Exercises: The set of rational numbers

In a pizzeria, a family pizza has been divided into $8$ identical slices. If Cian ate $3$ slices, express the part of the pizza he consumed as a fraction and clearly identify which is the numerator and which is the denominator. How is this fraction read?

Look at the following geometric figure and write down in your exercise book the fraction that represents the area shaded in orange.

Classify the following fractions as proper, improper, or equal to one:

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

1. b) $\dfrac{9}{9}$

1. c) $\dfrac{11}{3}$

1. d) $\dfrac{2}{5}$

We are organising a party and buying packs of soft drinks. Each pack has $6$ cans. If we need $\dfrac{12}{6}$ packs for the party, how many whole packs of soft drinks do we have to buy? Is this fraction a natural number?

Represent the following fractions visually in your exercise book, using squares or rectangles:

1. a) $\dfrac{3}{4}$

1. b) $\dfrac{5}{2}$ (Hint: you will need to draw more than one whole shape).

You can do it on paper and send a photo with your answer.

State what fraction of an hour the following time intervals represent (remember that one hour has $60$ minutes):

1. a) $15$ minutes.

1. b) $30$ minutes.

1. c) $10$ minutes.

1. d) $60$ minutes.

Copy the number line from the figure into your notebook (with 0, 1 and 2, and each unit divided into 4 equal parts) and place the following rational numbers on it:

$A = \dfrac{1}{2}$

$B = \dfrac{3}{4}$

$C = \dfrac{5}{4}$

You can do it on paper and send a photo with your answer.

Write the following natural numbers as fractions (you can use any denominator you prefer, as long as the result is correct):

1. a) $3$

1. b) $1$

1. c) $5$

1. d) $10$

Aoife has to determine what type of decimal the fraction $\frac{21}{60}$ generates without carrying out the division. On her exam she writes the following: "I decompose the denominator into prime factors: $60 = 2^2 \times 3 \times 5$. Since the prime number 3 appears in the decomposition, which is different from 2 and 5, the fraction necessarily generates a recurring decimal number." a) Explain in detail the error Aoife made in her reasoning. b) Correct the procedure by first simplifying the fraction and conclude, justifying your answer, what type of decimal number is actually obtained.

Design two different fractions that simultaneously satisfy the following three conditions: 1) The denominator of each fraction must be an integer between $10$ and $20$. 2) The value of both fractions must be strictly between $0.4$ and $0.6$. 3) One of the fractions must generate a terminating decimal number and the other must generate a recurring decimal number. Justify your answer by checking the prime factorisation of the denominators once the fractions are simplified to irreducible form and calculate their decimal expression.

Cian is studying rational numbers and states the following in his notebook:

1) "The number $A = \frac{-18}{-3}$ does not belong to the set of rational numbers ($\mathbb{Q}$) because the division yields $+6$, and $+6$ is an integer, not a fraction."

2) "The number $B = -\frac{5}{-2}$ is negative because it has two minus signs."

Identify the conceptual error Cian makes in each of his statements, explain in your own words why he is wrong (recalling the relationship between integers and rational numbers, as well as the sign rule in fractions), and write the simplified value and the final sign of $A$ and $B$.

A rational number can be written as a fraction $\frac{a}{b}$, where $a$ and $b$ are integers with $b \neq 0$. Find a rational number that simultaneously satisfies the following four conditions:

1) It is a strictly negative number.

2) Its value lies between $-2$ and $-1$ on the number line.

3) Its denominator $b$ is a prime number less than $6$.

4) The absolute value of its numerator is equal to the opposite of $-5$.

Write the fraction that satisfies these conditions, check step by step that it satisfies each of them, and explain between which two divisions on the number line it would be located if you divide each unit into as many parts as indicated by its denominator.

In mathematics, numbers are organised into "families" or sets. Look at the following Venn diagram representing the sets of Natural numbers ($\mathbb{N}$), Integers ($\mathbb{Z}$), and Rational numbers ($\mathbb{Q}$).

Place each of the following numbers in its corresponding place inside the diagram:

$-5$, $\frac{12}{4}$, $\frac{3}{7}$, $0$, $-\frac{10}{2}$, $1.25$, $-\frac{1}{4}$, $8$

Three friends are completing the "Battle Pass" of their favourite video game. Each has progressed a different fraction of the total:

• Cian: has completed $\frac{12}{18}$ of the total.

• Aoife: has completed $\frac{20}{30}$ of the total.

• Niamh: has completed $\frac{2}{3}$ of the total.

Work out whether the three friends are on the same level of the game by finding the irreducible fraction of each one.

Plot on the following number line the points $A$, $B$, $C$ and $D$ that correspond to these rational numbers. Each unit is divided into 4 equal parts.

$A = \frac{3}{2}$

$B = -1.25$

$C = -\frac{1}{4}$

$D = \frac{11}{4}$

In a study on mobile phone usage, the fraction of total storage occupied by different apps on Aoife's phone was analysed:

• Social Media: $\frac{3}{10}$ of the total.

• Games: $\frac{2}{5}$ of the total.

• Photos and Videos: $\frac{1}{4}$ of the total.

1. a) Write the fractions with a common denominator.
2. b) Order the categories from highest to lowest storage usage.
3. c) Is there any free space left on the phone? Explain your answer.

An influencer has $120,000$ followers. On Monday, $\frac{2}{5}$ of their followers liked their photo. On Tuesday, $\frac{3}{4}$ of those who liked it on Monday did so.

1. a) How many people liked it on Monday?
2. b) How many people liked it on Tuesday?
3. c) What fraction of the total followers represents the people who liked it on Tuesday?

For a birthday party, several pizzas of the same size were bought. At the end of the party, the following slices were left over:

• $\frac{3}{4}$ of a ham pizza.

• $\frac{5}{4}$ of pepperoni pizzas.

• $\frac{1}{2}$ of a vegetarian pizza.

1. a) Express the fraction of pepperoni pizza as a mixed number.
2. b) If we put all the leftovers together, do we have more or less than 3 whole pizzas? Justify your answer by calculating the sum.

Identify which of the following fractions will result in a terminating decimal and which in a recurring decimal (pure or mixed) without carrying out the division, looking only at the prime factorisation of their denominators:

1. a) $\frac{7}{20}$
2. b) $\frac{5}{12}$
3. c) $\frac{11}{25}$
4. d) $\frac{2}{9}$

(Hint: Remember the rule about prime factors 2 and 5 in the denominator).

In a jumping competition, the judges record the distances of four athletes. Some results are given as fractions and others as decimals:

• Athlete 1: $6.4$ meters.

• Athlete 2: $\frac{13}{2}$ meters.

• Athlete 3: $6.35$ meters.

• Athlete 4: $\frac{32}{5}$ meters.

1. a) Convert all the measurements to decimal form.

1. b) Which two athletes jumped exactly the same distance?

1. c) Order the athletes from the one who jumped the least to the one who jumped the most (if there is a tie, point it out).

At a weather station in the Wicklow Mountains (Met Éireann), the following temperature anomalies have been recorded on four consecutive days: $-\frac{9}{4}$ ºC, $-\frac{8}{3}$ ºC, $0.75$ ºC, and $\frac{7}{5}$ ºC. a) Without doing decimal divisions, reasonably estimate between which two consecutive integers each of the two negative anomalies lies and deduce which one corresponds to the lowest temperature (smallest value). b) Convert all fractions to their exact terminating or recurring decimal expression to check your previous estimate accurately. c) Order the four records from lowest to highest using the symbol $<$ and state which most specific number set (natural numbers $\mathbb{N}$, integers $\mathbb{Z}$, or rational numbers $\mathbb{Q}$) the result would belong to if you added the two positive anomalies.

To order the rational numbers $-\frac{3}{4}$, $-0.8$, $-\frac{5}{6}$, and $-\frac{7}{10}$ from least to greatest, two students propose different strategies: - Seán says: "It is better to calculate the decimal expression of each fraction by dividing and compare all the numbers in decimal form." - Aoife says: "It is preferable to write $-0.8$ as an irreducible fraction, reduce the four fractions to a common positive denominator, and compare their numerators." a) Apply Seán's method and write the resulting order. b) Apply Aoife's method to verify the result. c) Which of the two methods do you consider more appropriate and precise in this particular case? Justify your answer by analysing the type of decimal produced by the fraction $-\frac{5}{6}$ and the potential risk of rounding.

At a gauging station in the Shannon river basin, the decrease in flow rate in an irrigation canal is recorded during a drought episode. The technical report indicates that the accumulated change in level is $-\frac{37}{8}$ metres.

1. a) Before calculating any decimal digit, reasonably estimate between which two consecutive integers this rational number lies on the number line and justify which of the two integers it will be closer to.
2. b) Perform the division to obtain its exact decimal expression and check whether your estimate was correct.
3. c) Mentally draw or describe how you would locate this point exactly on the number line: explain into how many equal parts you would divide the segment between the two integers and how many of those parts you would count from the corresponding integer.

Met Éireann, the Irish Meteorological Service, recorded the following thermal anomalies (deviations from the usual average temperature) over four consecutive days in Valentia Observatory (Co. Kerry):

• Monday: $-\frac{7}{2}$ ºC

• Tuesday: $+\frac{5}{4}$ ºC

• Wednesday: $-\frac{11}{4}$ ºC

• Thursday: $-\frac{8}{5}$ ºC

1. a) Model the thermal evolution by ordering the four days from least to greatest anomaly (from the day with the coldest deviation to the day with the warmest) using the $<$ symbol.
2. b) On Friday, a thermal anomaly was measured whose absolute value was exactly half that of Monday, but with the opposite sign to that of Wednesday. Express Friday's anomaly as an irreducible fraction and reason which of the four previous days was closest to Friday's thermal record.

In an archaeological excavation at the Newgrange site, different finds have been classified using numerical labels. You must help the researchers organize these pieces into their corresponding boxes (Number sets).

Given the following list of labels:

$A = \frac{-24}{6}$, $B = \frac{15}{3}$, $C = \frac{7}{4}$, $D = \frac{0}{5}$, $E = \frac{-11}{2}$ and $F = \sqrt{144}$

State, giving reasons, which of these numbers belong to each of the following sets, keeping in mind that a number can belong to more than one set:

1. Natural Numbers ($\mathbb{N}$)

2. Integers ($\mathbb{Z}$)

3. Rational Numbers ($\mathbb{Q}$)

When designing a new memory chip, an engineer needs to place three sensors on a logic line. Sensor A is at position $\frac{2}{5}$ and sensor B is at position $\frac{3}{5}$.

The engineer needs to place a third sensor C exactly at the midpoint between A and B. In addition, he must place a sensor D at the midpoint between A and C.

Calculate the fractions that represent the positions of C and D and represent them on an appropriately divided number line.

Look at the following geometric figure. If the large square represents one unit ($1$), determine what fraction is represented by the section shaded in orange. To solve this, first identify what fraction of the total is represented by each of the pieces that make up the figure.

Three friends, Aoife, Cian and Darragh, are downloading a video game. After one hour, Aoife has downloaded $\frac{13}{20}$ of the total, Cian $\frac{19}{30}$ and Darragh has completed $\frac{5}{8}$.

1. Order the friends from greatest to least amount downloaded using the lowest common multiple method to compare the fractions.

2. If the video game takes up $120$ GB, how many GB does each person still have left to download? Express the results as fractions and integers.

Find the values of the unknowns $x$, $y$ and $z$ so that all the fractions in the following chain are equivalent to each other:

Once the values are found, explain which natural number has been used as a scale factor for expanding or simplifying at each step relative to the set's irreducible fraction.

A tailor has a piece of fabric measuring $15$ metres. He cuts a first piece measuring $3$ metres and $\frac{3}{4}$ of a metre. Afterwards, he cuts a second piece measuring half as much as the first piece.

1. Express the length of each cut as an improper fraction and as a mixed number.

2. What fraction of the original fabric is still left on the roll?

We say that the set of rational numbers is dense because we can always find a fraction between any two others, no matter how close together they are.

Find three fractions between $\frac{4}{7}$ and $\frac{5}{7}$ with denominator $28$. Order the five resulting fractions from smallest to largest.

In a robotics competition, the prototypes must travel along a track that is $10$ meters long. The robot "Zeta" has stopped when it still had $\frac{1}{5}$ of the total track left to travel. The robot "Sigma" has stopped when it had travelled $\frac{7}{9}$ of the total.

1. a) Represent the positions of both robots on the same number line, taking $0$ as the start and $1$ as the finish line (the whole track). Divide the unit of the line so that you can place both points exactly.

1. b) Which robot has stopped closer to the finish line? Justify your answer by comparing the fractions using cross-multiplication.

According to data from the Department of the Environment, Climate and Communications on water reserves in Ireland, two reservoirs present the following situation at the beginning of spring: - Reservoir A contains $175\text{ hm}^3$ out of a total capacity of $250\text{ hm}^3$. - Reservoir B contains $210\text{ hm}^3$ out of a total capacity of $336\text{ hm}^3$. a) Model the proportion of stored water relative to the total in each case using an irreducible fraction. b) Without doing the full division, prove by means of the factorisation of the irreducible denominators whether the decimal percentage of fullness of each reservoir corresponds to a terminating or recurring decimal. c) After a rainfall event, reservoir A increases its volume by $25\text{ hm}^3$ and reservoir B by $42\text{ hm}^3$. Express the new fullness fractions, compare them with reasoning, and determine which of the two reservoirs is now proportionally fuller.

In a class debate on the properties of the set $\mathbb{Q}$, two students make the following statements: - Cian states: "Any irreducible fraction whose denominator ends in an even digit always results in a terminating decimal number, because 2 is a divisor of the decimal base." - Aoife states: "If you add two rational numbers that are purely recurring decimals, the result can never be an integer." a) Prove that Cian's statement is false by proposing a specific counterexample and explain what condition he forgot to require for the prime factors of the denominator. b) Prove that Aoife's statement is also false by finding two fractions with purely recurring decimal expansions whose sum is exactly equal to the integer $2$. c) Construct a fraction $\frac{p}{q}$ with denominator $q = 60$ whose value is strictly between $1$ and $2$, and which results in a terminating decimal. Explain what mathematical condition the numerator $p$ must necessarily satisfy with respect to the prime factorisation of $60$.

In a laboratory practical, the voltage drops of two electrical circuits are analysed, expressed as negative rational numbers: $V_1 = -\frac{17}{24}\text{ V}$ and $V_2 = -\frac{11}{16}\text{ V}$. To determine which of the two circuits experiences a greater loss (that is, which of the two rational numbers is smaller), two team members propose different strategies:

• Aoife's method: Divide the numerator by the denominator in each fraction to obtain their decimal expressions and compare digit by digit.

• Seán's method: Reduce both fractions to a common denominator using the least common multiple and directly compare their integer numerators.

1. a) Apply both methods step by step to find out which of the two voltages is smaller.
2. b) Reflect on the two procedures: what added difficulty does Aoife's division present regarding repeating decimals? Justify why Seán's method is safer, more exact, and more representative of the algebraic properties of the set of rational numbers $\mathbb{Q}$.

One of the fundamental properties of the set of rational numbers $\mathbb{Q}$ compared to the integers $\mathbb{Z}$ is density: between two distinct rational numbers it is always possible to find another rational number, whereas between two consecutive integers there is no other integer.

1. a) Write the integers $-3$ and $-2$ as two fractions having the same positive denominator, so that you can immediately find at least three distinct rational numbers located strictly between them. Write those three rationals as irreducible fractions.
2. b) Solve the following inverse problem: find an irreducible fraction $x = \frac{p}{q}$ strictly between $-\frac{4}{5}$ and $-\frac{3}{5}$ knowing that:

1. Its denominator $q$ is a prime number between $10$ and $15$.

2. The sum of its integer numerator and its denominator is exactly $3$ (that is, $p + q = 3$).

3. The absolute value of its numerator is strictly less than $10$.

Prove in detail that the rational number you found satisfies each and every condition.