The Set of Rational Numbers

In a pizzeria, a family pizza has been divided into $8$ identical slices. If Marcos 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?

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?

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) $10$

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:

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

In an archaeological excavation at the Atapuerca 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 ($\mathbb{Q}$)

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

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.

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.

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.

A tailor has a piece of fabric measuring $15$ meters. He cuts a first piece measuring $3$ meters and $\frac{3}{4}$ of a meter. Afterward, 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?

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

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

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

• Álex: 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.

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

$$\frac{x}{6} = \frac{10}{12} = \frac{15}{y} = \frac{z}{30}$$

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.

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.

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 robotics competition, the prototypes must travel along a $10$-meter track. The robot "Zeta" stopped when it had $\frac{1}{5}$ of the total track left to cover. The robot "Sigma" stopped when it had covered $\frac{7}{9}$ of the total.

1. Represent the positions of both robots on the same number line. Divide the unit on the line so that you can plot both points accurately.

2. Which robot ended up closer to the finish line? Justify your answer by comparing the fractions using cross-multiplication.

In a study on cell phone usage, the fraction of total storage occupied by different apps on Lucía's phone was analyzed:

• 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?

Three friends, Ioana, Marcos and Rayan, are downloading a video game. After one hour, Ioana has downloaded $\frac{13}{20}$ of the total, Marcos $\frac{19}{30}$ and Rayan 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.

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.

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.

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 measurements to decimal form.
2. b) Which two athletes jumped exactly the same distance?
3. c) Order the athletes from the one who jumped the shortest distance to the one who jumped the longest.