Exercises: Fractions and decimals

Look at the following illustration representing the battery level of a mobile phone.

1. Write the fraction that represents the charged part of the battery relative to the total.

2. Simplify this fraction to obtain its irreducible fraction.

3. If the battery has a total capacity of $4000 \text{ mAh}$, how many mAh of energy are available?

State whether the following pairs of fractions are equivalent. To justify your answer, use cross-multiplication or simplify both fractions to their irreducible form.

1. a) $\frac{3}{5}$ and $\frac{9}{15}$
2. b) $\frac{4}{7}$ and $\frac{10}{21}$
3. c) $\frac{-2}{3}$ and $\frac{8}{-12}$

Represent approximately on the following number line the rational numbers indicated below.

Numbers to represent:

• $A = \frac{3}{2}$

• $B = -\frac{1}{2}$

• $C = \frac{3}{4}$

• $D = -1,5$

Perform the following combined operations with fractions. Remember to simplify the final result whenever possible and respect the order of operations (brackets first, then multiplication/division, and finally addition/subtraction).

1. a) $\frac{2}{3} + \frac{1}{4} \cdot \frac{2}{5}$
2. b) $\left( \frac{3}{2} - \frac{1}{4} \right) : \frac{5}{2}$
3. c) $\frac{5}{6} - \frac{1}{2} + \frac{2}{3}$

Classify the following decimal numbers into terminating decimals, pure recurring decimals, or mixed recurring decimals. Then, convert the first two (parts a and b) into their corresponding generating fraction.

1. a) $2,5$
2. b) $1,333...$
3. c) $4,1252525...$
4. d) $0,08$

A group of friends buys a giant pizza for dinner.

• Ana eats $\frac{1}{6}$ of the pizza.

• Carlos eats $\frac{2}{6}$ of the pizza.

• Beatriz eats $\frac{1}{3}$ of the pizza.

1. a) What fraction of the pizza have the three of them eaten altogether? (Remember to find a common denominator if necessary).
2. b) What fraction of the pizza is left over?

Calculate the irreducible fraction equivalent to each of the following percentages and decimal numbers.

1. a) $25\%$
2. b) $0,75$
3. c) $10\%$
4. d) $1,5$

In a school of 500 students, a survey was conducted on their favourite sports. The results are as follows:

• $\frac{1}{2}$ of the students prefer football.

• $\frac{3}{10}$ of the students prefer basketball.

• The rest of the students prefer other sports.

1. a) How many students prefer football?
2. b) How many students prefer basketball?
3. c) What fraction of the total number of students prefers "other sports"?

Mateo has to calculate the generating fraction of the mixed recurring decimal number $0,3\widehat{8}$ to solve a classroom activity. In his notebook, he writes the following:

$$0,3\widehat{8} = \frac{38}{90} = \frac{19}{45}$$

1. Identify the mistake Mateo made in the numerator and explain with reasoning why his procedure is incorrect.

2. Apply the correct deductive rule (subtracting the non-repeating part from the number without the decimal point) to find the generating fraction and simplify it to obtain the irreducible fraction.

3. Carry out the long division by hand of the numerator by the denominator of the fraction you obtained to verify that it indeed reproduces the decimal $0,3888\dots$

To assemble a shelf in the Technology workshop, Sofía needs to join two wooden battens together. The first batten is $1,\widehat{6}\text{ m}$ long, and the second batten is exactly $\frac{9}{4}\text{ m}$ long.

1. Before performing calculations with fractions, make a mental estimate by approximating each measurement to a decimal number with a single decimal place (tenths) and add them together. What is the approximate length of the two battens joined together?

2. Find the irreducible generating fraction of $1,\widehat{6}$ and calculate the exact sum with the fraction $\frac{9}{4}$, expressing the final result as an irreducible fraction.

3. Convert the resulting irreducible fraction into decimal form and compare it with your initial estimate. Was it a reasonable estimate? Justify what causes the small difference.

Classify the following numbers into the smallest number set to which they belong ($\mathbb{N}$, $\mathbb{Z}$ or $\mathbb{Q}$) and, for those that are decimal numbers (terminating or recurring), find their simplified generating fraction.

1. a) $4,25$
2. b) $-7$
3. c) $3,1\widehat{6}$
4. d) $5,\widehat{27}$
5. e) $-\frac{18}{3}$

Once you have obtained the fractions, order them from smallest to largest.

Perform the following combined operations with fractions, respecting the order of operations. Remember to simplify the final result to obtain the irreducible fraction.

$$ \left( \frac{3}{4} - \frac{1}{2} \right)^{-2} + \frac{5}{3} \cdot \left( \frac{2}{5} - 1 \right) - \frac{1}{2} : \frac{3}{4} $$

Javier has been checking the storage on his mobile phone, which has a total capacity of $128 \text{ GB}$. Look at the following distribution of his data:

1. The operating system and pre-installed apps take up $\frac{1}{8}$ of the total.

2. His holiday photos and videos take up $\frac{2}{5}$ of the remainder (what is left free after accounting for the system).

3. His games take up $\frac{1}{4}$ of what remained after saving the photos.

What fraction of the total represents the space he still has free? How many Gigabytes (GB) is that? Use the following diagram to help visualise the problem.

We have two different scales.

• Scale A: Used in a laboratory and when weighing a test tube gives a value of $24,5 \text{ g}$. The exact actual value is $\frac{49}{2} \text{ g}$.

• Scale B: Used in a warehouse and when weighing a box gives a value of $5,4 \text{ kg}$. The exact actual value is $\frac{16}{3} \text{ kg}$.

Calculate the absolute error and the relative error made in each measurement. Which of the two measurements is more precise? Justify your answer based on the relative error.

Calculate the exact value of the following expression by first converting all decimal numbers into their generating fractions. Do not use decimal approximations during the process.

$$ \frac{1,2 - 0,\widehat{6}}{1 - 0,8\widehat{3}} : 3^{-1} $$

Three friends, Ana, Beto and Carla, decide to buy a birthday present that costs $120 \text{ €}$.

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

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

• Carla pays the rest.

1. a) What fraction of the total does Carla contribute?
2. b) How many euros does each person pay?
3. c) If they had decided to pay equal shares, what fraction of the total would each person have paid, and how much money would that be?

A rectangular plot of land is to be fenced. We know that the length of its base is $\frac{15}{2}$ metres and its height is $\frac{2}{3}$ of the measure of the base.

1. a) Calculate the perimeter of the rectangle as a fraction and as a decimal.
2. b) Calculate the area of the plot in square metres ($\text{m}^2$).
3. c) If fencing costs $8,50 \text{ €}$ per metre, how much will it cost to fence the entire plot?

A bacterium reproduces by splitting in two every hour. Initially, in a culture there is a certain number of bacteria occupying $\frac{1}{64}$ of the total capacity of the Petri dish.

1. a) If the population doubles every hour, what fraction of the dish will be occupied after 3 hours? Express the result as a power with base 2.
2. b) How many hours in total will have to pass (from the beginning) for the dish to be completely full (that is, for the fraction to be 1)?
3. c) If initially there were 500 bacteria (corresponding to that $\frac{1}{64}$), how many bacteria will there be when the dish is at half capacity?

Find a positive irreducible fraction $\frac{n}{d}$ that simultaneously satisfies the following four conditions:

• The denominator $d$ is a single-digit number ($d < 10$).

• When dividing $n$ by $d$, a pure recurring decimal number whose period has a single digit is obtained.

• The value of the decimal number is strictly between $1,5$ and $2$ ($1,5 < \frac{n}{d} < 2$).

• The sum of the numerator and the denominator is equal to $25$ ($n + d = 25$).

• Justify which digits can under no circumstances be the denominator $d$ if we want the decimal to be a pure recurring decimal.

• Deduce the exact values of $n$ and $d$, ruling out with mathematical arguments the options that do not satisfy all the clues.

• Write the final fraction, verify that it is irreducible, and express it in decimal form indicating its period.

According to data from the hydrological bulletin of a Spanish river basin, a reservoir has a maximum capacity of $180\text{ hm}^3$. After the recent rains, the stored water is distributed as follows:

• The water intended for urban consumption represents $22,\widehat{2}\%$ of the reservoir's total capacity.

• The water reserved for agricultural irrigation is exactly equal to $\frac{5}{12}$ of the reservoir's total capacity.

• The remaining volume of stored water is reserved as the ecological flow of the river.

• Convert the urban consumption percentage ($22,\widehat{2}\%$) into an irreducible generating fraction. Remember that a percentage is equivalent to a fraction with a denominator of 100.

• Calculate how many cubic hectometres ($\text{hm}^3$) of water correspond exactly to each of the three uses: urban, agricultural, and ecological flow.

• To avoid environmental penalties, regulations require that the stored ecological flow exceed one fourth of the reservoir's total volume. Based on your calculations, determine with justification whether this requirement is met.

Calculate the exact value of the following expression, converting all decimal numbers beforehand into their irreducible generating fractions and simplifying the result step by step:

$$ \frac{1,1\widehat{6} - \left( \frac{1}{2} \right)^{-2} \cdot 0,1}{2 - \frac{1}{1 - \frac{1}{3}}} + \sqrt{0,\widehat{4}} $$

Note: Remember to respect the order of operations and work with fractions throughout the process to avoid rounding errors.

A cloud storage server experienced a series of massive data uploads and accidental deletions over the weekend.

1. On Saturday morning, the server was full up to $\frac{3}{4}$ of its total capacity.

2. At noon, an automatic script deleted $\frac{1}{5}$ of the data stored at that moment.

3. In the afternoon, new files were uploaded that occupied $\frac{2}{3}$ of the capacity that remained free after the deletion.

If at the end of the day there were still 120 GB of free space remaining on the server's hard drive:

a) What fraction of the total disk does the final free space represent?

b) What is the total storage capacity of the server in GB?

Observe the following geometric figure representing a square plot of land divided into different crop patches.

Calculate what fraction of the total area of the square is represented by the shaded region (orange colour). Express the result as an irreducible fraction.

In a chemistry laboratory, precision measurements are being taken. Two students, Álex and Bea, measure the length of a microscopic fibre.

• The actual value of the length of the fibre is $2,345 \cdot 10^{-4}$ metres.

• Álex obtains a measurement of $2,3 \cdot 10^{-4}$ metres.

• Bea obtains a measurement of $2,4 \cdot 10^{-4}$ metres.

Answer, justifying your answers with calculations:

a) Calculate the absolute error made by each student. Express the result in standard form.

b) Calculate the relative error of each measurement and express it as a percentage.

c) Based on the relative error, which of the two measurements is more precise and why?

d) If we truncate the actual value to ten-thousandths, does it coincide with rounding to ten-thousandths? Explain your answer.

Three taps ($A$, $B$ and $C$) fill a tank with water, and a drain ($D$) allows it to be emptied.

• Tap $A$ fills the tank in 4 hours working alone.

• Tap $B$ fills the tank in 6 hours working alone.

• Tap $C$ fills the tank in 12 hours working alone.

• Drain $D$ empties the full tank in 8 hours.

The tank is initially empty. All three taps ($A$, $B$, $C$) and the drain ($D$) are opened simultaneously. However, after 2 hours of operation, tap $A$ is turned off due to a fault, leaving the other two and the drain working.

Calculate the total time (in hours and minutes) it will take for the tank to fill completely from the start.

Find two rational numbers and two irrational numbers that lie strictly within the interval:

$$ \left( \frac{13}{9} \, , \, 1,4\widehat{5} \right) $$

Explain in detail the procedure you followed to find them and prove that they indeed belong to that interval.

A jeweller has two ingots of a gold and copper alloy.

• The first ingot has a purity such that for every 5 parts of gold there are 2 of copper.

• The second ingot is of lower quality, having 3 parts of gold for every 4 parts of copper.

The jeweller melts a certain amount of the first ingot with a certain amount of the second to obtain a new ingot of 420 grams. If he wants the final fraction of gold in the new ingot to be exactly $\frac{2}{3}$ of the total weight:

How many grams must he cut and melt from each of the two original ingots?

Let $x$ be a mixed recurring decimal number whose integer part is 0.

Knowing that:

• Its non-recurring part has 2 digits.

• Its recurring part has 1 digit.

• The irreducible generating fraction of this number has a denominator of 900.

• The number satisfies: $\frac{1}{8} < x < \frac{1}{7}$

Find all possible values of $x$ and express them as a fraction and as a decimal number.

On an exam, students are asked to calculate the exact value of the following combined operation:

$$M = \frac{0,\widehat{7} - 0,4\widehat{6}}{0,\widehat{3}}$$

Three 3rd-year ESO students propose different approaches:

• Hugo rounds each decimal beforehand to the hundredths ($0,78$; $0,47$, and $0,33$) and calculates using a calculator.

• Elena claims that the periods in the numerator can be subtracted directly, and then the resulting decimal numbers can be divided.

• David converts each term into its irreducible generating fraction and works with fractions step by step.

• Apply Hugo's method and calculate the decimal value he obtains.

• Apply David's method using irreducible generating fractions and simplify the result to an irreducible fraction. Obtain its exact recurring decimal expression.

• Critically evaluate the proposals:

a) Calculate the absolute error Hugo makes by comparing his result with David's exact value, and reflect on the risk of rounding before operating in situations requiring scientific or financial precision.

b) Justify why Elena's idea of operating directly with the recurring decimal digits is not a valid method in general for multiplication or division.

The aim is to design a numeric code from an irreducible fraction $\frac{a}{b}$ (with $a, b \in \mathbb{N}$ and $a < b$) that satisfies the following conditions:

• Its decimal expansion is mixed recurring, with a one-digit non-recurring part (anteperiod) and a two-digit period: $0,x\widehat{yz}$.

• The fraction lies on the real number line strictly between $0,15$ and $0,20$.

• The denominator $b$ is the smallest possible natural number that produces this decimal structure.

• Analyse the theoretical generating fraction $\frac{N - a}{990}$ and explain which prime factors the simplified denominator $b$ must necessarily contain. Prove why the smallest possible denominator cannot be $22$ and must be $b = 55$.

• Find the only numerator $a$ that makes $\frac{a}{55}$ irreducible and places it in the interval $(0,15;\; 0,20)$. Find its decimal expansion, identifying the non-recurring part and the period.

• Find the irreducible generating fraction of the number $0,1\widehat{9}$. In light of the result, classify it into the most restrictive number set to which it belongs ($\mathbb{N}$, $\mathbb{Z}$, or $\mathbb{Q}$), and write an example of an irrational number that lies strictly between your fraction $\frac{a}{55}$ and this number $0,1\widehat{9}$.