Understanding Fractions

Observe the following figure, which represents a bar of chocolate. What fraction of the bar is left if we have eaten the squares that are shaded in orange? Indicate which are the numerator and the denominator.

On a basketball team there are $12$ players. If $5$ of them are taller than $1,80$ feet, write the fraction that represents the players who are less than or equal to that height.

Observe the following figure and determine what fraction of the total area of the rectangle the orange shaded area represents. Keep in mind that to solve it you will have to imagine a subdivision of the figure into equal parts.

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

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

Represent the following fractions graphically in your notebook using rectangles or circles:

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

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

Match each improper fraction with its corresponding mixed number:

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

Write a fraction that simultaneously meets the following three conditions:

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

2. It is an improper fraction.

3. The 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.

Identify if 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 the case that is not equivalent, indicate which of the two fractions is greater.

Find a fraction equivalent to $\frac{4}{9}$ such that the sum of its numerator and its denominator is exactly $104$. Once found, determine if this fraction is proper or improper and justify whether its decimal value is greater or less than $0,5$.

Calculate the numerical value of these fractions (as an integer or 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 the simplest form you know).

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 do the minutes elapsed between 12:15 and 12:50 represent? Express the result as an irreducible fraction.
2. b) If the minute hand has covered $\frac{5}{6}$ of an hour, how many degrees has it turned? (Remember that a full turn is $360^\circ$).
3. c) If a school 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 if you have a full hour to study?

Represent the following rational numbers on the same number line. To do this, draw the line in your notebook and use a suitable scale (for example, let each unit measure $4$ in.):

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

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

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

1. d) $D = -1.25$

Observe the following number line where several points have been represented. 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, indicate what irreducible fraction the points marked with an orange circle represent. What mixed number would represent a point located $5$ divisions to the left of point $A$?

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

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

On a digital gaming platform, Chloe 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?

Three YouTubers are uploading a large video at the same time. After one hour, their progress bars indicate the following:

• Channel A: Has uploaded $\frac{5}{8}$ of the total.

• Channel B: Has uploaded $\frac{11}{16}$ of the total.

• Channel C: Has uploaded $\frac{3}{4}$ of the total.

Order the three YouTubers from least to greatest amount of video uploaded using the symbol $<$ and by first reducing the fractions to a common denominator.

In a robotics competition, the prototypes must complete a circuit of $60$ feet. Robot A has covered $\frac{7}{10}$ of the circuit, robot B has covered $45$ feet and robot C has covered $\frac{3}{4}$ of the distance that 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 circuit.

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 $12$ that is exactly at the midpoint between them.
3. c) Is this fraction greater or less than $\frac{5}{7}$? Justify your answer by comparing the cross products.

A young streamer has decided to organize his weekly broadcasting schedule as follows: he dedicates $\frac{2}{5}$ of the total time to strategy video games. Of the remaining time, he uses $\frac{1}{3}$ to chat with his followers (just chatting). If he still has $8$ hours a week left to prepare the scripts, what is the total time, in hours, that he dedicates to the channel per week? Represent using a bar model what fraction of the total the final $8$ hours represent.

An inheritance of $\$48.000$ is shared among three siblings. The eldest receives $\frac{3}{8}$ of the total. The middle sibling receives $\frac{5}{12}$ of the total. What fraction of the total corresponds to the youngest sibling? How much money does each one receive? Express the three amounts as fractions with a common denominator of $24$ to visually compare who received more.

A farmer has a square plot of $10.000$ $m^2$. He decides to dedicate $\frac{1}{4}$ 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 needs $20$ $m^2$ of space, how many trees will he be able to plant in one of those five parts?