The concept of a fraction

Look at the following figure, which represents a bar of chocolate. What fraction of the bar remains if we have eaten the squares shaded in orange? State the numerator and the denominator.

Represent the following fractions visually in your exercise book using rectangles or circles:

1. a) $\dfrac{2}{5}$

1. b) $\dfrac{5}{4}$

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

Look at the following figure and determine what fraction of the total area of the rectangle is represented by the area shaded in orange. Note that to solve it you will need to imagine subdividing the figure into equal parts.

In a basketball team there are $12$ players. If $5$ of them are over $1.80$ meters tall, write the fraction representing the players who are less than or equal to that height.

Calculate the numerical value of these fractions (as an integer or a decimal):

1. a) $\dfrac{15}{3}$

1. b) $\dfrac{1}{4}$

1. c) $\dfrac{18}{2}$

1. d) $\dfrac{3}{10}$

What fraction of an hour do the following time intervals represent? (Express the fraction in its simplest form).

1. a) $30$ minutes.
2. b) $15$ minutes.
3. c) $20$ minutes.
4. d) $10$ minutes.

Imagine a wall clock.

1. a) What fraction of a full turn is represented by the minutes elapsed between 12:15 and 12:50? Express the result as an irreducible fraction.
2. b) If the minute hand has moved through $\frac{5}{6}$ of an hour, how many degrees has it turned? (Remember that one full turn is $360^\circ$).
3. c) If one homework task takes you $\frac{3}{10}$ of an hour and another takes you $\frac{1}{4}$ of an hour, what fraction of an hour do you have left over if you have a full hour available to study?

Write the following fractions using figures and classify them as proper or improper:

1. a) Three eighths.
2. b) Seven halves.
3. c) Eleven elevenths.
4. d) Four ninths.

Match each improper fraction with its corresponding mixed number:

A group of friends bought several bottles of soda for a party. Each bottle has a capacity of $\frac{1}{4}$ of a liter. If they consumed a total of $19$ bottles, express this quantity as an improper fraction and as a mixed number. How many full liters have they drunk?

Three students are downloading the same video game on their computers at the same time. After one hour, their progress bars show the following:

• Computer A: Has downloaded $\frac{5}{8}$ of the total.

• Computer B: Has downloaded $\frac{11}{16}$ of the total.

• Computer C: Has downloaded $\frac{3}{4}$ of the total.

Order the three computers from smallest to largest amount of game downloaded using the symbol $<$, after first writing the fractions with a common denominator.

Identify whether the following pairs of fractions are equivalent using the cross-products method:

1. a) $\frac{12}{15}$ and $\frac{4}{5}$

1. b) $\frac{7}{3}$ and $\frac{21}{6}$

1. c) $\frac{10}{25}$ and $\frac{6}{15}$

For any pair that is not equivalent, state which of the two fractions is greater.

Find an equivalent fraction to $\frac{4}{9}$ such that the sum of its numerator and denominator is exactly $104$. Once found, determine whether that fraction is proper or improper and justify whether its decimal value is greater or less than $0.5$.

Represent the following rational numbers on the same number line. To do so, draw the line in your notebook and use a suitable scale (for example, so that each unit measures $4$ cm, so that each quarter of a unit measures $1$ cm):

1. a) $A = \frac{3}{4}$

1. b) $B = -\frac{1}{2}$

1. c) $C = \frac{9}{4}$

1. d) $D = -1.25$

Look at the following number line where several points are shown. Point $A$ corresponds to the number $-1$ and point $B$ corresponds to the number $0$. If the line segment between $A$ and $B$ is divided into $12$ equal parts, state which irreducible fraction is represented by the points marked with an orange circle. What mixed number would represent a point located $5$ divisions to the left of point $A$?

An NGO distributes the $960\,000$ pesos raised in a charity run among three projects. The first gets $\frac{3}{8}$ of the total. The second gets $\frac{5}{12}$ of the total. What fraction of the total goes to the third project? How much money does each one receive? Express the three quantities as fractions with a common denominator of $24$ to visually compare which project received the most.

Consider the fractions $\frac{2}{3}$ and $\frac{3}{4}$.

1. a) Calculate their decimal values.
2. b) Find a fraction with a denominator of $24$ that lies exactly halfway between them.
3. c) Is that fraction greater or smaller than $\frac{5}{7}$? Justify your answer by comparing the cross products.

On a digital gaming platform, Lucía has saved $180$ tokens. She decides to spend $\frac{4}{9}$ of her tokens on a new skin for her character and $\frac{2}{5}$ of what she has left on upgrading her protective gear. How many tokens has she spent in total and how many does she have left at the end?

To prepare a cookie recipe for $6$ people, $\frac{3}{4}$ kg of flour is needed. If we want to adapt the recipe to invite $15$ people:

1. a) What fraction of a kilogram of flour will we need?
2. b) Is that fraction greater or less than one? Justify whether it is a proper or improper fraction.

Write a fraction that satisfies all three of the following conditions simultaneously:

1. Its denominator is the prime number closest to $10$.

2. It is an improper fraction.

3. Its numerator is a multiple of $4$ between $13$ and $22$.

Is there more than one solution? Find them all and express them as a mixed number.

Marina has decided to organize her weekly study time as follows: she spends $\frac{2}{5}$ of her total time on math. Of the remaining time, she spends $\frac{1}{3}$ on English. If she still has $8$ hours left per week for the rest of her subjects, what is the total time, in hours, that she spends studying per week? Represent using a bar diagram what fraction of the total the final $8$ hours represent.

In a robotics competition, the prototypes must cover a track of $60$ meters. Robot A has covered $\frac{7}{10}$ of the track, Robot B has covered $45$ meters and Robot C has covered $\frac{3}{4}$ of the distance Robot A had left to finish. Order the robots from greatest to least distance covered and express the position of Robot C as a fraction of the total track.

A farmer has a square plot of $10,000$ $m^2$. He decides to devote $\frac{1}{4}$ of the area to growing vegetables. From the rest of the plot, he divides the land into $5$ equal parts to plant different fruit trees. What fraction of the total land does each of the parts dedicated to fruit trees represent? If each fruit tree requires $20$ $m^2$ of space, how many trees will he be able to plant in one of those five parts?