Exercises: Concept

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.

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.

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) $\dfrac{3}{10}$

Match each improper fraction with its corresponding mixed number:

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.

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.

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?

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.

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?

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.

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.

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.

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.

A mountain stage of the Rás Tailteann cycling race has a total distance of $176\text{ km}$. The television broadcast reports that the leading cyclists have already completed $\frac{5}{8}$ of the planned route.\na) Without performing the exact division yet, estimate whether or not they have passed the halfway point of the stage ($88\text{ km}$) by comparing the fraction $\frac{5}{8}$ with $\frac{1}{2}$. Justify your conclusion.\nb) Reasonably estimate between which two consecutive tens of kilometers ($100$ and $110\text{ km}$, $110$ and $120\text{ km}$, etc.) the distance covered must lie.\nc) Calculate the exact value of the kilometers covered by applying the fraction as an operator on the total distance.\nd) Check whether your previous estimate was accurate and calculate how many kilometers they still need to reach the finish line, expressing the remaining distance as an irreducible fraction of the total.

To organize a school field trip, the teaching staff must calculate $\frac{2}{5}$ of the $135$ 1st Year students enrolled in the school, who will travel in the first bus shift.\nTwo students from the class propose different strategies to calculate that figure:\n- Aoife says: «First I divide $135$ by $5$ to calculate how much one-fifth of the total is, and then I multiply that result by $2$».\n- Darragh says: «First I multiply $135$ by $2$, and then I divide the resulting number by $5$».\na) Work through both methods step by step and check whether both lead to the same result.\nb) In Aoife's method, what specific meaning does the intermediate result obtained after the first operation ($135 : 5$) have?\nc) Now suppose that you need to calculate $\frac{2}{5}$ of a remaining budget of $19\text{ €}$. If you apply Aoife's method, you get a decimal number in the first step ($19 : 5 = 3.8$), while with Darragh's method the first step yields an integer ($19 \times 2 = 38$). Which of the two methods would you choose in this second case to perform the calculation mentally more comfortably, and why?

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

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 $0.5$.

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 $48,000$ € 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.

In a robotics competition, the prototypes must cover a track of $60$ metres. Robot A has covered $\frac{7}{10}$ of the track, Robot B has covered $45$ metres 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?

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.

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?

According to official hydrological data, a reservoir has a maximum storage capacity of $240\text{ hm}^3$. After a drought period, the reservoir stores only $\frac{3}{10}$ of its total capacity. Over a week with heavy rainstorms, rivers contribute a volume of water to the reservoir equivalent to $\frac{1}{4}$ of its maximum capacity. During that same week, to meet the needs of urban consumption and irrigation, $18\text{ hm}^3$ are released.\na) Model the situation by writing a single arithmetic expression combining the fractions and the provided data to determine the final volume of stored water.\nb) Calculate how many $\text{hm}^3$ of water the reservoir contains at the end of that week.\nc) Express the final water volume as an irreducible proper fraction of the total capacity of the reservoir.\nd) The basin's drought protocol stipulates that an area enters a 'normal status' only if the reservoir exceeds half of its total capacity. Justify mathematically whether the reservoir has already reached the normal status or remains below it.

In a food packaging facility, jars of cooked legumes are going to be marketed from a large tank. Two packaging formats are being considered:\n- The standard format has a capacity of $\frac{3}{4}\text{ kg}$.\n- The family format has a capacity of $1 + \frac{1}{8}\text{ kg}$.\nThe production manager notices that if the family format is used, exactly $6$ fewer jars are needed than if the standard format is used to pack the entire contents of the tank.\na) Express the capacity of the family format as an improper fraction and convert the capacity of the standard format into an equivalent fraction with denominator $8$. How many eighths of a kilogram does each format hold?\nb) From the fractions obtained in eighths, reason how many standard jars are needed to pack exactly the same amount of legumes contained in $2$ family jars.\nc) Deduce how many jars of each type would be needed for the difference between both formats to be $6$ jars in total. Explain your deduction clearly.\nd) How many kilograms of legumes were in the tank? Express the amount as a mixed number and as an exact decimal number.