Exercises: Fractions and decimals

Look at the following illustration representing the battery level of a cell 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) $\frac{2}{3} + \frac{1}{4} \times \frac{2}{5}$
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 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 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.

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

Javier has been checking the storage on his cell 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 vacation 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 visualize 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 $\text{RD\$}\ 7\,200$.

• 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 pesos 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 $\frac{15}{2}$ meters 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 meters ($\text{m}^2$).
3. c) If fencing costs $\text{RD\$}\ 510$ per meter, 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?

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:

$$ \frac{1.1\widehat{6} - \left( \frac{1}{2} \right)^{-2} \times 0.1}{2 - \frac{1}{1 - \frac{1}{3}}} + \sqrt{0,\widehat{4}} $$

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 color). Express the result as an irreducible fraction.

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

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

• The actual value of the length of the fiber is $2.345 \times 10^{-4}$ meters.

• Alex obtains a measurement of $2.3 \times 10^{-4}$ meters.

• Bea obtains a measurement of $2.4 \times 10^{-4}$ meters.

Answer, justifying your answers with calculations:

a) Calculate the absolute error made by each student. Express the result in scientific notation.

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.