Exercises: The set of rational numbers

The following image represents the loading bar of a YouTube video that is being processed. What fraction of the video has already loaded? What fraction is left to load? Express both fractions in the simplest possible form (irreducible fraction).

Of the $320$ followers Lucy has on her photography account, $\frac{3}{8}$ are from her school, $\frac{1}{4}$ are from her neighborhood and the rest are people she has met on her travels. Calculate:

1. a) The number of followers who go to the same school as her.
2. b) The number of followers who live in her neighborhood.

To prepare a fruit smoothie, the recipe indicates that $\frac{2}{5}$ of a gallon of milk should be used. If we want to prepare a larger amount maintaining the proportion, write three equivalent fractions to $\frac{2}{5}$ by amplification.

Simplify the following fractions until you obtain the irreducible fraction in each case:

1. a) $\dfrac{12}{18}$

1. b) $\dfrac{20}{50}$

1. c) $\dfrac{15}{60}$

Compare the following pairs of rational numbers using the signs $<$ (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) $\dfrac{1}{3} \square \dfrac{4}{12}$

Copy the table into your notebook and mark with an "X" the sets to which each number belongs (a number can belong to several sets):

Draw a number line and represent the following rational numbers on it approximately. Indicate whether they are greater than, less than or equal to one:

1. a) $\dfrac{1}{2}$

1. b) $\dfrac{5}{4}$

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

Three friends, Ashley, Ben, and Carlos, have ordered $2$ family pizzas for dinner. If they want to share them so that everyone eats exactly the same amount, what fraction of a pizza does each person get? Is it a proper or an improper fraction? Reason your answer.

A content creator has an external hard drive with a total capacity of $500$ GB. They have used $\frac{3}{20}$ of the space for video clips and $\frac{2}{5}$ for music and sound effects.

1. a) Calculate what fraction of the hard drive the total occupied space represents.
2. b) What fraction of the drive is still free?
3. c) How many gigabytes (GB) of space are left free?

Find the unknown term $x$ 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) $\frac{27}{81} = \frac{1}{x}$

Simplify the following fractions until you obtain the irreducible fraction. For part c, use the prime factor decomposition of the numerator and the denominator.

1. a) $\frac{48}{72}$

1. b) $\frac{150}{375}$

1. c) $\frac{420}{1260}$

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

Identify the fractions that correspond to the points $A$, $B$ and $C$ marked on the following number line:

Classify the following numbers, indicating which set or sets they belong to (Natural numbers $\mathbb{N}$, Integers $\mathbb{Z}$ or Rational numbers $\mathbb{Q}$):

1. a) $-5$

1. b) $\frac{18}{3}$

1. c) $0,75$

1. d) $-\frac{1}{4}$

1. e) $\frac{20}{5}$

Match each decimal number with its corresponding simplified generating fraction:

1. $0,125$

2. $2,5$

3. $1{,}\hat{3}$

4. $0{,}\overline{45}$

1. a) $\frac{4}{3}$
2. b) $\frac{5}{11}$
3. c) $\frac{1}{8}$
4. d) $\frac{5}{2}$

Three friends are completing a soccer sticker album. Alex has stuck in $\frac{1}{3}$ of the total stickers, Beatrice has stuck in $\frac{2}{5}$ of the total and Daniel has the rest of the stickers.

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

In a high mountain weather station, the following temperature variations with respect to the seasonal mean in degrees Fahrenheit have been recorded:

• Monday: $-\frac{9}{4} \text{ } ^\circ\text{C}$

• Tuesday: $-2,1 \widehat{6} \text{ } ^\circ\text{C}$

• Wednesday: $-| -2,2 \text{ } | ^\circ\text{C}$

• Thursday: $-\frac{22}{10} \text{ } ^\circ\text{C}$

Represent these four values on a number line and order them in increasing order (from smallest to largest) using the $<$ symbol. Justify your answer by converting all the numbers to their irreducible fraction form.

Calculate the irreducible generating fraction of the following repeating decimals. Then, find a rational number between them that has the number $11$ as its denominator.

1. a) $2,0 \widehat{45}$
2. b) $2, \widehat{045}$

A communications satellite sends an adjustment signal every $0,12\widehat{6}$ seconds. A receiving station is programmed to pick up signals only at intervals that are exact multiples of $\frac{19}{150}$ of a second.

Reason mathematically whether the receiving station will be able to pick up the satellite's signal at some point. To do this, express the signal time as an irreducible fraction and compare it with the station's interval.

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

• Zone A takes up $\frac{2}{5}$ of the total.

• Zone B takes up $\frac{3}{7}$ of what is left after marking out Zone A.

• Zone C takes up the rest of the surface area.

If the total surface area of the swimming pool is $140 \text{ ft}^2$, calculate what fraction of the total Zone C represents and how many square feet correspond to it. Express the result of the surface area of Zone C as an exact decimal number.

Three friends, Alex, Beatrice and Chris, are taking part in an obstacle course. Alex has completed $\frac{13}{20}$ of the course, Beatrice $64\%$ and Chris has covered $0,6\widehat{3}$ of the total.

1. a) Indicate who is in first position and who is last.
2. b) If the total course is $4500$ feet, how many feet does Chris have left to reach the finish line?
3. c) Express each person's position as an irreducible fraction with the same denominator.

Find a proper fraction whose sum of its terms (numerator and denominator) is $24$ and which is equivalent to a pure repeating decimal whose repeating part has a single digit. How many possible solutions exist that meet this condition? Justify your answer by finding at least two of them.

A rectangular plot of land $120,45$ feet long by $90,3$ feet wide is to be divided into equal squares of the largest possible surface area.

1. a) Express the dimensions of the land as irreducible fractions.
2. b) Calculate the side length of each square, knowing that it must be an exact number of inches.
3. c) How many squares will be obtained in total?

Consider the following rational numbers:

1. a) Without performing the division, can you be sure what type of decimal the fraction $x$ will generate? Justify your answer by analyzing the prime factors of the denominator.
2. b) Prove whether $x$ and $y$ are equivalent.
3. c) Find the irreducible fraction of the sum $x + y$. Express the result as a mixed repeating decimal.