Exercises: Fractions and decimals

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

1. Write the fraction that represents the charged part of the battery with respect 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 the cross-multiplication method 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 the following rational numbers approximately on the number line 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 (parentheses first, then multiplications/divisions, and finally additions/subtractions).

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 as terminating decimals, repeating decimals or mixed repeating 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.

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

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

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

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

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

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

In a high school of 500 students, a survey was carried out about their favorite sports. The results are as follows:

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

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

• The rest of the students prefer other sports.

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

Classify the following numbers into the smallest numerical set to which they belong ($\mathbb{N}$, $\mathbb{Z}$ or $\mathbb{Q}$) and, for those that are decimals (terminating or repeating), 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 the fractions are obtained, 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.

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

Ethan has been checking the storage of his cell phone, which has a total capacity of $128 \text{ GB}$. He observes the following distribution of his data:

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

2. The photos and videos from his vacation take up $\frac{2}{5}$ of the rest (what is left free after counting the system).

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

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

We have two different scales.

• Scale A: Used in a laboratory, and when weighing a test tube it gives us 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 it gives us 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, first converting all decimal numbers to their generating fraction. Do not use decimal approximations during the process.

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

Three friends, Ashley, Ben, and Chloe, decide to buy a birthday present that costs $\$120 \text{ \$}$.

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

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

• Chloe pays the rest.

1. a) What fraction of the total does Chloe contribute?
2. b) How many dollars does each one pay?
3. c) If they had decided to pay equally, what fraction of the total would each 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 $\frac{15}{2}$ feet and its height measures $\frac{2}{3}$ of the base measurement.

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 feet ($\text{ft}^2$).
3. c) If a foot of fencing costs $\$8,50 \text{ \$}$, how much will it cost to fence the whole plot?

A bacterium reproduces by dividing into two every hour. Initially, in a culture there is a certain number of bacteria that occupies $\frac{1}{64}$ 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 of 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, the fraction is 1)?
3. c) If at the beginning there were 500 bacteria (corresponding to that $\frac{1}{64}$), how many bacteria will there be when the dish is at half its capacity?

Calculate the exact value of the following expression, previously converting all decimal numbers to their irreducible generating fraction 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 has suffered 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 that was stored at that time.

3. In the afternoon, new files were uploaded that took up $\frac{2}{3}$ of the capacity that had been left free after the deletion.

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

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

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

Look at the following geometric figure representing a square plot of land divided into different cultivation parcels.

Calculate what fraction of the total area of the square the shaded area (orange color) represents. Express the result as an irreducible fraction.

Hint: You can assume that the dividing lines connect vertices with midpoints or midpoints with each other recursively.

In a chemistry laboratory, precision measurements are being carried out. Two students, Alex and Mia, measure the length of a microscopic fiber.

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

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

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

Answer by 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 match rounding to ten-thousandths? Explain.

Three faucets ($A$, $B$, and $C$) pour water into a tank, and a drain ($D$) allows it to be emptied.

• Faucet $A$ fills the tank in 4 hours if it runs alone.

• Faucet $B$ fills the tank in 6 hours if it runs alone.

• Faucet $C$ fills the tank in 12 hours if it runs alone.

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

The tank is initially empty. The three faucets ($A$, $B$, $C$) and the drain ($D$) are opened simultaneously. However, after 2 hours of operation, faucet $A$ is closed due to a fault, leaving the other two and the drain running.

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

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 have followed to find them and show that they do indeed belong to this interval.

A jeweler 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 jeweler 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 repeating decimal number whose integer part is 0.

Knowing that:

• Its non-repeating part has 2 digits.

• Its repeating part has 1 digit.

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

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

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