Percentages

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.

To calculate $25\,\%$ of $\text{€}\ 180$, 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 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?

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.

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 $100\%$.

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?

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?

In a trendy shoe shop, a pair of trainers costing $€85$ 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.

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?

Cian has bought a keyboard for his computer for $€121$, 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, Cian concludes that the keyboard cost $121 - 25.41 = 95{,}€59$ without VAT.

1. a) Find the conceptual error Cian 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 $95{,}€59$ and you applied a $21\,\%$ increase to it. Do you obtain the initial $€121$?
3. c) Correct Cian'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.

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