Exercises: Rational numbers

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-metre 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 metres 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.

Classify the following decimal numbers into terminating, pure recurring, or mixed recurring. Then, express the terminating decimals in the form of an irreducible fraction.

1. a) $3,25$
2. b) $7,1111\dots$
3. c) $0,25$
4. d) $5,0424242\dots$

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.

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 mobile 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) $\frac{3}{4} \cdot \frac{8}{9}$

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

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

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

Marta has bought a large rectangular pizza for dinner. On opening the box, she sees that the pizza is cut into 24 equal slices.

• Marta eats 4 slices.

• Her brother Luis eats 6 slices.

• Her parents eat 8 slices between them.

• Write down the fraction of pizza eaten by each (Marta, Luis, and parents). Simplify them fully.

• What fraction of the pizza is left over? Express the result as an irreducible fraction.

Marcos is solving a powers exercise in class and makes a very common mistake. The following is written in his notebook:

$$\left(-\frac{2}{5}\right)^{-3} = -\left(\frac{2}{5}\right)^3 = -\frac{2^3}{5^3} = -\frac{8}{125}$$

His classmate Lucía warns him that the final result is incorrect because he applied the meaning of the negative exponent and the handling of the sign of the base incorrectly.

1. Explain with reasoning which two conceptual mistakes Marcos made in his working.

2. Write out the correct working step by step, applying the properties of powers of rational numbers, and state the resulting irreducible fraction.

Before calculating the exact value of the following expression:

$$E = \left(1 - \frac{2}{3}\right)^3 \cdot \left(\frac{5}{2}\right)^2$$

1. Estimate: Without doing long numerical calculations on paper, reflect and answer with justification:

2. Is the base $\left(1 - \frac{2}{3}\right)$ a number less than or greater than $1$? When raised to the power of three, does the result become larger or closer to zero?

3. Is the base $\frac{5}{2}$ greater than or less than $2$? Approximately what is its square?

4. With these two deductions, estimate whether the final value of $E$ will be less than $1$, between $1$ and $2$, or greater than $2$.

5. Check: Now perform the exact calculation of $E$ following the order of operations, obtain the irreducible fraction, and calculate its decimal value rounded to two decimal places to check whether your initial estimation was correct.

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 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 visualise the problem:

Classify the following decimal numbers (terminating, pure recurring, or mixed recurring) and find their generating fraction. Once you have obtained the fractions, perform the following operation and simplify the result:

$$ 1,2\widehat{5} - 0,\widehat{6} $$

In the Physics laboratory, we measured the length of a table and the thickness of a coin. The results were:

• Length of the table: actual value $120 \text{ cm}$, measured value $118 \text{ cm}$.

• Thickness of the coin: actual value $2 \text{ mm}$, measured value $2,5 \text{ mm}$.

Calculate the absolute error and the relative error (as a percentage) for each measurement. Which of the two measurements was more precise? Justify your answer based on the relative error.

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.

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 rest.

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

A rubber ball is dropped from a height of 27 metres. On each bounce, the ball reaches a height equivalent to $\frac{2}{3}$ of the previous height.

1. a) Write down the fraction and the height in metres it reaches after the first bounce.
2. b) Write down the fraction and the height in metres it reaches after the second bounce.
3. c) To what height will it rise after the third bounce?

We have an oil drum with a capacity of 120 litres.

We want to pack all the oil into two types of containers:

• We fill 40 bottles of $\frac{3}{4}$ of a litre each.

• We pack the rest of the oil into $1,5$-litre containers.

How many 1.5-litre containers will we need to pack the remaining oil?

Note: Remember that 1.5 litres can be expressed as the fraction $\frac{3}{2}$.

On a 3rd year ESO exam, the following combined operation with powers must be simplified as much as possible:

$$A = \frac{\left(\frac{2}{3}\right)^3 \cdot \left(-\frac{4}{9}\right)^{-2}}{\left(\frac{8}{27}\right)^{-1}}$$

Two students propose different ways to arrive at the irreducible fraction:

• Sergio's method: Calculate the natural numbers directly (raising $2^3 = 8$, $3^3 = 27$, inverting $(-4/9)$ to $(-9/4)^2 = 81/16$, etc.), multiply the resulting fractions, and finally simplify by dividing by the greatest common divisor.

• Irene's method: Decompose all the numbers that appear into prime factors ($4 = 2^2$, $9 = 3^2$, $8 = 2^3$, $27 = 3^3$), express each power with bases $2$ and $3$, handle the signs of the bases, and operate by adding or subtracting exponents using the properties of powers.

• Solve the expression using Irene's method, detailing each step.

• Compare both methods and justify why Irene's is much more convenient and reliable than Sergio's when working with rational numbers and high powers.

Due to a prolonged dry period in southern Spain, technicians from the Ministry for the Ecological Transition are analysing the water reserves of a reservoir with a maximum capacity of $160\text{ hm}^3$. At the beginning of June, the reservoir is at $\frac{3}{4}$ of its total capacity.

During the months of June, July, and August, the reservoir receives no rainfall and, due to irrigation and evaporation, the volume retained at the end of each month becomes exactly $\frac{4}{5}$ of the volume at the end of the preceding month.

1. Model the situation by writing a single mathematical expression using powers of fractions that represents the fraction of the reservoir's total maximum capacity remaining at the end of the three months.

2. River basin regulations require activating an emergency protocol if reserves drop below $65\text{ hm}^3$. Calculate the exact volume of water available at the end of August (as an irreducible fraction and as a decimal value with a comma) and determine with justification whether it is necessary to activate this protocol.

Calculate the exact value of the following combined expression. To do this, you must first convert all decimal numbers to their irreducible generating fraction and simplify each step before operating. Be careful with the order of operations and negative signs.

$$ \frac{2^{-2} - \left( 1,1\widehat{6} - \frac{5}{6} \right)^{-1} \cdot \sqrt{\frac{16}{81}}}{0,\widehat{3} \cdot \left( 1 - \frac{3}{2} \right)^2 + 2,5} $$

Note: Remember that $a^{-n} = \frac{1}{a^n}$.

A group of 3rd ESO students is organising the funding for their end-of-year trip. They have designed a fundraising plan in stages:

1. In the first stage (raffle ticket sales), they managed to raise $\frac{2}{5}$ of the total cost of the trip.

2. In the second stage (e-sports tournament), they raised $\frac{3}{4}$ of what they still needed after the first stage.

3. In the third stage (charity market), they raised $\frac{2}{3}$ of the new remaining amount.

If after these three stages they still need 150 € to cover the total cost:

1. a) What fraction of the total does the amount raised in the third stage represent?
2. b) What is the total cost of the trip?
3. c) How much money did they raise in the e-sports tournament?

Use the following bar chart to help you visualise the remaining fractions:

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

Two 3D printers are working to print the parts for a technology project.

• Printer model A (older) would take 6 hours to print all the parts on its own.

• Printer model B (latest generation) would take only 4 hours to complete the same job on its own.

To speed up the process, both are set to work simultaneously. However, after 1 hour and a half of working together, printer B breaks down and stops.

1. a) What fraction of the total job have they completed together before the breakdown?
2. b) What fraction of the job remains to be done?
3. c) How much extra time will printer A take to finish the remaining job on its own? Express the result in hours and minutes.

We have a square piece of conductive material for a circuit. Successive cuts are made following a fractal geometric pattern:

1. The original square is divided into 4 equal squares, and the top left square is removed (coloured black).

2. Of the 3 remaining green squares, each is again divided into 4 equal squares, and the top left square of each is removed.

Look at the figure generated after the second step:

1. a) Write down the combined fraction operation that calculates what part of the total area has been removed (shaded areas) after the second step.
2. b) If the side length of the original square is $1 \text{ m}$, how many $\text{cm}^2$ of "white" material (not removed) remain after the second step?

Find an irreducible fraction $\frac{a}{b}$ that is the result of evaluating:

$$ 2,3\widehat{1} - 0,\widehat{5} + \frac{1}{11} \cdot 0,5 $$

If you try to perform this operation by converting to decimals and calculating, you will obtain an approximation error. Prove algebraically, using generating fractions, that the result is exact.

Next, calculate the absolute error and the relative error (in %) that would be incurred if we approximated the number $2,3\widehat{1}$ simply as $2,31$ before starting to calculate.

A biologist is preparing a bacterial culture and needs a very specific room temperature.

The laboratory thermostat has a somewhat unusual scale marked in fractions of a degree.

The ideal temperature $T$ must satisfy the following distance (modulus) condition:

$$ \left| T - \frac{25}{4} \right| \leq \frac{3}{8} $$

1. a) Represent on the number line the set of all possible temperature values $T$ (interval).
2. b) Write down the solution interval using inequalities and fractions.
3. c) Would a temperature of $6,65^\circ \text{C}$ be valid? Justify your answer by calculating with fractions.

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

We have three juice tanks with different capacities:

• Tank A: Filled to $\frac{3}{5}$ of its capacity.

• Tank B: Filled to $\frac{5}{6}$ of its capacity.

• Tank C: Empty, with a capacity of 100 litres.

We know that the total capacity of tank A is $\frac{2}{3}$ of the capacity of tank B.

If we pour the current contents of A and B into tank C, the mixture occupies exactly $\frac{11}{20}$ of the capacity of C.

Calculate the maximum capacity (total volume) of tanks A and B.

A highly elastic rubber ball is dropped from a height of 243 cm.

Each time it bounces on the ground, the ball rises to a height that is equal to $\frac{2}{3}$ of the height from which it fell previously.

1. a) Write down the sequence of heights (in the form of powers and fractions) reached by the ball on the first, second, and third bounce.
2. b) To what height will it rise after the fifth bounce? Express the result as an irreducible fraction and then convert it to a decimal.
3. c) What total distance will the ball have travelled (adding drops and bounces) just as it touches the ground for the fourth time?

We have the following equality involving a power with an unknown integer exponent $x$:

$$\left(\frac{2}{3}\right)^x \cdot \left(\frac{9}{4}\right)^{-2} : \left(\frac{3}{2}\right)^{-3} = \frac{27}{8}$$

1. Express all the powers and the final result as powers of a single base fraction of the form $\frac{3}{2}$. Apply the properties of negative exponents and power of a power.

2. Set up a simple equation with the exponents and deduce the integer value of $x$ with justification.

3. Now find a positive irreducible fraction $\frac{a}{b}$ and a strictly negative integer exponent $n < 0$ such that multiplying $\left(\frac{a}{b}\right)^n$ by the result of the original expression ($\frac{27}{8}$) yields exactly one ($1$). Justify your choice.

The technology department of a secondary school buys a batch of laptops valued at $9\,600\text{ €}$. Due to intensive use and technological obsolescence, the equipment undergoes continuous depreciation: each year it loses one third of the book value it had the previous year.

The school is considering two options once the equipment reaches a certain age:

• Trade-in Plan at 3 years ($t = 3$): Sell the batch to a refurbishment company for its exact book value at that time.

• Trade-in Plan at 4 years ($t = 4$): Keep the laptops for one more year at the school and sell them after 4 years for their book value on that date, but adding a fixed institutional grant for digital recycling of $300\text{ €}$ for the school.

• Write the formula that expresses the book value $V(t)$ as a function of the elapsed years $t$, using a power with an irreducible rational base.

• Calculate the exact book value of the batch after $3$ years and after $4$ years, simplifying the fractions before obtaining the amount in euros.

• Analyse which of the two plans yields higher economic revenue for the school and calculate the exact financial difference between both options.