent to the original one.
2. b) Using the values from your counterexample, determine whether the new fraction is greater than or less than the original one.
3. c) Explain why Aoife's reasoning is a conceptual mistake: justify what kind of relationship (additive or multiplicative) defines equivalent fractions, and explain why adding $k$ to both terms of a proper fraction makes the resulting value approach 1 instead of remaining constant.