Exercises: Percentage calculations

In a chemistry laboratory, $250$ ml of saline solution with a salt concentration of $0,9\%$ is prepared. How many millilitres of pure salt does this solution contain?

A forest has lost $15\%$ of its trees due to a wildfire. If forestry officials have counted $360$ burnt trees, how many trees did the forest have before the fire?

A mobile phone battery has a total capacity of $4500$ mAh. If, after a day of heavy use, the diagnostic app indicates that $1260$ mAh remains, what percentage of battery remains available on the device?

A family pays $85$ € per month for their broadband and TV package. If the telecoms provider informs them that next month there will be a price increase of $4\%$ due to tariff updates, how much will they pay per month from now on?

A pair of trainers with a tag price of $120$ € is discounted by $35\%$ for the end-of-season sale. Calculate the final price to be paid at the till.

After applying a salary increase of $6\%$ agreed in the collective agreement, a worker's net monthly salary is $1643$ €. What was her salary before the pay rise was applied?

I paid $34$ € for a digital video game that was discounted by $15\%$ during a special promotion. What was the original price of the video game before the discount?

A capital of $4500$ € is deposited into a bank account offering a simple annual interest rate of $2,5\%$. Determine how much interest will be earned after $4$ years.

Sergio has received an invoice for the repair of his electric bicycle for a final amount of $242$ €, with $21\%$ VAT already included. To find out how much the repair shop charged him for the work before taxes, Sergio carries out the following calculation in his notebook:

1. a) Explain in your own words why Sergio's reasoning is wrong, pointing out to which amount the $21\%$ VAT should be applied and why directly subtracting that percentage does not reverse the original increase.
2. b) Write down the growth factor (variation index) corresponding to the $21\%$ increase and use it to set up the correct calculation to determine the original price without VAT.
3. c) Check the validity of your result by calculating $21\%$ VAT on the amount obtained and adding it to it.

In several autonomous communities in Spain, the monthly public transport pass for young people had an established fare of $20$ € per month. With the national package of energy-saving measures, the Ministry of Transport approved a $30\%$ discount, and subsequently, the regional government applied an additional discount of $20\%$ on the already discounted price.

1. a) Determine the multiplier (variation index) for each of the two discounts and calculate the overall multiplier resulting from chaining both price reductions.
2. b) Calculate the final price a student pays for their monthly pass after the consecutive application of both discounts.
3. c) In a class debate, Marta states: "Since they lowered it by $30\%$ and then by $20\%$, the total discount is $50\%$, so the pass should cost exactly $10$ €". Use the overall multiplier to mathematically refute Marta's statement and calculate the real equivalent single discount percentage.

An appliance shop applies a discount of $20\%$ to a high-end television with an initial price of $650$ €. At the till, you present your VIP loyalty card, which gives you an additional discount of $10\%$ on the already reduced price. Calculate the final price you will pay for the television and determine the single percentage discount equivalent to these two successive reductions.

The value of some stock market shares experienced a rise of $15\%$ during January and subsequently suffered a drop of $12\%$ during February due to market instability. What was the overall percentage change over the two-month period? Have the shares gained or lost value compared to the start of the year?

If the side length of a square is increased by $25\%$, by what percentage will its total area increase compared to the original area?

An amount of $8000$ € is deposited into an investment fund offering a compound interest rate of $3,2\%$ per annum. Calculate the final balance accumulated after $5$ years and the total interest earned, assuming that no money is withdrawn during that time.

When weighing a chemical sample with an actual mass of $45,60$ g, an uncalibrated laboratory scale reads $45,85$ g. Calculate the absolute error of the measurement and the relative error, expressing the latter as a percentage rounded to two decimal places.

In a winery, $40$ litres of wine with an alcohol content of $12\%$ are mixed with $60$ litres of another wine with an alcohol content of $15\%$. What is the alcohol percentage of the resulting mixture?

A latest-generation laptop costs $847$ € with $21\%$ VAT included. If the shop advertises that all its products have a discount of $30\%$ off the original price (before tax is applied), what was the original price of the laptop without discount and excluding VAT?

An algae population in a research pond grows at a rate of $8\%$ per day. If the current estimated population is $50000$ individuals, how many individuals will there be after exactly one week (seven days)? Round the answer to the nearest integer.

According to reports from the Ministry for the Ecological Transition on water reserves in Spain, a reservoir with a maximum capacity of $500\text{ hm}^3$ was at $30\%$ of its total capacity at the end of summer. During an autumn storm, the volume of water stored in that reservoir experienced an increase of $40\%$.

1. a) Before performing calculations with decimals, make a reasoned estimate: a spokesperson states in the press that, by adding $40\%$ to the $30\%$ it already had, the reservoir has reached $70\%$ of its total capacity. Reason mentally why this estimate is incorrect and estimate between which two tens of percent (for example, between $40\%$ and $50\%$) the real occupancy must lie after the rains.
2. b) Using the growth factor (variation index) corresponding to the increase, calculate the exact volume of stored water (in $\text{hm}^3$) that the reservoir contains after the storm.
3. c) Calculate what percentage of the maximum capacity this final volume represents and check whether the result matches your estimate from part a).

The urban planning department of a town council plans to remodel a square public plaza to exactly double its current landscaped area (that is, to achieve an area increase of $100\%$).

1. a) Knowing that the area of a square depends on the square of its side length ($A = l^2$), formulate the mathematical relationship that links the area's multiplier ($I_A$) with the side length's multiplier ($I_l$).
2. b) Knowing that doubling the area implies that $I_A = 2$, set up the corresponding equation and find the exact value of $I_l$. By what percentage must the length of each side of the square be increased? (Round the percentage to one decimal place).
3. c) A member of the municipal committee initially proposed: "To get twice the landscaped area, it is enough to increase each side by $50\%$". Check what percentage increase in area would actually be obtained with this proposal and explain why the area is not doubled.

Find how many full years must pass for an initial investment of $12000$ € deposited in an account offering a compound interest rate of $4,5\%$ per annum to reach or exceed a value of $18000$ €.

When designing a new cylindrical soft drink container, the packaging department decides to increase the base radius by $20\%$ and decrease the height by $15\%$ relative to the dimensions of the original container. What is the exact percentage change in the volume of the new container compared to the old one? State whether it is an increase or a decrease.

House prices in a certain city dropped by $6\%$ three years ago, rose by $5\%$ two years ago, and rose again by $3\%$ over the last year. If a flat currently has a market value of $185000$ €, what was its price three years ago before all these percentage changes? Round to the nearest cent.

A perfumer has two essences in their laboratory: one with an aromatic concentration of $15\%$ and another with a concentration of $40\%$. How many millilitres of each essence must be mixed to obtain exactly $200$ ml of perfume with a final aromatic concentration of $25\%$?

A financial institution offers a fixed-term deposit with a nominal annual interest rate (TIN) of $6\%$, but specifies that interest will be paid monthly (the interest earned is added to the principal each month). Calculate the Annual Equivalent Rate (AER) for this deposit, expressed as a percentage rounded to two decimal places.

Prove algebraically that if the price of a product $C$ is increased by $p\%$ (where $p > 0$) and, subsequently, the new price is discounted by the same $p\%$, the final price will always be strictly less than the initial price $C$.

A delivery van loses $14\%$ of its value each year compared to the previous year due to depreciation. If after $6$ years of use its residual value is estimated at $12500$ €, calculate its original purchase price and the total percentage of its initial value lost over those $6$ years.

An investor purchases a block of shares for $25000$ €. After holding the investment for $8$ years, without adding or withdrawing any capital, they sell the shares for $38500$ €. What was the average annual rate of return (annual compound interest rate) obtained from their investment? Express the result as a percentage to two decimal places.

A graphic designer working as a freelancer in Spain is going to purchase computer equipment quoted at $1500$ € (tax base, excluding taxes). The retailer offers him a $12\%$ professional discount, and $21\%$ VAT must mandatorily be applied to the purchase. At the time of issuing the invoice, three calculation methods are considered:

• Method 1: Apply the $12\%$ discount to the $1500$ € and, to the resulting amount, add the $21\%$ VAT.

• Method 2: Apply the $21\%$ VAT to the $1500$ € and, to the resulting amount with taxes, apply the $12\%$ discount.

• Method 3: Subtract the percentages directly ($21\% - 12\% = 9\%$) and apply a single increase of $9\%$ to the $1500$ €.

1. a) Model Methods 1 and 2 using the corresponding multipliers (variation indices). Justify, by applying the properties of multiplication, whether one of the two methods benefits the buyer more or if both are economically equivalent.
2. b) Analyze Method 3 algebraically by calculating its multiplier and comparing it to the overall multiplier obtained in the previous methods. Explain the conceptual error of subtracting the percentages and calculate the difference in euros compared to the correct price.
3. c) If the freelancer needs to program a spreadsheet to invoice his own clients with any discount $d\,\%$ and $21\%$ VAT, write the simplified expression of the single multiplier that must multiply the tax base.

According to data published by the National Statistics Institute (INE), the annual cumulative inflation (CPI) in Spain over a three-year period recorded the following consecutive variations: $+6,5\%$ in the first year, $+5,7\%$ in the second year, and $+3,1\%$ in the third year. During that same three-year period, a worker's collective agreement stipulated a fixed wage increase of $2,5\%$ each year.

1. a) Calculate the compound multiplier (variation index) of the cost of living (CPI) after the three years and determine the total cumulative percentage price increase (round to two decimal places).
2. b) Calculate the cumulative wage multiplier for the worker at the end of the three years and find her total percentage wage increase.
3. c) Purchasing power quantifies real buying capacity based on income and prices. Its multiplier is calculated using the quotient of the wage multiplier and the CPI multiplier: $I_{\text{purchasing}} = \frac{I_{\text{wage}}}{I_{\text{CPI}}}$. Calculate this index, determine the percentage change in the worker's purchasing power at the end of the three-year period, and interpret its real impact on her household finances.
4. d) If this worker's starting net monthly salary was $1600$ €, how much does she earn at the end of the third year, and what salary should she receive to maintain exactly the same purchasing power she had at the beginning?