Exercises: Rational numbers

Look at the following graphical representation of a fraction. Your task is to:

1. Write the irreducible fraction that represents the part colored in orange with respect to the whole 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-foot obstacle race. At a given moment, their positions are as follows:

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

• Ben has covered $\frac{3}{4}$ of the total.

• Chloe has covered $\frac{7}{10}$ of the total.

• Calculate exactly how many feet each one has covered.

• Order the fractions from smallest to largest using the $<$ symbol.

• Approximately locate the position of Ashley (A), Ben (B), and Chloe (C) on the number line shown below.

Classify the following decimal numbers as terminating, pure repeating, or mixed repeating. Then, express the terminating ones as an irreducible fraction.

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

Hint: Remember that to convert a terminating decimal to a fraction, you write the number without the decimal point in the numerator and a 1 followed by as many zeros as there are decimal places in the denominator.

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 rest is desert.

• How many square pixels does the forest occupy?

• How many square pixels does the water occupy?

• Calculate what irreducible fraction of the map the desert represents.

Perform the following operations with fractions. Remember that to add or subtract you must first find the least 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 of your cell phone is distributed as follows:

• Photos take up $\frac{2}{5}$ of the space.

• Apps take up $\frac{1}{3}$ of the space.

• What fraction of the total do the photos and apps take up 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 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 rules of signs and that an integer has a denominator of 1.

Emma has bought a large rectangular pizza for dinner. When she opens the box, she sees that the pizza is cut into 24 equal slices.

• Emma eats 4 slices.

• Her brother Luke eats 6 slices.

• Her parents eat 8 slices between them.

• Write the fraction of pizza that each person has eaten (Emma, Luke, and parents). Simplify them fully.

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

Calculate the value of the following combined operation, respecting the order of operations (parentheses, powers, multiplications/divisions, additions/subtractions). You must simplify the final result until you 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 cell phone has an unknown total storage capacity. The operating system takes up $\frac{1}{6}$ of the total memory. Of what is left free, your apps and games take up $\frac{3}{5}$. 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 graphical diagram to help you visualize the problem:

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

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

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

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

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

Calculate the absolute error and the relative error (as a percentage) of 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 this, use Thales's theorem or the division of line segments to locate the points exactly, not approximately.

Use the following diagram as a reference for your construction in your notebook:

Three friends buy a birthday present that costs $\$60$.

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

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

• Chloe pays the rest.

1. a) What fraction of the total does Chloe pay?
2. b) How much money in dollars does each one contribute?

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

1. a) Write the fraction and the height in feet it reaches after the first bounce.
2. b) Write the fraction and the height in feet 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 gallons.

We want to bottle all the oil in two types of containers:

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

• The rest of the oil is bottled in $1,5$-gallon jugs.

How many 1.5-gallon jugs will we need to bottle the remaining oil?

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

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.

$$ \frac{2^{-2} - \left( 1,1\widehat{6} - \frac{7}{6} \right)^{-1} \cdot \sqrt{\frac{16}{81}}}{0,\widehat{3} \cdot \left( 1 - \frac{3}{2} \right)^2 + 2,5} $$

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

A group of 8th grade students is organizing the funding for their end-of-year trip. They have designed a fundraising plan in stages:

1. In the first stage (lottery 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 managed to raise $\frac{2}{3}$ of the new remainder they had left.

If after these three stages they are still $\$150$ short of covering 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 visualize the remainder fractions:

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 run simultaneously. However, after 1.5 hours of working together, printer B breaks down and stops.

1. a) What fraction of the total work have they completed together before the breakdown?
2. b) What fraction of the work is left pending?
3. c) How much extra time will printer A take to finish the remaining work 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 (colored black).

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

Look at the figure generated after the second step:

1. a) Write 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 of the original square measures $1 \text{ m}$, how many $\text{cm}^2$ of "white" material (not removed) are left after the second step?

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

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

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

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

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

The laboratory thermostat has a slightly strange scale marked in fractions of a degree.

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

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

1. a) Represent on the number line the set of all possible values of temperature $T$ (interval).
2. b) Write 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.

We have three juice tanks with different capacities:

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

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

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

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 very elastic rubber ball is dropped from a height of 243 inches.

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 previously fell.

1. a) Write the sequence of heights (in the form of powers and fractions) that the ball reaches 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 traveled (adding descents and ascents) just as it touches the ground for the fourth time?