Rational Numbers

In a strategy video game, the map is divided into different zones according to the terrain.

The total map occupies 1 200 square pixels.

• The forest occupies $\frac{1}{3}$ of the map.

• The water occupies $\frac{1}{4}$ of the map.

• The remainder is desert.

• How many square pixels does the forest occupy?

• How many square pixels does the water occupy?

• Work out what irreducible fraction of the map represents the desert.

Look at the following graphical representation of a fraction. Your task is to:

1. Write down the irreducible fraction that represents the section coloured orange compared to the total circle.

2. State whether the fraction $\frac{9}{24}$ is equivalent to the one represented in the image. Justify your answer by simplifying both fractions.

Three friends are competing in a 100-meter obstacle race. At a given moment, their positions are as follows:

• Ana has covered $\frac{2}{5}$ of the total distance.

• Beto has covered $\frac{3}{4}$ of the total distance.

• Carla has covered $\frac{7}{10}$ of the total distance.

• Calculate how many meters each person has covered exactly.

• Order the fractions from smallest to largest using the symbol $<$.

• Mark the approximate positions of Ana (A), Beto (B), and Carla (C) on the number line shown below.

Represent the rational numbers $\frac{7}{4}$ and $-\frac{2}{3}$ on the number line.

To do so, use Thales' Theorem or the division of segments to place the points exactly, not approximately.

Draw in your notebook a line like the following one, with the integers marked (it is best for the unit to be large, for example 6 cm), and carry out your construction on it:

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

Perform the following operations with fractions. Remember that to add or subtract, you must first find the lowest common multiple (LCM) of the denominators. Express the result as an irreducible fraction.

1. a) $\frac{2}{3} + \frac{1}{6}$

1. b) $\frac{4}{5} - \frac{1}{2}$

1. c) $\frac{3}{4} + \frac{5}{12} - \frac{1}{3}$

The storage on your cell phone is divided as follows:

• Photos occupy $\frac{2}{5}$ of the space.

• Applications occupy $\frac{1}{3}$ of the space.

• What fraction of the total do photos and applications occupy together?

• If the rest of the space is free, what fraction represents the free space?

• If the total capacity of the phone is 60 GB, how many GB are left free?

Calculate the following products and divisions of fractions. Remember to simplify the final result if possible.

1. a) $\frac{3}{4} \times \frac{8}{9}$

1. b) $\frac{2}{5} : \frac{4}{15}$

1. c) $\frac{5}{6} \times (-2)$

Note: Remember the rule of signs and that an integer has a denominator of 1.

Three friends buy a birthday present costing 1,200 pesos.

• Ana contributes $\frac{1}{3}$ of the total price.

• Beto contributes $\frac{2}{5}$ of the total price.

• Carla pays the rest.

1. a) What fraction of the total does Carla pay?
2. b) How much money in pesos does each person contribute?

Calculate the value of the following combined operation, following the order of operations (brackets, powers, multiplication/division, addition/subtraction). You must simplify the final result to obtain the irreducible fraction.

$$ \left( \frac{3}{2} - \frac{1}{4} \right)^2 : \frac{5}{8} - \left( 1 - \frac{3}{5} \times \frac{10}{3} \right) $$

Your cell phone has an unknown total storage capacity. The operating system occupies $\frac{1}{6}$ of the total memory. Of what remains free, your applications and games occupy $\frac{3}{5}$. If in the end you have 32 GB free for your photos and videos:

1. a) What fraction of the total represents the free memory for photos?
2. b) What is the total storage capacity of the phone in GB?

Use the following diagram to help you visualize the problem: