Exercises: Problems with fractions

In a Year 9 class of 28 students, a survey was conducted about their leisure preferences and it was found that $\frac{3}{7}$ of the class prefers playing sports to watching series.

Calculate how many students prefer playing sports and how many prefer watching series, assuming that everyone chose one of the two options.

Your mobile phone has a long-lasting battery. After using it all morning to listen to music, you notice that the battery indicator shows you have $25\%$ charge remaining, which is equivalent to the fraction $\frac{1}{4}$.

If the phone indicates that the remaining $\frac{1}{4}$ corresponds to 3 hours of use:

1. a) How many hours of use does the battery have when fully charged ($100\%$)?
2. b) How many hours have you been listening to music?

Use the following visual diagram to help work out the answer.

Two friends, Alex and Bea, are competing to see who completes the weekly challenges of a language app first. In the figure, each full bar represents the total number of challenges.

• Alex has completed $\frac{2}{3}$ of the challenges.

• Bea has completed $\frac{5}{8}$ of the challenges.

Represent both fractions by reducing them to a common denominator and determine: who is currently winning the competition?

For the end-of-year trip, students are selling raffle tickets.

• Group A has sold $\frac{1}{3}$ of the total tickets.

• Group B has sold $\frac{2}{5}$ of the total.

• Group C has sold the rest.

1. a) Calculate what fraction of the total tickets Group A and Group B have sold combined.
2. b) What fraction of the tickets did Group C have to sell?

After a birthday party, there is cake left over. Looking at the image, we see that $\frac{1}{4}$ of the original cake remains on the plate. If the remaining piece weighs 300 grams:

1. a) How much did the whole cake weigh before the party started?
2. b) How many grams of cake have the guests eaten?

A streamer wants to do a 4-hour live stream. They have planned the time as follows:

• They will spend $\frac{1}{5}$ of the time chatting with the chat (Just Chatting).

• They will spend $\frac{3}{4}$ of the time playing their favourite game.

• The rest of the time will be spent watching funny videos.

Calculate how many minutes will be dedicated to each of the three activities.

A water tank used for watering a school vegetable garden has a total capacity of 1 200 litres.

During a very hot week, $\frac{5}{6}$ of the water in the tank was used.

1. a) How many litres of water have been used?
2. b) Represent the situation graphically and calculate how many litres remain in the tank.

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

Salma has a music playlist with 120 songs.

• $\frac{1}{2}$ of the songs are Pop.

• $\frac{1}{3}$ of the songs are Rock.

• The rest are Electronic Music.

1. a) How many songs are Pop and how many are Rock?
2. b) What fraction of the total do the Electronic Music songs represent? (Remember to simplify your answer if possible).

Marcos has received $60$ € as a birthday gift and plans his weekend expenses. On Saturday, he spends $\frac{1}{3}$ of the total money on buying a book. On Sunday, he decides to spend $\frac{1}{2}$ of what he had left on a concert ticket. To find out how much money he has left over, Marcos performs the following operation: «Since I spent one third and then one half, in total I spent $\frac{1}{3} + \frac{1}{2} = \frac{5}{6}$ of my money, so I have $\frac{1}{6}$ of $60$ € left, which is $10$ €».\na) Identify the conceptual error Marcos made by directly adding both fractions and explain why $\frac{1}{2}$ does not refer to the same reference amount as $\frac{1}{3}$.\nb) Correct the procedure by calculating, step by step, what fraction of the actual total he spent on Sunday, what fraction of the total he has left at the end, and exactly how much money he has in his pocket.

A hiker completes a $30\text{ km}$ stage of the «Senda del Oso» in Asturias. During the morning, she covers $\frac{3}{5}$ of the total route. In the afternoon, due to fatigue, she only travels $\frac{1}{4}$ of the distance she had left to cover, at which point she decides to stop and rest until the next day.\na) Before carrying out the calculations with exact kilometres, make a reasoned estimate: has the hiker covered more or less than three quarters ($75\,\%$) of the complete stage in total? Justify your prediction.\nb) Calculate exactly what fraction of the total stage she covered in the afternoon and how many kilometres she still needs to cover to reach the end.\nc) Check whether your initial estimate in section a) was correct by calculating the exact difference in kilometres with respect to $75\,\%$ of the stage.

Imagine you have a monthly mobile data plan. During the first week of the month, you have used $\frac{1}{3}$ of your total data streaming videos. During the second week, you have spent $\frac{2}{5}$ of what you had left browsing social media.

1. a) What fraction of your total data have you consumed in these two weeks?
2. b) If you still have 2 GB left for the rest of the month, how many GB was your data plan initially?

A group of students is organising an end-of-year trip. To fund it, they decide to sell raffle tickets. The first month they sell $\frac{3}{8}$ of the total tickets. The second month, they make an effort and sell $\frac{2}{3}$ of the tickets they had left to sell.

1. a) What fraction of the total tickets do they still have left to sell after the second month?
2. b) If in the end they have 150 unsold tickets left over, how many tickets did they have at the start for the raffle?

In a perfume factory, they have a large tank with 24 litres of lavender essence. They need to bottle all the contents into small bottles for sale.

1. a) If each bottle has a capacity of $\frac{3}{4}$ of a litre, how many bottles can they fill completely?
2. b) If they change format and use smaller bottles of $\frac{3}{8}$ of a litre, how many bottles will they need now? Explain the relationship between the result in part a) and part b).

A very elastic rubber ball is dropped from a certain height. Due to energy loss, each time it bounces it rises to only $\frac{3}{5}$ of the height from which it fell on the previous bounce.

After bouncing twice (falls, bounces, rises, falls again, bounces and rises), the ball reaches a height of 54 cm.

1. a) What fraction of the initial height does the height after the second bounce represent?
2. b) From what initial height was the ball dropped?

In the final of a basketball tournament, the free throw statistics of two players are compared. Player A scored 12 out of the 15 free throws he took. Player B took 20 free throws and scored 17.

1. a) Write down the fraction of successful shots for each player and simplify them if possible.
2. b) Express the fractions with a common denominator to determine which player had better accuracy (a higher proportion of successful shots).
3. c) If Player A had taken the same 20 shots as Player B while maintaining his success rate, how many baskets would he have scored?

A community allotment of $600 m^2$ is divided up to grow different vegetables. $\frac{1}{4}$ of the plot is dedicated to tomatoes and $\frac{2}{5}$ of the plot to peppers. The rest of the land is divided equally to plant lettuces and carrots.

1. a) What fraction of the total allotment is dedicated to lettuces?
2. b) How many square metres does the carrot crop occupy?

You are downloading a new video game that takes up a lot of hard drive space. When the progress bar reaches $\frac{2}{7}$ completed, your internet connection drops. Upon reconnecting, the download continues and progresses $\frac{3}{5}$ of what was left to download.

1. a) What fraction of the total game has been downloaded after the second stage?
2. b) If there are still 4 GB left to complete the download, how much space does the video game take up in total?

In a 3rd ESO class, a vote is held to choose the destination for the end-of-year trip. $\frac{1}{3}$ of the students vote to go to an amusement park. $\frac{2}{9}$ vote to go to the mountains. The rest of the class, which is 12 students, vote to go to the beach.

1. a) What fraction of the class voted to go to the beach?
2. b) How many students are there in total in the class?
3. c) How many students voted for the amusement park and how many for the mountains?

According to data from the hydrological bulletin of the Ministry for the Ecological Transition, a peninsular reservoir held a certain water reserve at the beginning of spring. Throughout the season, irrigation and urban consumption used $\frac{3}{8}$ of the stored water. Subsequently, during the summer months, $\frac{1}{5}$ of the water that still remained in the reservoir evaporated. Finally, to maintain the ecological flow of the river, $20\text{ hm}^3$ were released, leaving a final volume of $60\text{ hm}^3$ in the reservoir.\na) Model the situation by writing a single chained expression with fraction operations representing the fraction of the initial water that remained just before releasing the ecological flow.\nb) Determine in a reasoned manner how many cubic hectometres ($hm^3$) of water were stored in the reservoir at the beginning of spring.\nc) If the absolute maximum capacity of this reservoir is $250\text{ hm}^3$, what percentage of its total capacity was it at the beginning of spring? Express the result as an irreducible fraction and as a percentage.

To organize the school library, 3rd year ESO students are in charge of cataloguing a batch of newly received books. On Monday, they catalogue $\frac{2}{7}$ of the total books. On Tuesday, they catalogue $\frac{3}{5}$ of those that remained unrecorded. On Wednesday, they complete the work by cataloguing the last $40$ books. To find out how many books the batch had in total, two students propose different approaches:\n- Lucía proposes calculating day by day the remaining fraction with respect to the total by multiplying the remaining fractions: $1 - \frac{2}{7} = \frac{5}{7}$, and then calculating the fraction of the total represented by Wednesday's books.\n- Sara proposes setting up a linear equation by letting $x$ be the total number of books: $\frac{2}{7}x + \frac{3}{5}\left(x - \frac{2}{7}x\right) + 40 = x$.\na) Solve the problem using Lucía's arithmetic chained fractions method to find the total number of books.\nb) Solve the problem using Sara's algebraic equation method and verify that the same solution is reached.\nc) Compare both methods and justify which of the two seems more direct to you and carries less risk of making errors with denominators in this type of problem.

A water tank for irrigating an organic vegetable patch is filled using two taps. Tap A, which has a higher flow rate, can fill the empty tank in 4 hours. Tap B, which is older and slower, takes 6 hours to fill it completely. One day, the tank is empty and the farmer opens both taps at the same time, but after an hour and a half (1.5 h), Tap A breaks down and stops running water.

Calculate what fraction of the tank is left to be filled at that moment and how long Tap B, working alone, will take to finish filling it.

Three friends, Alex, Bea and Carlos, decide to buy a joint birthday present. Alex contributes $\frac{2}{5}$ of the total price of the present. Bea decides to pay $\frac{3}{4}$ of what was left to pay after Alex's contribution. Finally, Carlos pays the remaining $15$ € to complete the price.

Determine:

1. a) What fraction of the total did Carlos pay?
2. b) What was the total price of the present?
3. c) How much money did each of the other two friends contribute?

A competition basketball is dropped from a certain height. Due to energy loss, on each bounce the ball rises to $\frac{2}{3}$ of the height from which it fell on the previous bounce. After hitting the ground for the third time, the ball rises to a height of $56$ cm.

Calculate the initial height from which the ball was dropped. Express the result in metres.

We are preparing two paint mixtures for an urban art mural.

• Tin A contains a mixture of $1\frac{1}{2}$ litres, of which $\frac{1}{3}$ is blue paint and the rest is white paint.

• Tin B contains $\frac{3}{4}$ of a litre of a mixture where $\frac{2}{5}$ is blue paint and the rest is white.

If we pour the contents of both tins into a large bucket to create a single mixture:

1. a) What total volume of paint do we have?
2. b) What fraction of the total volume of the new mixture is blue paint?

A cyclist is competing in a mountain stage. He has covered $\frac{3}{7}$ of the total route when he stops for refreshments. He continues his ride and covers $\frac{2}{5}$ of what was left until he reaches a mountain pass. At that point, he checks his GPS and sees that he still has $18$ km left to reach the finish line.

How many kilometres is the stage in total and how many kilometres did he cover in the first section before the refreshment stop?

At a secondary school, an end-of-year trip is being organised. Initially, $\frac{1}{6}$ of the 3rd ESO students decide not to go. Of the remaining students who were interested, $\frac{2}{15}$ pull out because the dates do not suit them. If $130$ students end up going on the trip:

1. a) What fraction of the total number of 3rd ESO students finally went on the trip?
2. b) How many students are there in total in 3rd ESO?
3. c) How many students pulled out because of the dates?

A farmer has a rectangular plot of land. He devotes $\frac{2}{5}$ of the area to growing wheat. Of the remaining land, he uses $\frac{3}{4}$ to grow maize. The rest of the plot, which is $600\text{ m}^2$, is left fallow (uncultivated).

Given that the plot has a width of $40$ metres, calculate the length of the plot.

A famous video game streamer observes how the views for one of her old videos decrease over time.

In the first month, the video receives a certain number of views.

In the second month, the views decrease by $\frac{1}{4}$ compared to the previous month.

In the third month, the views drop by $\frac{2}{5}$ compared to the second month.

If in the third month the video had $10.800$ views.

How many views did the video get in the first month?

In a municipal sports centre, members' registrations across three mutually exclusive areas are analysed: swimming, weight room, or instructor-led classes.\nThe following conditions are known:\n1. A unit fraction $\frac{1}{n}$ (with $n$ an integer greater than $1$) of the members practises swimming. Furthermore, it is known that less than half of the members practise this sport and that $n$ is an even number.\n2. Of the remaining members, another unit fraction $\frac{1}{m}$ uses the weight room.\n3. The remaining $180$ members attend instructor-led classes, which represents exactly $\frac{3}{8}$ of the centre's total registered members.\na) Using the data from the instructor-led classes, calculate the total number of members in the sports centre.\nb) Set up the corresponding fraction equation and deduce the only integer values of $n$ and $m$ that make the problem consistent, proving why there can be no other solutions.\nc) Write the inverse «direct problem»: write the complete problem statement just as you would pose it to a classmate, using the values of $n$, $m$, and the total you have found, so that the unknown to be calculated is the final $180$ members.

Two Andalusian agricultural cooperatives, OleoSur and MonteOlivo, have obtained exactly the same olive harvest in kilograms at the start of the season. Each adopts a different commercial strategy to sell its production in two stages:\n- OleoSur sells $\frac{2}{5}$ of its total harvest in November. In January, it sells $\frac{1}{3}$ of what it had left after the first sale.\n- MonteOlivo sells $\frac{1}{3}$ of its total harvest in November. In January, it sells $\frac{2}{5}$ of what it had left after that first sale.\na) Before calculating kilograms, several members of MonteOlivo claim that they will have more olives left for self-consumption than OleoSur because in November they sold a smaller fraction of the total ($\frac{1}{3} < \frac{2}{5}$). Prove algebraically what fraction of the total harvest each cooperative retains at the end of January and indicate whether this claim is true or false.\nb) If MonteOlivo has exactly $24\,000\text{ kg}$ of unsold olives left over, calculate how many kilograms each cooperative harvested in total and how many kilograms each of them sold in November and in January.\nc) Based on the results obtained, explain in a reasoned manner the following apparent paradox: how is it possible for both cooperatives to have exactly the same final amount of olives left over if in neither of the two months did they sell the same number of kilograms?

Topic challenge. Is the weather forecast really reliable? Become a climate data analyst by checking the official AEMET website for your municipality.

1. Data collection: Note down the forecasted probability of precipitation for the next $4$ days. Convert each percentage into an irreducible fraction and also express it in decimal form, identifying the type of decimal obtained.

2. Approximation and errors: Record the forecasted maximum temperatures (with decimals if any, or take the trouble to measure the actual temperature with a thermometer at the same time). Round the values to the nearest unit and calculate the absolute error and relative error made in each approximation.

3. Probability calculation: Assume that the daily probability of a correct forecast is $p = \frac{4}{5} = 0,8$, and that each day's outcome is independent of the others. Using the properties of powers of rational numbers, calculate the probability of getting the forecast right for the $4$ consecutive days: $(\frac{4}{5})^4$. Compare its fractional and decimal result.

4. Final product: Gather your tables and calculations in a digital spreadsheet and write a brief conclusion arguing the reliability of the forecasts.

Assessment criteria:

• Accuracy in converting between fractions and decimals.

• Correct calculation and interpretation of absolute and relative errors.

• Rigorous application of powers of rational numbers to probabilistic calculation.

• Organization, clarity, and proper use of the spreadsheet.