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.

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

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.

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.

In the school canteen, $3$ identical pies are going to be shared equally among $4$ diners. When organizing the sharing, Lucas states: «Each of us gets $\frac{4}{3}$ of a pie, because there are $4$ of us and we have $3$ pies».\na) Identify the conceptual error that Lucas makes when assigning the numbers to the numerator and denominator, and explain what each of these terms represents in the fraction.\nb) Classify the fraction proposed by Lucas ($\frac{4}{3}$) as proper or improper. Based on that classification, justify why his claim is impossible in practice if they only have $3$ pies.\nc) Write the correct fraction that corresponds to each diner and calculate its exact decimal value to check how much pie each one eats.

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 soft drink for a party. Each bottle has a capacity of $\frac{1}{4}$ of a litre. If they consumed a total of $19$ bottles, express this quantity as an improper fraction and as a mixed number. How many full litres have they drunk?

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.

On a digital gaming platform, Aoife 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.

In a municipal athletics sports school, it is known that $\frac{3}{5}$ of the enrolled students are girls. The administration registry confirms that there are exactly $27$ girls enrolled.\na) From these data, deduce how many people in total are enrolled in the sports school. Explain in detail the reasoning you follow: what quantity represents $\frac{1}{5}$ of the total and how the full total ($\frac{5}{5}$) is obtained.\nb) What fraction of the total do the enrolled boys represent, and how many boys are there?\nc) If someone claimed that in this same school there are $20$ girls enrolled while keeping the fraction of $\frac{3}{5}$ exactly, reason why this situation would be numerically impossible in reality.

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.

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) $D = -1.25$

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.

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.