Exercises: Percentages

In an e-sports tournament, $20\%$ of the $150$ participants play on a tablet, while the rest play on a computer. Calculate the exact number of players using a tablet to compete.

The loading bar of your favourite video game shows that $75\%$ of the total download has been completed. If the full file takes up $40$ GB, how many gigabytes have been downloaded so far?

Work out the following percentages mentally using the division strategies learned in class (divide by $2$ for $50\%$, by $4$ for $25\%$ and by $10$ for $10\%$):

1. a) $50\%$ of $84$ viewers on a livestream.
2. b) $25\%$ of a $400$ gram pizza.
3. c) $10\%$ of a travel pass costing $35$ euros.

Look at the following grid representing the plan of a leisure area in a shopping centre. Each small square represents $1\%$ of the total. If the orange shaded area is reserved for arcade machines, what percentage of the total does this area occupy? Express the result also as an irreducible fraction.

A pair of trainers that cost $60$ euros has a $15\%$ discount today. How many euros do you save by buying them today? What is the final price you will pay for them?

Your mobile phone indicates that it has $12\%$ battery left. If the total capacity of the battery is $4500$ mAh (milliampere-hours), how many mAh of power are left in the device exactly?

In a survey carried out in your Year 7 class, $18$ out of the $30$ students confirmed that they prefer horror films to comedy films. Calculate the percentage of students who prefer horror films.

Calculate, showing your reasoning at each step:

1. a) $200\%$ of $45$. (Think: $200\%$ of an amount is double its value.)
2. b) $150\%$ of $60$.
3. c) $0,5\%$ of $800$.
4. d) $300\%$ of $12$.

In a sports shop in your neighbourhood, a school backpack costs $49{,}90\text{ €}$ and has a $15\,\%$ discount.

1. a) Before doing any calculations on paper, make a mental estimate of the money you will be discounted: round the price to the nearest ten and mentally find $10\,\%$ and half of that value for $5\,\%$. Between which two whole numbers of euros will the discount lie?
2. b) Calculate the exact final price of the backpack by applying the corresponding discount multiplicative factor: $\left(1 - \dfrac{15}{100}\right)$.
3. c) Compare your exact result with the estimate you made in section a). Is the final result reasonable? Justify why estimating the result beforehand helps us detect possible mistakes at the checkout.

To calculate $25\,\%$ of $180\text{ €}$, three classmates propose different ways:

• Alba says: "I divide $180$ by $4$, because $25\,\%$ is equivalent to a quarter of the total."

• Bruno says: "I multiply $180$ by $0{,}25$, which is the decimal expression of the ratio $\dfrac{25}{100}$."

• Carla says: "I first calculate $10\,\%$ of $180$, double it to get $20\,\%$, and add half of $10\,\%$ to complete the $25\,\%$."

1. a) Check that all three methods give exactly the same result by performing the operations proposed by each one.
2. b) If you had to do the calculation in your head, without using pencil or paper, which of the three methods would you choose? Explain why it seems the most agile to you.
3. c) If the discount were not $25\,\%$, but $17\,\%$, and you could use a calculator, which of the three approaches would be the most direct and efficient? Justify your answer.

The battery of Lucía's tablet has a total capacity of $5000$ mAh. If after playing "Roblox" for a while she has $18\%$ battery left, how many mAh of power does she still have? What percentage of the battery has she already used?

In the vote to choose the activities for the school festival, $1200$ students took part. $65\%$ chose the sports tournament and the rest preferred the robotics workshop. How many students voted for the robotics workshop?

To organise the end-of-year trip, Year 7 students have sold raffle tickets. If they have already sold $360$ tickets, which represents $75\%$ of the total they had, how many tickets did they have in total to sell? How many do they still have left to sell?

An adventure video game takes up $12$ GB in a console's memory. If the system indicates that this game takes up $15\%$ of the total available memory, how many GB of capacity does the console's memory have in total?

In a multiple-choice mathematics exam with $40$ questions, a student answered $28$ questions correctly, got $8$ wrong and left $4$ blank. Calculate the percentage of correct answers, wrong answers and unanswered questions relative to the total. Check that the sum of the three percentages is $100\%$.

The price of a laptop in a shop is $450$ € excluding taxes. If $21\%$ VAT (Value Added Tax) must be applied, what will be the final price we will have to pay for the laptop?

Marta has posted a video on TikTok. In the first two hours, she has received $180$ likes. If she knows that $12\%$ of the people who watch the video like it, how many views has the video had in total during those two hours?

A neighbourhood had $1.200$ residents ten years ago and today has $3.000$.

1. a) By how many residents has the population increased?
2. b) What percentage increase does this represent compared to the number of residents ten years ago? Is it a percentage greater than or less than $100\%$? Interpret the result.

Percentages can also be smaller than $1\%$:

1. a) A saline solution contains $0,9\%$ salt. How many grams of salt are in a $500$ g bag of saline solution?
2. b) A batch of $2.000$ kg of flour contains $0,05\%$ impurities. How many kilograms of impurities does it contain?

Marcos has bought a keyboard for his computer for $121\text{ €}$, a price that already includes $21\,\%$ VAT. To find out how much money corresponds to VAT and how much to the original price of the product (taxable base), he does the following operation:

Therefore, Marcos concludes that the keyboard cost $121 - 25{,}41 = 95{,}59\text{ €}$ without VAT.

1. a) Find the conceptual error Marcos has made. To which amount is the $21\,\%$ VAT applied: to the original price or to the final invoiced price?
2. b) Check what would happen if the price without VAT were really $95{,}59\text{ €}$ and you applied a $21\,\%$ increase to it. Do you obtain the initial $121\text{ €}$?
3. c) Correct Marcos's calculation by using division by the corresponding multiplicative factor to determine the real price of the keyboard before tax and the exact amount of VAT.

According to the peninsular electricity balance of Red Eléctrica de España, on a winter day the total energy demand was $650\text{ GWh}$ (gigawatt-hours). Of that demand, $60\,\%$ was covered by renewable energy. Furthermore, within the total renewable generation of that day, $65\,\%$ corresponded to wind production thanks to a windstorm.

1. a) Model the situation using chained operations with percentages and calculate how many $\text{GWh}$ of renewable energy were generated in total during that day.
2. b) Determine how many $\text{GWh}$ were specifically contributed by wind turbines.
3. c) What percentage did wind energy represent of the country's total energy demand that day? Deduce this final percentage by operating directly with the percentage rates, without needing to resort to the values in $\text{GWh}$.

In a trendy shoe shop, a pair of trainers costing $85 \text{ €}$ has a discount of $20\%$. However, as it is the last day of the sale, an additional discount of $10\%$ is applied to the already reduced price. What is the final price of the trainers? Is it the same as applying a single discount of $30\%$? Justify your answer by calculating both cases.

Álex bought a computer to play his favourite video games. The final price he paid, including $21\%$ VAT, was $1452 \text{ €}$. His friend tells him that the computer cost $1200 \text{ €}$ before taxes. Is his friend right? Perform the necessary calculations to find the base price (excluding VAT) of the computer.

In a survey carried out at your school about social media use, $45\%$ of the students prefer Instagram, $35\%$ prefer TikTok and the remaining 48 students choose other platforms. How many students are there in total at the school? How many of them prefer Instagram?

A movie streaming service costs $9,99 \text{ €}$ per month. If you decide to pay for the whole year at once, they offer a $15\%$ discount. However, this month there is a special promotion, and on top of that already reduced annual price, they offer a further fixed discount of $10 \text{ €}$. How much will you save in total per year if you take advantage of both offers compared to paying month by month?

The storage of a mobile phone is $128 \text{ GB}$. The operating system takes up $12,5\%$ of the total. If Álex has already used up $60\%$ of the remaining free space, how many gigabytes does he still have available to install new apps?

In a theatre with a capacity of $750$ people, $12\%$ of the tickets were left unsold today. Of the tickets sold, $65\%$ were for adults and the rest for minors under 18 years old. How many minors under 18 attended the theatre today?

A soft drink brand has decided to reduce the sugar content of its main drink. Previously, a $330 \text{ ml}$ can contained $35 \text{ g}$ of sugar. Now they have launched a version with a $40\%$ reduction in sugar and, in addition, they have increased the size of the can by $10\%$.

1. How many grams of sugar does the new can have?

2. What percentage of sugar does the weight of the sugar represent relative to the total volume in the new can? (Assume $1 \text{ ml} = 1 \text{ g}$ for the soft drink).

In the school council elections, candidate A received $30\%$ of the votes, candidate B received $25\%$, and candidate C received the remaining votes. If candidate C won with 28 votes more than candidate B, how many students voted in total? How many votes did each candidate receive?

The number of members in a youth book club has increased this year by $150\%$ compared to last year, when there were $24$ members.

1. a) A student says: "an increase of $150\%$ means that there are now $150$ members". Explain their mistake.
2. b) Calculate how many members there are now. Remember: an increase of $150\%$ means adding $150\%$ of the initial number to it.
3. c) If next year the club wants to triple last year's member count, what percentage increase does it need compared to last year?

In a youth bookstore in Valencia, a comic book had a certain initial price. On the occasion of Book Day, a $15\,\%$ discount was applied to it. Two weeks later, due to the print run selling out, the shop increased the already discounted price by $15\,\%$, leaving it set at a final price of $19{,}55\text{ €}$.

1. a) A customer claims that the comic book costs the same as at the beginning again because "dropping by $15\,\%$ and then increasing by $15\,\%$ cancels out the change". Explain in detail why this reasoning is incorrect using the concepts of reference base and factors of change.
2. b) Write a mathematical expression that relates the unknown original price ($P$) to the final price through the corresponding multiplicative factors, and calculate the price the comic book had before any discount.
3. c) Calculate the overall real percentage variation that the price of the comic book underwent between the initial value and the final price of $19{,}55\text{ €}$.

A hypermarket presents two different offers for the same brand of olive oil, whose original price is $8\text{ €}$ per bottle:

• Promotion A: "Buy 3, pay for 2: take 3 bottles and pay for only 2".

• Promotion B: "Second item at $70\,\%$ off" (for every two bottles purchased, the second has a $70\,\%$ discount).

1. a) If a family plans to buy a pack of 6 bottles:

• Model and calculate the total cost and the overall discount percentage obtained with Promotion A on the original amount of the 6 bottles.

• Model and calculate the total cost and the overall discount percentage obtained with Promotion B on the original amount of the 6 bottles.

1. b) Justify in a reasoned manner which of the two promotions is more economical to purchase the 6 bottles.
2. c) If a consumer goes to the shop intending to purchase only 2 bottles and the establishment does not allow mixing promotions, what is the real discount percentage per bottle that would apply with each of the two offers? Justify which one is best for them in that case.

Unit challenge. Imagine you are organizing a healthy meal for your school group. You are going to design a budget proposal using a spreadsheet or GeoGebra with real data.

1. Data collection and ratio: Visit a supermarket website and write down the price of $3$ ingredients (for example, a $1,5$ kg bag of rice, vegetables, and fruit). Calculate the cost-per-kilogram ratio and the unit cost per serving.

2. Direct proportionality and percentages: Model the cost of the meal in the spreadsheet according to the number of diners ($4$, $8$, and $12$ people). Apply a $15\,\%$ discount coupon to the basket and then add the $10\,\%$ reduced VAT.

3. Inverse proportionality: Renting the community kitchen has a fixed cost of $60$ €. In another table, calculate how much each participant must pay depending on whether $2$, $4$, $6$, or $12$ people attend. Plot the graph of this relationship.

4. Final proposal: Write a brief report comparing the final cost per person according to group size and explain how sharing fixed costs influences it.

Assessment criteria:

• Correct identification and accurate calculation of ratios and percentages.

• Proper mathematical distinction and modeling of direct and inverse proportionality.

• Rigorous use of digital tools (table and graph) and real data.

• Clarity and reasoned justification in the report's conclusions.