Exercises: Percentage increases and decreases

The price of a ticket for the "Tecno-Vibes" music festival is $55\text{ €}$. If the organizers decide to raise the price by $12\text{ \%}$ due to high demand, what will the new ticket price be? Also state the change index that was applied.

A sportswear shop announces a seasonal clearance sale with a discount of $25\text{ \%}$ on all its trainers. If a pair of running shoes originally cost $110\text{ €}$, how much will we pay for them after the discount? Calculate the corresponding index of variation.

In the year 2022, a monthly subscription to a streaming platform cost $12,50\text{ €}$. In 2023, the price rose to $14,75\text{ €}$. Calculate the index of variation and the percentage increase that the subscription experienced.

After applying a discount of $15\text{ \%}$, a laptop costs $680\text{ €}$. What was the original price of the laptop before the promotion? Use the concept of index of variation to solve it.

The salary of a young programmer increases by $5\text{ \%}$ at the end of their first year under contract. However, in the second year, due to a crisis in the company, their salary is reduced by $5\text{ \%}$. Do you think their final salary is the same as the initial one? Justify your answer by calculating the overall index of variation.

An electronic component has a retail price of $242\text{ €}$ with $IVA$ $21\text{ \%}$ VAT already included. Determine the price of the product before tax (taxable base).

A shopping centre holds a "VAT-free Day". This means that from the final retail price (which already includes $21\text{ \%}$ $IVA$ VAT), the value equivalent to the tax is discounted.

If a television displays a price tag of $484\text{ €}$ ($IVA$ included), what will its price be during the promotion? Check whether this promotion is equivalent to a discount of $21\text{ \%}$ on the final price.

A cinema ticket costs $9\text{ €}$. If you pay with a youth card, you get a discount of $20\text{ \%}$. But if it is also "cheap Wednesday", the already discounted price is reduced by an additional $10\text{ \%}$.

1. a) Calculate the final price of the ticket on a Wednesday for a youth card holder.
2. b) Calculate the total index of variation and the actual percentage discount on the initial price.

A tech store announces its seasonal sales. A pair of wireless headphones that cost $120$ € has a $15 \%$ discount. After applying the discount, when buying them through the website, a $5 \%$ surcharge for shipping handling fees is applied to the discounted price. The diagram summarizes the process.

1. a) Calculate the final price of the headphones.

1. b) What is the overall change index of this operation?

Lucía buys a laptop for her studies whose final invoice with $21\,\%$ VAT included amounts to $847\text{ €}$. To find out the price of the computer before taxes, she calculates $21\,\%$ of $847\text{ €}$, obtaining $177,87\text{ €}$, and then subtracts that amount: $847 - 177,87 = 669,13\text{ €}$. Identify and explain with reasoned arguments the conceptual error made by Lucía, define the appropriate growth factor to reverse this percentage, and calculate the actual price of the laptop before applying the tax.

The youth public transport pass in an autonomous community had a monthly fare of $40\text{ €}$. At the beginning of the year, it experienced a $15\,\%$ increase on its price and, six months later, to compensate for family expenses, a $15\,\%$ discount was applied to the already increased price.\na) Before carrying out numerical calculations on paper, estimate with justification whether the final resulting price will be exactly $40\text{ €}$, greater than $40\text{ €}$, or less than $40\text{ €}$.\nb) Perform the formal mathematical calculation using growth factors to check your estimate, state the overall growth factor, and determine the effective percentage variation with respect to the original price of $40\text{ €}$.

A tech store launches a new model of wireless headphones at a price of $120$ €. Due to high demand, the following month they decide to increase its price by $15$ %. However, when the "Black Friday" week arrives, they apply a discount of $20$ % to the increased price. Calculate the final price of the headphones and determine the overall index of variation of the price.

A young influencer dedicated to sustainable design had $15.400$ followers on her account at the beginning of the year. Following an awareness campaign about recycling, her number of followers increased to reach $19.096$. What has been the percentage increase in her community? If she expects growth to be $8$ % over the current figure next quarter, how many followers will she have then?

In the January sales, a pair of trainers has a discount of $35$ %. If the discounted price on the tag is $57,20$ €, what was the price of the trainers before the sales started? State also the total saving made in euros.

The price of a premium subscription to a streaming platform is $15,99$ € per month, VAT included. Knowing that the VAT rate applied to these digital services in Spain is $21$ %, calculate:

1. a) The taxable base of the bill (the price before tax).
2. b) The exact amount paid in VAT.
3. c) If the platform announces an increase of $5$ % in the taxable base for next year, what will be the new monthly price including VAT?

A small local shop decides to update its product prices in line with the cost of living. At the beginning of the year, it raises all prices by $4$ %. In summer, due to a special promotion, it applies a discount of $10$ % to the updated prices.

1. a) Calculate the overall index of variation.
2. b) What was the total percentage change compared to the original price at the end of last year?
3. c) If a product originally cost $25$ €, what is its price after both changes?

Due to improvements in transport infrastructure, the travel time between two cities has been reduced by $18$ % this year. If the journey currently takes $82$ minutes, how long did it take last year? Express the result in hours and minutes.

The annual gross salary of a worker underwent the following changes over a three-year period: in the first year it rose by $2$ %, in the second year it remained frozen ($0$ % change), and in the third year it fell by $3$ % due to company restructuring.

1. a) Calculate the overall index of variation at the end of the three years.
2. b) If the initial salary was $24.500$ €, what is the salary after the third year?
3. c) By what percentage should the salary rise in the fourth year to recover exactly the initial purchasing power of $24.500$ €?

An investor buys shares of a technology company. During the first month, the value of the shares rises by $20$ %. During the second month, the value falls by $20$ %. The investor claims: "I'm back where I started, I've neither gained nor lost".

1. a) Prove mathematically whether the investor is right or not, using variation indices.

1. b) If he initially invested $5.000$ €, what is the value of his investment after the two months?

1. c) Calculate the overall percentage change of the investment.

A sports shop offers three alternatives during the sales to clear seasonal items whose initial price is $P$ euros:\n- Method 1: Apply a single, direct discount of $25\,\%$.\n- Method 2: First apply a $15\,\%$ discount and then an additional $10\,\%$ discount on the discounted price.\n- Method 3: First apply the $10\,\%$ discount and, on that amount, the $15\,\%$ discount.\na) Express the final price obtained with each method in terms of the initial price $P$ using growth factors.\nb) Reasonably analyze whether Method 2 and Method 3 lead to the same final price and explain which mathematical property this is due to.\nc) Justify which of the three methods is most advantageous for the customer and calculate the equivalent effective discount percentage applied in Method 2.

According to data from the National Statistics Institute (INE) on the rental market in a provincial capital, the average price per square meter varied as follows over three consecutive years: in 2021 it increased by $6\,\%$; in 2022 it suffered a further increase of $8\,\%$; and in 2023 it experienced a drop of $5\,\%$.\na) Model the price evolution by calculating the growth factor for each year and the overall growth factor corresponding to the full three-year period.\nb) Determine the cumulative percentage variation experienced by the rental price between the beginning of 2021 and the end of 2023.\nc) If the monthly rent of an apartment cost $750\text{ €}$ at the beginning of 2021, calculate its monthly rent at the end of 2023. Likewise, if a tenant paid $870\text{ €}$ per month at the end of 2023, deduce how much they paid at the beginning of 2021.

A smartphone technology components company has experienced a series of changes in its production costs over three quarters. In the first quarter, the cost increased by $12\%$. In the second quarter, due to process optimisation, the cost decreased by $8\%$. For the third quarter, management has set a target for the final cost to be exactly $5\%$ lower than the initial cost before the first quarter. Calculate the exact percentage decrease that must be applied in the third quarter to reach this target.

A young freelance programmer charged a total of $2.450$ € for a project after applying a $21\%$ VAT increase and, subsequently, a $15\%$ income tax deduction on the amount already increased by VAT. However, the client made a mistake and should have applied the income tax deduction only to the taxable base (before tax). Calculate:

1. The original taxable base of the project.

2. The difference in money that the programmer should claim or refund to correct the error.

In an urban fashion store, successive discounts are applied. First, a general discount of $20\%$ is announced on all items. During the "clearance week", an additional discount of $30\%$ is applied to the already discounted price. A customer has a VIP coupon that gives an extra discount of $10\%$ on the latest price obtained.

Is this process of three discounts equivalent to a single discount of $60\%$? Justify your answer by calculating the overall index of variation and determine what the initial price of a hoodie was if it ends up costing $39,20$ € after the three discounts.

The rent price index in a large city increased by $15\%$ last year and is expected to increase by another $8\%$ this year. At the same time, the average salary of young people in the city increased by $2\%$ last year and a $5\%$ rise is forecast for this year.

The diagram summarizes the data: it starts from an index of $100$ in year 0 and shows the increase for each year.

Calculate the overall change index of rents and of salaries after the two years. Using these data, determine by what percentage the purchasing power of a young person who spends their entire salary exclusively on rent has actually changed.

A supermarket chain advertises a "3 for 2" offer on tubs of detergent. At the same time, its direct competitor offers buy one get the second at half price (a $50\%$ discount on the second unit).

1. Calculate the actual percentage discount offered by each promotion when buying the minimum number of tubs needed to take advantage of it.

2. If the original price of a tub in both stores is $12$ €, which offer is more advantageous, and what is the percentage saving of the better offer compared to the worse one?

Due to inflation, the price of a video game console model has increased by $25\%$ since its launch. To promote sales on Black Friday, the shop wants to apply a discount such that the final price is exactly the same as the original launch price.

1. What percentage discount must the shop apply to the inflated price?

2. If after applying this discount the price is $450$ €, what was the price of the console before the price rise due to inflation?

An investor splits their savings into two financial products. In Product A, they invest $60\%$ of their capital and make a profit of $15\%$. In Product B, they invest the rest and suffer a loss of $8\%$.

Calculate the overall index of variation for their total capital and the net percentage gain or loss. If at the end of the year their total capital amounts to $53.100$ €, what was their initial capital and how much money did they lose specifically in Product B?

A bacterial population decreases by $20\%$ every hour due to the application of an antibiotic. However, every 3 hours, the surviving bacteria develop resistance that causes their number to suddenly increase by $10\%$. If at the start of the experiment there was a culture of $500.000$ bacteria, calculate how many bacteria there will be after 6 hours and find the overall index of variation of the population over that period of time.

An extra virgin olive oil agricultural cooperative increased the farm-gate price of its product by $30\,\%$ in the first half of the year due to drought. For the second half of the year, with the aim of recovering sales in the domestic market, the board of directors needs to apply a percentage variation $x\,\%$ to the increased price so that the resulting final price represents an exact overall increase of $10,5\,\%$ compared to the price the oil had before the year began.\na) Set up an equation based on growth factors that relates the initial increase, the unknown variation $x$, and the set overall factor.\nb) Solve the equation, clarify whether the variation $x$ corresponds to an increase or a discount, and determine its exact percentage.\nc) Generalize the reasoning: if a quantity experiences an increase of $r\,\%$ ($r > 0$), deduce algebraically the expression for the discount percentage $d\,\%$ that must then be applied to the resulting amount to recover the initial amount exactly.

During a two-year period, year-on-year inflation in Spain (measured through the CPI) recorded an increase of $5,7\,\%$ in the first year and $3,1\,\%$ in the second year. In that same period, the collective bargaining agreement of a transport company applied a wage increase to its employees of $4\,\%$ in the first year and $2,5\,\%$ in the second year.\na) Mathematically model through growth factors both the cumulative evolution of the cost of living (CPI) and the cumulative evolution of wages.\nb) Wage purchasing power is modeled by the quotient between the wage growth factor and the CPI growth factor: $I_p = \frac{I_v^{\text{wage}}}{I_v^{\text{CPI}}}$. Formally justify whether the workers have gained or lost purchasing power at the end of the two years and calculate the exact percentage of that gain or loss.\nc) If an employee earned a net monthly salary of $1\,700\text{ €}$ before this two-year period began, calculate what their salary should be at the end of the third year to recover the original purchasing power exactly, assuming that the inflation forecast for the third year is $2\,\%$.