Exercises: Simple and compound interest

Calculate the simple interest produced by a principal of $3000$ € invested at $4\%$ per annum for $5$ years. What will be the final amount at the end of this period?

A bank grants a loan of $1500$ € at $6\%$ annual simple interest. If the loan is repaid over a period of $8$ months, determine the total amount that must be paid to the bank as interest.

Calculate the final amount obtained by depositing $5000$ € into a savings account offering $3\%$ annual compound interest for $4$ years.

Find the principal initially invested at a simple interest rate of $5\%$ per annum if after $3$ years it generated $450$ € in profit.

A new motorbike has a price of $4000$ €. If it loses $10\%$ of its value each year, calculate what its exact value will be after $3$ years.

A principal of $2000$ € has grown to $2240$ € after remaining invested for $4$ years at simple interest. Find the annual interest rate applied.

A mobile phone battery loses $15\%$ of its charge for every hour of gaming. If you start playing with $100\%$ charge, determine what percentage of battery will remain after $5$ hours.

Calculate the simple interest generated by $10000$ € invested at $2,5\%$ per annum for $146$ days. Assume a year has $365$ days to perform the calculations.

Marcos wants to calculate the final capital he will obtain after depositing $4000$ € for $3$ years in a fixed-term deposit account with a compound interest of $2,5\%$ per year. On his exam, he writes the following work:

1. a) Explain the conceptual error Marcos made and what type of money growth he assumed in his approach.
2. b) Write the correct formula he should have applied, justifying why there should be a power instead of a product, and calculate the actual final capital he will obtain, rounding to the nearest euro cent.

A small neighborhood shop in Sevilla purchases a computer checkout system for $1200$ €. It is estimated that this equipment suffers a continuous depreciation of $20\%$ per year due to rapid technological obsolescence.

1. a) Before performing any operations with a calculator, make a reasoned mental estimate: will the equipment be worth more or less than half of its original price ($600$ €) after $3$ years have passed? Justify your conjecture by analyzing the approximate loss of value year by year.
2. b) Apply the depreciation formula ($V_f = V_0 \cdot (1 - \frac{r}{100})^t$) to calculate its exact value after $3$ years. Compare the result obtained with your previous estimate and assess whether your intuitive reasoning was correct.

Compare the return on a principal of $8000$ € invested for $6$ years at a rate of $5\%$ per annum under two different methods: simple interest and compound interest. What is the exact difference in euros between the interest generated by both methods?

A person wishes to have $10000$ € in $5$ years to buy a vehicle. If their bank offers a deposit with $4\%$ annual compound interest, how much money must they deposit today to achieve their goal exactly?

An initial principal of $6000$ € has grown to $7146,10$ € after $3$ years of investment. Knowing that the interest has been reinvested each year (compound interest), determine the annual interest rate applied.

An amount of $4000$ € is invested at a simple interest rate of $3\%$ per annum for $4$ years. After that time, the total final amount is withdrawn and reinvested in another financial product offering $4\%$ annual compound interest for $2$ more years. Calculate the overall final amount that will be obtained at the end of the $6$ years.

Determine how many years a principal must remain invested at a simple interest rate of $8\%$ per annum for its value to double.

An investor deposits $5000$ € at an annual compound interest rate of $6\%$, but with monthly compounding periods. Calculate the final amount they will have in their account after $2$ years.

A family spends $1200$ € per month on food. If the average inflation is estimated at $2,5\%$ per year, how much will they spend per month in $4$ years to maintain exactly the same level of consumption? Assume that price increases follow the compound interest formula.

Salma invests $3000$ € at $4\%$ per annum and Bruno invests $2500$ € at $6\%$ per annum, both at simple interest. Set up an equation and calculate in how many years they will have exactly the same accumulated final amount.

A person recently sold a second-hand industrial machine for $8403,50$ €. They know for certain that the machine was purchased new exactly $4$ years ago and that during this time it has suffered accumulated depreciation at a constant rate of $15\%$ per year.

1. a) Set up the equation for the value of the machine as a function of time and clearly identify the annual reduction factor that affects the initial price $V_0$.
2. b) Solve algebraically for the unknown value and calculate how much the machine cost completely new, rounding to the nearest euro.
3. c) If the depreciation had been calculated as a linear loss of $15\%$ per year on the initial purchase price (simple model), would it have cost the same? Reason which model of loss of value is more detrimental to the seller in the early years.

A family in Zaragoza decides to start a savings plan for their newborn daughter's future studies. They deposit an initial capital of $5000$ € in a bank at a compound interest of $3,2\%$ per year. At the same time, the grandparents hand over $200$ € in cash each year to be kept in a piggy bank at home (without generating any interest).

1. a) Mathematically model the capital accumulated in the bank $C_b(t)$ and the money accumulated in the piggy bank $C_h(t)$ as a function of elapsed years $t$.
2. b) Calculate how much money will be in each of the two funds when the girl turns $18$ years old.
3. c) If the grandparents had deposited the total $3600$ € ($18 \cdot 200$ €) from day one into a savings account at a simple interest of $2\%$ per year for those same $18$ years, would they have reached the amount obtained by the parents' bank fund? Justify your conclusion with the corresponding calculations.

Set up the corresponding equation and use logarithms to determine how many years must elapse for a principal of $5000$ € to grow to $7000$ € if invested at an annual compound interest rate of $3,5\%$.

A person deposits $2000$ € into an account offering $4\%$ annual compound interest. After exactly $2$ years, they make a new deposit of $1500$ € into the same account. If they make no withdrawals, calculate the total accumulated amount they will have after $5$ years from the first deposit.

To finance a project, a company evaluates two loan options for an amount of $10000$ € to be repaid in a single payment after $3$ years. Bank A offers a simple interest rate of $6\%$ per annum, but charges an arrangement fee of $1,5\%$ on the principal, paid at the time of granting. Bank B offers a compound interest rate of $5\%$ per annum with no fee. Determine which option is cheaper at the end of the period and what the difference in euros is between both.

Calculate what annual compound interest rate must be applied to a principal for its value to triple in exactly $15$ years. Express the result as a percentage rounded to two decimal places.

Carlos invests $4000$ € at an annual compound interest rate of $5\%$. On the same day, Diana invests $5000$ € at an annual compound interest rate of $3\%$. Use logarithms to determine after how many years the accumulated amount for Carlos will equal Diana's.

A high-end computer system currently costs $1500$ €. Due to technological obsolescence, its nominal value depreciates by $15\%$ each year. At the same time, inflation causes money to lose $2\%$ of its purchasing power annually. Calculate what the real value (adjusted for the loss of purchasing power) of the system will be after $4$ years.

An investor wishes to accumulate a final amount of $20000$ € in $4$ years. To achieve this, they plan to make an initial deposit of an amount $C$ today, and a second deposit of the same amount $C$ in exactly $2$ years. If the bank account offers a yield of $3\%$ annual compound interest, set up the corresponding equation and find the value of the contribution $C$.

The following graph shows the evolution of the same initial principal invested at simple interest (blue bars) and compound interest (green bars) with the same annual interest rate over the first three years. By observing the exact values on the graph, deduce the initial principal and the interest rate. Next, calculate what the exact difference in euros between both methods will be after $10$ years.

You need to determine the minimum number of complete years that must pass for an initial investment of $10\,000$ €, placed at $4\%$ annual compound interest, to generate an accumulated final capital of at least $15\,000$ €. To solve this question, two classmates propose two different approaches:

• Method A (Numerical iterative with table): Calculate the capital year by year by successively multiplying by the growth factor until exceeding $15\,000$ €.

• Method B (Analytical with logarithms): Set up the corresponding exponential equation and solve for time $t$ using logarithms.

1. a) Solve the problem applying both methods: for Method A, show the capital values in the years immediately before and after exceeding the goal; for Method B, obtain the exact decimal value of $t$.
2. b) Evaluate both methods: what advantages and disadvantages does each approach present if the goal were to find the time needed to multiply the capital by ten instead of increasing it by $50\%$? Justify which of the two methods is more efficient and generalizable in financial mathematics.

A courier company in Valencia renews its fleet by purchasing a low-emission van worth $36\,000$ €, which suffers an estimated depreciation of $12\%$ per year. At the same time, to anticipate its future replacement, the company deposits $12\,000$ € into a corporate investment account that yields $4\%$ annual compound interest.

1. a) Write the algebraic equation that models the exact moment $t$ (in years) when the residual market value of the vehicle will equal the capital accumulated in the investment account.
2. b) Algebraically simplify said equation by grouping the terms containing the unknown $t$ into a single quotient of powers of the form $\left(\frac{a}{b}\right)^t = k$.
3. c) Solve for $t$ by applying the properties of logarithms and calculate its value rounded to two decimal places. From this data, determine in approximately which year and month that financial break-even will be reached.

Unit Challenge. Cost Management for a School Trip.

Imagine you are coordinating the budget for a weekend class trip.

1. Real data and direct proportionality: Check the website of Red Eléctrica de España (REE) for today's average price of a kWh ($p$ in €/kWh). With a forecast consumption of $140$ kWh for the group, calculate the total cost of electricity by applying direct proportionality.

2. Inverse proportionality function: Facility rental incurs a fixed cost of $k = 420$ €. In GeoGebra or a spreadsheet, model the function $f(x) = \frac{420}{x}$, where $x$ is the number of attendees. Generate a table and plot the curve for $x$ between $6$ and $25$ participants, explaining the behaviour of the function.

3. Proportional divisions: The cost of shared transport ($180$ €) is funded through an inversely proportional division based on the volunteer hours contributed by the three organising committees: $2$, $3$, and $6$ hours. Determine the amount to be paid by each committee.

4. Final report: Draft a proposal with the table, the graph, and a conclusion justifying the minimum number of students that must attend so that the individual fixed cost is below $25$ €.

Evaluation criteria:

• Accurate retrieval of real REE data and correct solving of direct and inverse proportionality.

• Appropriate modelling of the hyperbola $f(x) = k/x$ using a digital tool.

• Mathematical rigor in the inverse proportional division.

• Clarity and justification of the final proposal.