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?

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:

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

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.

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.

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

$B = -1.25$

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

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

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.

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).

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.

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).

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.