Exercises: Compound interest and percentage change

Write down the multiplier corresponding to each of these percentage changes:

1. a) An increase of $25\ \%$.

1. b) A decrease of $15\ \%$.

1. c) An increase of $3\ \%$.

1. d) A decrease of $60\ \%$.

1. e) An increase of $130\ \%$.

What percentage increase or decrease corresponds to each of these multipliers?

1. a) $1,45$

1. b) $0,82$

1. c) $2,1$

1. d) $0,05$

1. e) $1,06$

Apply the corresponding multiplier to calculate the final value:

1. a) A festival ticket costing $80\ \text{€}$ goes up by $15\ \%$.

1. b) A pair of trainers costing $64\ \text{€}$ is discounted by $25\ \%$.

1. c) The rent of $950\ \text{€}$ per month for premises goes up by $4\ \%$.

A saver deposits $2.000\ \text{€}$ in a deposit account at $4\ \%$ per annum compound interest; that is, each year the interest is added to the principal and also generates interest.

1. a) What does the principal become at the end of the first year?

1. b) What about at the end of the second year?

1. c) Why is the increase in the second year greater than that in the first year, if the percentage is the same?

Percentages can be greater than $100\ \%$:

1. a) The population of a new neighbourhood has increased by $150\ \%$ in ten years. What is the multiplier? If there were $1.200$ residents, how many are there now?

1. b) A club's point balance has tripled. What percentage increase does this correspond to?

1. c) Is "increasing by $200\ \%$" the same as "doubling"? Reason your answer with an example.

Percentages can also be less than $1\ \%$:

1. a) An institution charges a fee of $0,3\ \%$ for each urgent transfer. How much does it charge for a transfer of $4.500\ \text{€}$?

1. b) In an Olympic swimming pool containing $2.500.000\ \text{L}$ of water, $0,02\ \%$ corresponds to the water renewed daily due to chlorination and balancing. How many litres is that?

1. c) Express $0,5\ \%$ as a multiplier and as a fraction.

In a clothing store during the sales season, a coat initially costs $90\ \text{€}$. In the first week, a $20\ \%$ discount is applied, and in the second week, an additional $15\ \%$ discount is applied to the already discounted price.

Marcos calculates the final price by adding both percentages: $20\ \% + 15\ \% = 35\ \%$, so he applies a single variation index of $1 - 0,35 = 0,65$ and claims that the coat costs $90 \cdot 0,65 = 58,50\ \text{€}$.

1. a) Identify and explain with reasoned arguments Marcos's conceptual error. To which amount should the second discount actually be applied?
2. b) Write down the variation indices for each discount and calculate, using chained indices, the correct final price of the coat.
3. c) Without just looking at the number obtained, reason why the actual price ends up being more expensive than what Marcos calculated.

A family's two-monthly water bill has increased by $5\ \%$ due to an extraordinary drought surcharge, reaching a final amount of $42\ \text{€}$.

1. a) Before carrying out complex calculations, make a mental estimate: was the price before the increase approximately $40\ \text{€}$, $35\ \text{€}$, or less? Justify your answer by calculating mentally what a $5\ \%$ increase on $40\ \text{€}$ would represent.
2. b) Write down the variation index associated with the increase and calculate the exact original price by dividing the final amount by this index.
3. c) Check whether the exact result validates your estimate. If you directly subtracted $5\ \%$ ($42 \cdot 0,95$) from the final $42\ \text{€}$, would you get the original price? Perform the numerical check and reason why they do not match.

A shop increases the price of its products by $20\ \%$ and, one month later, discounts them by $20\ \%$.

1. a) A pair of headphones initially cost $50\ \text{€}$. How much do they cost after the price increase? And after the discount?

1. b) Calculate the overall multiplier by multiplying the individual multipliers.

1. c) Have the prices returned to their initial value? What was the overall percentage change?

Grandparents deposit $3.000\ \text{€}$ in a savings account at $5\ \%$ per annum compound interest for their granddaughter.

1. a) Write the expression that gives the final amount after $4$ years using a power.

1. b) Calculate the final amount using a calculator.

1. c) How many euros correspond to interest?

After a discount of $25\ \%$ is applied, a coat costs $54\ \text{€}$.

1. a) What is the multiplier for the discount?

1. b) Calculate the initial price by dividing by the multiplier.

1. c) Check the result by applying the discount to the initial price you obtained.

Compare two ways of investing $1.000\ \text{€}$ for $3$ years at $6\ \%$ per annum, as shown in the figure:

• Simple interest: each year the same interest is generated on the initial principal.

• Compound interest: each year the interest is added to the principal.

1. a) Calculate the final amount with simple interest.

1. b) Calculate the final amount with compound interest.

1. c) What is the difference? Where does this difference come from?

A bank offers a deposit account of $1.200\ \text{€}$ at $6\ \%$ per annum with monthly interest payments (monthly compound interest).

1. a) What percentage of interest is applied each month?

1. b) Write the expression for the final amount after $2$ years, taking into account that there are $24$ monthly periods.

1. c) Calculate the final amount and compare it with the amount that would be obtained with annual interest payments: $1.200 \cdot 1,06^2$. Which is greater?

The shares of a technology company rose by $40\ \%$ in the first year, fell by $30\ \%$ in the second and rose by $10\ \%$ in the third.

1. a) Calculate the multiplier for each year and the overall multiplier for the three years.

1. b) If a share portfolio was worth $2.500\ \text{€}$ at the beginning, how much is it worth at the end of the third year?

1. c) What was the overall percentage change? Why does it not match the sum of the three percentages?

A youth savings account offers an annual compound interest rate of $3\ \%$. A person deposits $1.500\ \text{€}$ and decides not to touch the money for $4$ years. Two students propose two different approaches to calculate the accumulated capital at the end of the period:

• Elena proposes the step-by-step method: calculate $3\ \%$ of $1.500\ \text{€}$ and add it; to the result obtained, calculate $3\ \%$ again and add it, repeating the process four consecutive times.

• Lucía proposes the global method: set up the compound interest formula using the variation index: $C_f = 1.500 \cdot 1,03^4$.

1. a) Prove algebraically that both methods yield the same result by factoring out the common term in the first two years of Elena's method.
2. b) Compare both approaches: what are the advantages and disadvantages of each? If the period were $20$ years, which one would you choose and why?
3. c) Calculate the final accumulated capital and the total interest earned after $4$ years using the method you consider most efficient.

According to census data from the National Statistics Institute (INE), a rural municipality in Zamora had $1.200$ inhabitants as of January 1, 2020. Due to depopulation, its population is decreasing at an estimated constant rate of $4\ \%$ per year.

1. a) Write the algebraic expression that models the population $P(t)$ as a function of the elapsed years $t$ since 2020, clearly indicating the annual variation index used.
2. b) Use the model to estimate the municipality's population on January 1, 2024 (round the result to the nearest whole number of people).
3. c) What total percentage of its population will the town have lost over those $4$ years? Compare this overall percentage with the direct multiplication $4\ \text{years} \cdot 4\ \% = 16\ \%$ and mathematically reason why the actual cumulative loss is less than $16\ \%$.

The price of a city transport pass has decreased by $20\ \%$ thanks to a public subsidy.

1. a) If a single ticket cost $1,55\ \text{€}$, how much does it cost now?

1. b) If the subsidy is withdrawn and the price returns to the initial level, by what percentage must the current price increase to return to the original price?

1. c) Explain, using multipliers, why the answer is not $20\ \%$, and state the general rule: after a decrease of $r\ \%$, what percentage increase returns the price to the start?

Two banks offer a one-year deposit for $5.000\ \text{€}$:

• Bank A: $4,9\ \%$ annual interest with semiannual interest payments (half of the percentage each half-year, that is, $2,45\ \%$, with compound interest).

• Bank B: $5\ \%$ annual interest with a single interest payment at the end of the year.

1. a) Calculate the final capital in bank A, taking into account that $2,45\ \%$ is applied each half-year.

1. b) Calculate the final capital in bank B.

1. c) The APR (Annual Percentage Rate, known in Spain as the TAE, Tasa Anual Equivalente) is the real annual percentage that a deposit yields, taking into account the frequency of payments. What is the APR of each offer? Which bank is better?

Decision-making task. Inés has saved $900\ \text{€}$ working at weekends and wants to invest it for $3$ years to buy a computer when she finishes secondary school. Compare three options, as shown in the figure:

• Option 1: youth account at $2\ \%$ simple interest per annum.

• Option 2: fixed deposit at $1,8\ \%$ compound interest per annum.

• Option 3: Treasury bills at $2,5\ \%$ compound interest per annum, but with a penalty of $15\ \text{€}$ if withdrawn before $3$ years.

1. a) Calculate the final amount for each option.

1. b) What is the difference in euros between the best and the worst option?

1. c) Inés thinks she might need the money before $3$ years for a summer course. Does that change your recommendation? Justify your decision.

The price of the set menu in a neighbourhood has followed inflation: it rose by $6\ \%$ in 2023, $3,5\ \%$ in 2024, and $2\ \%$ in 2025. In 2022, the set menu cost $11\ \text{€}$.

1. a) Calculate the price of the menu at the end of each year using multiplier factors.

1. b) What is the overall percentage increase of the menu over the three years?

1. c) Workers' wages in the neighbourhood have only increased by $9\ \%$ over that period. Have they gained or lost purchasing power to pay for the menu? Justify your answer by comparing multiplier factors.

An agricultural cooperative applied for an $8.000\ \text{€}$ loan to be repaid in a single payment after $3$ years using an annual compound interest system to renew its machinery. At the end of the $3$-year period, the cooperative settles the debt by paying a total of $9.528,13\ \text{€}$ (initial capital plus accrued interest).

1. a) Set up the compound interest equation where the only unknown is the annual variation index $I_v$.
2. b) Solve for $I_v$ using the cube root and determine the nominal annual interest rate $r\ \%$ applied by the financial institution (round $r$ to the nearest whole number).
3. c) If the contract had agreed to settle interest monthly instead of annually at that same rate $r\ \%$, write the complete expression to calculate the total amount to be repaid after $36$ months. Reason whether with monthly compounding the cooperative would pay more or less money than with annual compounding.

Over a period of two consecutive years, inflation in Spain according to the CPI recorded an increase of $6\ \%$ in the first year and $4\ \%$ in the second year. During that same period, Irene's collective bargaining agreement set a salary increase of $5\ \%$ in the first year and another $5\ \%$ in the second year. Irene states: "Since $5 + 5 = 10\ \%$ and inflation also adds up to $6 + 4 = 10\ \%$, my purchasing power has not changed at all."

1. a) Identify Irene's error. Calculate the cumulative variation index of prices (inflation) and the cumulative variation index of Irene's salary after the two years.
2. b) The purchasing power variation index is defined as the quotient between the cumulative salary index and the cumulative price index: $I_p = \frac{I_{\text{sueldo}}}{I_{\text{precios}}}$. Calculate this value to four decimal places and deduce the exact percentage of purchasing power that Irene has actually gained or lost.
3. c) If Irene's salary was $1.500\ \text{€}$ at the beginning of the two-year period, calculate her final salary after the two years. Then, calculate what single percentage increase should be applied to that final salary to recover exactly the purchasing power she had at the beginning.

Unit Challenge. Imagine you are organizing the financing of a school cooperative and you must present a realistic economic plan in a spreadsheet or digital document.

1. Real data and variation: Check the official current year-on-year inflation rate (CPI) on the INE website. Apply this percentage change to update a baseline budget of $1500$ €.

2. Direct and inverse modelling: Write the direct function of transport cost according to distance ($y = 0,18x$). Then, using GeoGebra or a spreadsheet, plot the inverse proportionality function of the cost per member when renting a $360$ € stand based on the number of participants ($y = 360/x$).

3. Proportional distribution and interest: Distribute $450$ € in profits directly proportionally to the hours dedicated by three members ($10$, $15$, and $25$ hours). Then, compare in a table the final capital obtained by depositing a fund of $1000$ € for $3$ years at $2,5$ % simple interest versus compound interest.

4. Final proposal: Write a report justifying which savings formula is most advantageous and attach your tables and graphs.

Assessment criteria:

• Mathematical precision in the calculations of percentages, proportional distribution, and interest.

• Technical accuracy in modelling and graphing inverse proportionality.

• Rigour in integrating real INE data and coherence in the report's conclusions.