Rational numbers
Your phone shows 75 % battery, at the pizzeria you order three quarters of a pizza each, and the GPS says you have covered two thirds of the journey. All of these are fractions: numbers used to talk about parts. In this section you will learn to handle them with confidence, which is the basis of almost everything that comes next.
What is a rational number?
A fraction $\frac{a}{b}$ has two parts: the denominator $b$ says how many equal parts we split the whole into, and the numerator $a$ says how many we take.
The same fraction can be read in three ways, and it is worth keeping all of them in mind:
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Part of a whole: $\frac{3}{4}$ of a pizza is 3 slices of a pizza cut into 4.
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Quotient: $\frac{3}{4} = 3 : 4 = 0{,}75$. That is why your battery at 75 % is at $\frac{3}{4}$.
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Operator: $\frac{3}{4}$ of 200 € is $200 \cdot \frac{3}{4} = 150$ €.
Integers are also rational numbers: $5 = \frac{5}{1}$ and $-2 = \frac{-4}{2}$. And a fraction can be negative: $-\frac{3}{4} = \frac{-3}{4} = \frac{3}{-4}$ (the sign can go wherever you like, but it is usual to put it in front).
Play around with the numerator and the denominator. Notice what happens when the numerator is greater than the denominator: the fraction is worth more than one whole (an improper fraction).
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 $\frac{1}{3}$…
In a strategy video game, the map is divided into different zones according to the terrain.
The total map occupies 1 200 square pixels.
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The forest occupies $\frac{1}{3}$ of the map.
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The water occupies $\frac{1}{4}$ of the map.
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The remainder is desert.
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How many square pixels does the forest occupy?
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How many square pixels does the water occupy?
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Work out what irreducible fraction of the map represents the desert.
Equivalent fractions
Half a pizza, two quarters or four eighths: it is the same amount of pizza, even though it is cut differently. These are equivalent fractions.
If you multiply or divide the numerator and the denominator by the same number, you get an equivalent fraction. Dividing is called simplifying. When it can no longer be simplified, you have the fraction in its simplest form: the cleanest way of writing that number.
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…
Look at the following graphical representation of a fraction. Your task is to:
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Write down the irreducible fraction that represents the section coloured orange compared to the total circle.
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State whether the fraction $\frac{9}{24}$ is equivalent to the one represented in the image. Justify your answer by simplifying both fractions.
Comparing and ordering fractions
Who has eaten more: the person who eats $\frac{3}{4}$ of a bar of chocolate or the one who eats $\frac{5}{7}$? At first glance it is not clear. Look at the bars:
By a small margin, $\frac{3}{4}$ is greater. But you will not always be able to draw a picture, so you need a method.
Three friends are competing in a 100-metre obstacle race. At a given moment, their positions are as follows: Ana has covered $\frac{2}{5}$ of the total distance. Beto…
Three friends are competing in a 100-metre obstacle race. At a given moment, their positions are as follows:
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Ana has covered $\frac{2}{5}$ of the total distance.
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Beto has covered $\frac{3}{4}$ of the total distance.
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Carla has covered $\frac{7}{10}$ of the total distance.
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Calculate how many metres each person has covered exactly.
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Order the fractions from smallest to largest using the symbol $<$.
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Mark the approximate positions of Ana (A), Beto (B), and Carla (C) on the number line shown below.
Representing fractions on the number line
Every rational number has its exact place on the number line. To place $\frac{7}{4}$:
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If it is improper, take out the whole numbers: $\frac{7}{4} = 1 + \frac{3}{4}$. It lies between 1 and 2.
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Divide the stretch from 1 to 2 into 4 equal parts (the denominator).
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Count 3 of those parts (the numerator of the remainder).
And what if you have to divide a segment into exactly 7 parts, without a ruler with a scale? You use Thales' theorem: you draw an auxiliary ray from 0, mark 7 equal steps with the compasses, join the last mark to 1 and draw parallel lines. You will see this in the geometry chapter.
Try dragging the point: you will see which fraction and which decimal correspond to each position.
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…
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.
Adding and subtracting fractions
Adding fractions is like adding coins from different countries: before putting them together, you have to convert them all into the same currency. Here the "currency" is the denominator.
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…
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.
- a) $\frac{2}{3} + \frac{1}{6}$
- b) $\frac{4}{5} - \frac{1}{2}$
- c) $\frac{3}{4} + \frac{5}{12} - \frac{1}{3}$
The storage on your mobile phone is divided as follows: Photos occupy $\frac{2}{5}$ of the space. Applications occupy $\frac{1}{3}$ of the space. 1. What fraction of the…
The storage on your mobile phone is divided as follows:
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Photos occupy $\frac{2}{5}$ of the space.
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Applications occupy $\frac{1}{3}$ of the space.
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What fraction of the total do photos and applications occupy together?
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If the rest of the space is free, what fraction represents the free space?
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If the total capacity of the phone is 60 GB, how many GB are left free?
Multiplying and dividing fractions
Multiplying is easier than adding: numerator times numerator and denominator times denominator. The drawing explains why: $\frac{2}{3}$ of $\frac{3}{4}$ is the region where the two strips overlap.
Calculate the following products and divisions of fractions. Remember to simplify the final result if possible. a) $\frac{3}{4} \cdot \frac{8}{9}$ b)…
Calculate the following products and divisions of fractions. Remember to simplify the final result if possible.
- a) $\frac{3}{4} \cdot \frac{8}{9}$
- b) $\frac{2}{5} : \frac{4}{15}$
- c) $\frac{5}{6} \cdot (-2)$
Note: Remember the rule of signs and that an integer has a denominator of 1.
Three friends buy a birthday present costing 60 €. Ainhoa contributes $\frac{1}{3}$ of the total price. Beto contributes $\frac{2}{5}$ of the total price. Carla pays the…
Three friends buy a birthday present costing 60 €.
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Ainhoa contributes $\frac{1}{3}$ of the total price.
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Beto contributes $\frac{2}{5}$ of the total price.
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Carla pays the rest.
- a) What fraction of the total does Carla pay?
- b) How much money in euros does each person contribute?
Combined operations
When operations are mixed, there is a compulsory order. If you do not follow it, each person would get a different result from the same calculation.
Within the same level, work from left to right.
Calculate the value of the following combined operation, following the order of operations (brackets, powers, multiplication/division, addition/subtraction). You must…
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.
$$ \left( \frac{3}{2} - \frac{1}{4} \right)^2 : \frac{5}{8} - \left( 1 - \frac{3}{5} \cdot \frac{10}{3} \right) $$Your mobile 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…
Your mobile 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:
- a) What fraction of the total represents the free memory for photos?
- b) What is the total storage capacity of the phone in GB?
Use the following diagram to help you visualise the problem: