Percentage increases and decreases

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

A sportswear store announces a seasonal clearance sale with a discount of $25\text{ \%}$ on all its sneakers. If a pair of running shoes originally cost $\$110$, 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 $\$12.50$. In 2023, the price rose to $\$14.75$. 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 $19,096$. What was 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 movie theater ticket costs $\$9$. If you pay with a youth discount 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 discount card holder.
2. b) Calculate the total index of variation and the actual percentage discount on the initial price.

An electronics store announces its seasonal sales. A pair of wireless headphones that cost $120$ dollars has a $15 \%$ discount. After applying the discount, when buying them through the website, a $5 \%$ surcharge for shipping and 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?

A tech store launches a new model of wireless headphones at a price of $120$ dollars. 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.

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 $12$ dollars, which offer is more advantageous, and what is the percentage savings of the better offer compared to the worse one?

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

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

If a television displays a price tag of $\$484$ ($IVA$ 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.

Sophia buys a laptop for her studies whose final invoice with $21\,\%$ sales tax 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 Sophia, define the appropriate growth factor to reverse this percentage, and calculate the actual price of the laptop before applying the tax.

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

In the January sales, a pair of sneakers has a discount of $35$ %. If the discounted price on the tag is $57.20$ dollars, what was the price of the sneakers before the sales started? State also the total saving made in dollars.

The price of a premium subscription to a streaming platform is $15.99$ dollars per month, sales tax included. Knowing that the sales tax rate applied to these digital services in the state is $21$ %, calculate:

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

A young freelance programmer charged a total of $2,450$ dollars for a project after applying a $21\%$ sales tax increase and, subsequently, a $15\%$ income tax withholding on the amount already increased by sales tax. However, the client made a mistake and should have applied the income tax withholding only to the pre-tax amount (before tax). Calculate:

1. The original pre-tax amount of the project.

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

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.

The youth public transit pass in a state had a monthly fare of $\$40$. At the beginning of the year, it experienced a $15\,\%$ increase in 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 $\$40$, greater than $\$40$, or less than $\$40$.\nb) Perform the formal mathematical calculation using growth factors to check your estimate, state the overall growth factor, and determine the effective percentage change with respect to the original price of $\$40$.

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 $24,500$ dollars, 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 $24,500$ dollars?

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 $5,000$ dollars, what is the value of his investment after the two months?

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

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 optimization, 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.

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 store 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 store apply to the inflated price?

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