Percentage increases and decreases

The price of a ticket for the "Tecno-Vibes" music festival is $\text{€}\ 55$. 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.

In the year 2022, a monthly subscription to a streaming platform cost $\text{€}\ 12.50$. In 2023, the price rose to $\text{€}\ 14.75$. Calculate the index of variation and the percentage increase that the subscription experienced.

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?

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 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?

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.

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

Aoife buys a laptop for her studies whose final invoice with $21\,\%$ VAT included amounts to $€847$. To find out the price of the computer before taxes, she calculates $21\,\%$ of $€847$, obtaining $€177.87$, and then subtracts that amount: $847 - 177.87 = €669.13$. Identify and explain with reasoned arguments the conceptual error made by Aoife, define the appropriate growth factor to reverse this percentage, and calculate the actual price of the laptop before applying the tax.

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 Ireland 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?

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 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.

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?

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.

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?

During a two-year period, year-on-year inflation in Ireland (measured through the CPI by the CSO) 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 $€1\,700$ 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 $2\,\%$.

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.

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 $24,500$ €?