Exercises: Percentages

Calculate mentally:

a) $25\%$ of $240$.

b) $10\%$ of $380$.

c) $50\%$ of $96$.

d) $75\%$ of $800$.

Look at the grid in the figure, made up of $100$ squares.

a) What percentage of the squares are shaded?

b) What percentage are left unshaded?

Of the $30$ students in a class, $18$ are girls. What percentage of the class are girls? And boys?

A handball team has won $12$ of the $20$ matches it has played. What is its win percentage so far?

A pair of trainers costing $\text{€}\ 60$ is reduced by $15\%$. How many euros do you save? How much do you pay?

A pair of headphones costs $\text{€}\ 50$ excluding VAT. If VAT is $21\%$, what is the final price?

Express each percentage as a decimal and as an irreducible fraction:

a) $35\%$

b) $4\%$

c) $120\%$

Vera has a monthly data allowance of $12$ GB and has already used $30\%$. How many gigabytes does she have left?

In a village of $2\,500$ inhabitants, $8\%$ are under $15$ years old. How many people under $15$ years old live in the village?

A travelcard costs $\text{€}\ 20$ and is going to go up by $5\%$. What will the new price be?

What percentage does $45$ represent out of $180$?

Calculate:

a) $150\%$ of $60$.

b) $0.5\%$ of $800$.

Complete in your notebook:

a) If you get a $30\%$ discount, you pay ____ $\%$ of the price.

b) If a price goes up by $12\%$, you pay ____ $\%$ of the original price.

c) If you have used $45\%$ of the battery, you have ____ $\%$ left.

In a survey of $400$ young people, $64\%$ state that they do sport every week. How many young people do sport?

Noa had $\text{€}\ 240$ in savings and spent $25\%$ on a gift for her mother. How much money does she have left?

In a nature park with $12\,000$ trees, $35\%$ are pine trees. How many pine trees are there in the park?

A family dinner at a restaurant has a total cost of $68.40\text{ \euro}$. The family decides to leave a voluntary tip of $10\,\%$.

1. a) Before doing any written calculations, mentally estimate between which two integer numbers of euros the tip will lie, rounding the bill amount to the nearest tens. Explain how you reasoned your estimation.
2. b) Calculate the exact tip and the final total amount to pay using the corresponding decimal factor.
3. c) Compare your exact calculation with the estimate you made at the beginning: was it a useful approximation to make a quick decision at the table? Justify your answer.

Lucas wants to buy headphones with a marked price of $50\text{ \euro}$. The shop announces a $20\,\%$ discount in the audio section and, in addition, when paying, Lucas presents a membership coupon that gives an extra $10\,\%$ discount. Lucas claims with conviction: «Since $20\,\% + 10\,\% = 30\,\%$, they will deduct $30\,\%$ off $50\text{ \euro}$; that is, I save $15\text{ \euro}$ and I will only pay $35\text{ \euro}$».

1. a) Identify the conceptual error Lucas makes by directly adding the two percentages and explain in your own words why it is not correct to do so.
2. b) Calculate the actual price Lucas will pay by applying the discounts consecutively using multiplying factors.
3. c) State how much more or less money Lucas will pay compared to what he had mistakenly calculated.

A jacket that cost $\text{€}\ 74.90$ is reduced by $30\%$. How many euros are discounted? What is the final price?

I paid $\text{€}\ 59.50$ for a pair of trousers discounted by $15\%$. What was the price before the discount?

A cooking channel increased its followers by $25\%$ in the last year and now has $1\,800$ followers. How many followers did it have a year ago?

The price of a comic has gone from $\text{€}\ 12$ to $\text{€}\ 15$. What percentage has it gone up by?

A basketball player has scored $17$ of the $25$ free throws she took this season. What is her success percentage?

To buy an electric bike, Cian deposits $€1\,200$ into an account paying a simple interest rate of $3\%$ per annum for $2$ years. How much interest will he receive? How much money will he have in total at the end?

A stationer's has a "3 for 2" offer (buy three, pay for two) on felt-tip pens that cost $\text{€}\ 4.50$ each. If you buy $3$ felt-tip pens:

a) How much do you pay in total?

b) What percentage discount does the offer represent compared to the price without discount? Round to two decimal places.

A comic shop offers the second item at $50\%$ of its price. If a volume of manga costs $\text{€}\ 9.95$:

a) How much do two volumes cost?

b) What total percentage discount is applied on the purchase of the two volumes?

A video game that cost $\text{€}\ 60$ went up by $10\%$ during the Christmas campaign, and after the holidays, the new price dropped by $10\%$.

a) What was the price after the increase?

b) What is the price after the drop?

c) Has it returned to the original price? If not, what is the total percentage change?

An online shop sells a rucksack for $\text{€}\ 95$ excluding VAT. It applies a discount of $20\%$ and then adds $21\%$ VAT to the discounted price. What is the final price you pay?

In a survey of $1\,500$ young people, $40\%$ state that they play team sports, and of those, $25\%$ play basketball. How many young people play basketball? What percentage of the total do they represent?

A reservoir with a capacity of $240\ \text{hm}^3$ was at $35\%$ after the summer. Autumn rains have filled it to $58\%$ of its capacity.

a) How many cubic hectometres of water did it hold after the summer?

b) How many cubic hectometres has it gained from the rain?

State whether these statements are true or false, supporting your answers with numerical examples:

a) If a price goes up by $50\%$ and then goes down by $50\%$, it returns to the initial price.

b) Applying two consecutive discounts of $10\%$ is equivalent to a single discount of $19\%$.

c) $5\%$ of an amount is half of $10\%$ of that same amount.

d) If $90\%$ of the students passed an exam, $1$ in every $10$ failed.

The chart shows the comics that a shop sold between Monday and Friday last week.

a) What percentage of the week's sales were made on Friday?

b) By what percentage did sales on Friday increase compared to Monday?

Two gyms have the following offers:

• Gym A: $\text{€}\ 30$ per month, with a $20\%$ discount if you pay for the full year in advance.

• Gym B: $\text{€}\ 25$ per month, plus a one-off joining fee of $\text{€}\ 40$.

a) Calculate how much a full year costs at each gym.

b) Which one would you choose for one year? What if you only want to join for $6$ months (in which case Gym A does not apply the annual discount)? Justify both answers.

A recycling bin collects $800$ kg of packaging. $45\%$ of the weight is plastic and, of that plastic, only $60\%$ can be recycled.

a) How many kilograms of plastic are in the bin?

b) How many kilograms are actually recycled?

c) What percentage of the total bin contents is recycled?

A sports shop is clearing out its trainers with a $15\,\%$ discount on the tag price $P$. Two employees debate the best way to calculate the final discounted price:

• Aoife proposes: «I first calculate the discount by multiplying $P \times 0.15$ and then subtract it from the original price: $P - (P \times 0.15)$».

• Seán proposes: «I directly multiply the original price by the decimal factor corresponding to the discount: $P \times 0.85$».

1. a) Prove algebraically, by factoring out $P$ in Aoife's expression, why both methods lead to the exact same result.
2. b) If you have to mentally calculate the discount on trainers that cost $60\text{ €}$, which method do you choose and why do you find it more convenient?
3. c) If you need to program the cash register to update the prices of $500$ items instantly, which of the two methods is more efficient for the computer processor in terms of the number of operations performed? Justify your answer.

According to tourist occupancy data in rural accommodation in Ireland published by the CSO, a rural cottage recorded $200$ overnight stays during the month of April. In May, coinciding with a bank holiday weekend, bookings increased by $35\,\%$ compared to April. In June, due to the end of the school year and exams, bookings decreased by $20\,\%$ compared to May.

1. a) Model the trend of bookings by writing a single chained operation with multiplying factors that allows you to directly calculate the overnight stays in June from those in April.
2. b) Calculate the number of bookings recorded in June and interpret whether the final occupancy increased or decreased compared to the initial figure in April.
3. c) Determine the overall cumulative percentage change between April and June. Explain why this final result does not match the direct sum $+35\,\% - 20\,\% = +15\,\%$.

The ticket price for a water park dropped by $20\%$ at the end of the summer. When the next season arrives, the park wants to restore the exact original price. By what percentage must the reduced price increase? Explain why the answer is not $20\%$.

A bookshop's sales fell by $35\%$ in February compared to January. In March, sales recovered to reach the January level again. By what percentage did March sales increase compared to February? Round your answer to two decimal places.

Aoife bought a pair of headphones on a website for €96.80. This price already included a discount of $20\%$ on the base price and a $21\%$ VAT rate applied after the discount. What was the base price of the headphones, without discount or VAT?

Seán wants to buy a mountain bike and is studying several simple interest deposit options:

a) What principal must he deposit at $4\%$ per annum for $3$ years to get $€270$ in interest?

b) At what annual percentage rate must he invest $€2\,500$ for $2$ years to earn $€200$ in interest?

c) For how long must he keep $€1\,800$ at $5\%$ per annum to earn $€360$ in interest?

A sports shop advertises: "Sale: $50\%$ off plus an extra $20\%$ off: a total discount of $70\%$!". A pair of trainers at the shop costs $\text{€}\ 120$. Explain the error in the advert, calculate the actual percentage discount applied and the correct final price of the trainers.

A savings account offers simple interest of $0.4\%$ per month. If you deposit $\text{€}\ 3\,000$:

a) How much interest do you earn each month?

b) How many full months must pass to accumulate $\text{€}\ 180$ in interest?

Nora's family is comparing three mobile phone plans:

a) Calculate the cost of a full year for each plan, taking into account the terms.

b) Nora uses around $45$ GB per month. Which plan suits her best? Justify your answer using the calculations from the previous section.

c) Her father uses $15$ GB per month. Which plan do you recommend to him? How much does he save per year compared to the most expensive one that covers his needs?

Tickets for a music festival have increased by $5\%$ each year for three consecutive years. If three years ago they cost $\text{€}\ 40$:

a) How much do they cost now? Round to the nearest cent.

b) What has been the total percentage increase over the three years? Does it match $15\%$? Give reasons for your answer.

Two shops sell the same rucksack for $\text{€}\ 80$. Shop A reduces it directly by $25\%$. Shop B first applies a discount of $10\%$ and then, on the resulting price, another discount of $15\%$.

a) Calculate the final price of the rucksack in each shop.

b) In which shop is it better to buy it? Explain, showing your calculations, why the two offers are not equivalent.

State whether the following statements are true or false, justifying each answer with a numerical example:

a) An amount cannot increase by more than $100\%$.

b) $200\%$ of an amount is double that amount.

c) If the price of a product goes from $\text{€}\ 50$ to $\text{€}\ 150$, it has increased by $200\%$.

d) It is impossible to reduce a price by $110\%$.

In a secondary school, $360$ students play a federated sport, which represents $37.5\%$ of the total.

a) How many students are there in the school?

b) How many do not play a federated sport?

Here is Alex's family's electricity bill for one month:

• Consumption: $\text{€}\ 62.40$

• Contracted power: $\text{€}\ 18.20$

• Electricity tax: $5.11\%$ on the sum of consumption and contracted power

• VAT: $21\%$ on the previous total

a) Calculate the amount of electricity tax, rounding to the nearest cent.

b) Calculate the total amount of the bill.

c) The family installs solar panels and their consumption drops by $40\%$. If the contracted power and taxes are calculated in the same way, what will the total amount of the new bill be?

In a technology shop, the selling price of a tablet is changed in two consecutive stages: first, an increase of $x\,\%$ is applied to the original price and, subsequently, a discount of $x\,\%$ is applied to the new, increased price. After both changes, it is found that the final price of the tablet is exactly $9\,\%$ lower than its starting price.

1. a) Model the situation by setting up an equation in terms of $x$ that equates the product of the two variation factors to the resulting overall variation factor.
2. b) Solve the equation and find what percentage $x$ was applied in each of the two stages.
3. c) If the retailer had first applied the $x\,\%$ discount and then the $x\,\%$ increase, would they have obtained a different final price? Justify your answer by analysing the mathematical properties of the multiplication of factors, without needing to make up specific numbers.

For a school canteen, a batch of $6$ jugs of olive oil needs to be purchased, with a standard price of $12\text{ \euro}$ per jug. The purchasing manager compares three current offers from different suppliers:

• Supplier A: «$3 \times 2$» promotion (for every $3$ jugs you buy, you only pay for $2$).

• Supplier B: «Second unit at a $60\,\%$ discount» (buying in pairs, the first jug is paid in full and the second has a $60\,\%$ discount on its individual price).

• Supplier C: «Direct $25\,\%$ discount on the entire purchase» when buying $6$ or more jugs.

1. a) Model using numerical expressions the total cost that would be paid at each supplier when purchasing the batch of $6$ jugs, and calculate the resulting average price per jug for each option.
2. b) Determine what actual percentage discount on the original price the «$3 \times 2$» promotion is equivalent to for a batch of $6$ jugs, and compare it with the overall percentage discounts of the other two suppliers.
3. c) Which supplier is the most economical for buying the $6$ jugs? Justify whether your choice would change if the school only needed to buy $5$ jugs, by calculating the total cost it would entail for the two suppliers that proved to be the most competitive.