Exercises: Problems with fractions

In an 8th grade class there are 28 students. A survey was carried out on their leisure preferences and it was discovered that $\frac{3}{7}$ of the class prefers playing video games to watching TV shows.

Calculate how many students prefer video games and how many prefer watching TV shows, assuming that everyone has chosen one of the two options.

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

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

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

Use the following visual diagram to help deduce the answer.

Two friends, Alex and Mia, are competing to see who completes the achievements of a new video game first.

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

• Mia has completed $\frac{5}{8}$ of the achievements.

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

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

• Group A has sold $\frac{1}{3}$ of the total raffle 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 raffle tickets Group A and Group B have sold together.
2. b) What fraction of the raffle tickets has Group C had to sell?

After a birthday party, there is some cake left over. Looking at the image, we see that $\frac{1}{4}$ of the original cake is left on the plate. If the leftover 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 live stream lasting 4 hours. He has planned the time as follows:

• He will dedicate $\frac{1}{5}$ of the time to chatting with viewers (Just Chatting).

• He will dedicate $\frac{3}{4}$ of the time to playing his favorite game.

• He will use the rest of the time to watch funny videos.

Calculate how many minutes he will dedicate to each of the three activities.

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

During a very hot week, $\frac{5}{6}$ of the water in the tank has been consumed.

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

Maria 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 the result if possible).

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

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

A group of students is organising an end-of-year trip. To finance 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 beginning for the raffle?

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

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

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

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

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 of the 15 free throws they took. Player B took 20 free throws and scored 17.

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

A community urban garden of $600 m^2$ is divided to grow different vegetables. $\frac{1}{4}$ of the land is dedicated to tomatoes and $\frac{2}{5}$ of the land to peppers. The rest of the land is divided into equal parts to plant lettuces and carrots.

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

You are downloading a new video game that takes up a lot of space on the hard drive. When the download bar is $\frac{2}{7}$ complete, your internet connection drops. Upon reconnecting, the download continues and advances $\frac{3}{5}$ 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 for the full download to finish, how much space does the video game take up in total?

In an 8th-grade 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 a theme 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 theme park and how many for the mountains?

A water tank for watering an organic garden is filled by 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 pouring water.

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

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

Determine:

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

A competition basketball is dropped from a certain height. Due to the loss of energy, 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 $56$ in.

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

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

• Can A contains a mixture of $1\frac{1}{2}$ gallons, where $\frac{1}{3}$ is blue paint and the rest is white paint.

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

If we pour the contents of both cans 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 corresponds to the blue paint?

A cyclist is doing a mountain stage. They have covered $\frac{3}{7}$ of the total route when they stop for refreshments. They continue their ride and cover $\frac{2}{5}$ of what was left until they reach a mountain pass. At that point, they check their GPS and see that they still have $18$ miles left to reach the finish line.

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

An end-of-year trip is being organized at a middle school. Initially, $\frac{1}{6}$ of the 8th-grade students decide not to go. Of the rest of the students who were interested, $\frac{2}{15}$ finally drop out because the dates do not suit them. If in the end $130$ students go on the trip:

1. a) What fraction of the total 8th-grade students finally went on the trip?
2. b) How many students are there in total in 8th grade?
3. c) How many students dropped out because of the dates?

A farmer has a rectangular plot of land. They dedicate $\frac{2}{5}$ of the surface area to growing wheat. Of the remaining land, they use $\frac{3}{4}$ to grow corn. The rest of the plot, which is $600\text{ ft}^2$, is left fallow (uncultivated).

Knowing that the plot has a width of $40$ feet, calculate the length of the plot.

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

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

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

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

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

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