Exercises: The set of rational numbers

In a pizzeria, they have divided a family pizza into $8$ identical slices. If Marcus has eaten $3$ slices, express the part of the pizza he has 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 in your notebook the fraction that represents the orange shaded area.

Classify the following fractions into 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 preparing a party and buying packs of soft drinks. Each pack has $6$ cans. If we need to use $\dfrac{12}{6}$ of 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 graphically in your notebook, using squares or rectangles:

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

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

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

1. a) $15$ minutes.

1. b) $30$ minutes.

1. c) $10$ minutes.

1. d) $60$ minutes.

Draw a number line in your notebook and place the following rational numbers on it:

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

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

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

Write the following natural numbers as a fraction (you can use whichever 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 within 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 favorite video game. Each one has progressed a different fraction of the total:

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

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

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

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

Represent on the following number line the points $A$, $B$, $C$ and $D$ corresponding to these rational numbers:

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

$B = -1.25$

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

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

In a study on mobile phone use, the fraction of total storage taken up by different applications on Chloe's phone has been 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) Reduce the fractions to a common denominator.
2. b) Order the categories from largest to smallest storage used.
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?

Several pizzas of the same size were bought for a birthday party. At the end of the party, the following pieces were left over:

• $\frac{3}{4}$ of a ham pizza.

• $\frac{5}{4}$ of pepperoni pizzas.

• $\frac{1}{2}$ of a vegetable 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 working out the sum.

Identify which of the following fractions will result in a terminating decimal and which in a repeating decimal (pure or mixed) without performing the division, by looking only at the prime factorization 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 of factors 2 and 5 in the denominator).

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

• Athlete 1: $6.4$ feet.

• Athlete 2: $\frac{13}{2}$ feet.

• Athlete 3: $6.35$ feet.

• Athlete 4: $\frac{32}{5}$ feet.

1. a) Convert all measurements to decimal form.
2. b) Which two athletes have jumped exactly the same distance?
3. c) Order the athletes from the one who jumped the least to the one who jumped the most.

In an archaeological excavation at the Cahokia site, different findings 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 with reasons which of these numbers belong to each of the following sets, bearing in mind that a number can belong to more than one:

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

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

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

In the design of 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, they must place a sensor D at the midpoint between A and C.

Work out the fractions that represent the positions of C and D and represent them on a suitably divided number line.

Look at the following geometric figure. If the large square represents one unit ($1$), determine what fraction the orange shaded part represents. To solve this, first identify what fraction of the total each of the pieces making up the figure represents.

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

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

2. If the video game takes up $120$ GB, how many GB does each one 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:

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

Once the values have been found, explain which natural number has been used as a multiplying or dividing factor in each step with respect to the irreducible fraction of the set.

A tailor has a piece of fabric of $15$ yards. He cuts a first piece that measures $3$ yards and $\frac{3}{4}$ of a yard. Then, he cuts a second piece that measures half of what the first one measured.

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 two others, no matter how close they are.

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

In a robotics competition, the prototypes must cover a track of $10$ feet. The robot "Zeta" has stopped when it had $\frac{1}{5}$ of the total track left to cover. The robot "Sigma" has 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 of the line so that you can place both points exactly.

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