Exercises: Simple interest

You have decided to save part of what you received for your birthday in a youth account offering $3 \%$ annual simple interest. If you deposit an initial principal of $400 €$ and keep the money in the bank for $2$ years, how many euros of interest will you have generated by the end of that time?

An investment platform for minors allows your parents to open a savings fund in your name. They have deposited $1500 €$ at $2,5 \%$ annual simple interest. If the money stays there for $4$ years until you start university, what will be the final amount in the account (the initial money plus interest)?

You want to buy a new laptop and you are short of $600 €$. Your uncle lends you that money at a very low simple interest rate of $1,2 \%$ per annum, on condition that you repay him in 6 months. Calculate the interest you will have to pay him at the end of that period.

(Remember to use the formula adapted for months: $I = \frac{C \cdot r \cdot t}{1200}$)

A youth association has deposited the proceeds from a charity fair, a total of $2400 €$, in a deposit account yielding $1,8 \%$ per annum. If they need to withdraw the money after 100 days to fund a summer activity, how much interest will they have generated?

(Use the 360-day commercial year for the calculation).

Marta has saved $200 €$ over $3$ years and the bank has paid her a total of $12 €$ in interest. At what annual interest rate was her money deposited?

Through a banking app, Kofi sees that he has earned $15 €$ in interest in one year with a deposit paying $1,5 \%$ annual simple interest. What initial principal did Kofi deposit one year ago to obtain that return?

Imagine you have two options to keep $500 €$ from a prize for $2$ years:

• Option A: An account at $4 \%$ annual simple interest.

• Option B: An account that gives you a single payment of $35 €$ interest at the end of the two years.

Calculate the interest for Option A and determine which of the two is more profitable for you.

Calculate how long a principal of $1000 €$ must be deposited at an annual rate of $2 \%$ so that the interest produced is $60 €$. Express the result in years.

Lucía has saved $600\text{ €}$ and deposits them in an interest-bearing bank account at an annual simple interest rate of $4\%$ for $9$ months. To calculate the profit she will earn, Lucía performs the following calculation in her notebook:

Upon seeing the result, Lucía claims with great confidence that in less than a year she has gained almost half of what she deposited. Identify the conceptual error Lucía has made, explain in a reasoned way why her result is absurd considering that the rate is annual, and correctly calculate the interest she will receive.

Carlos deposits $800\text{ €}$ into a fixed-term bank deposit for $3$ years with an annual simple interest return of $2,8\%$.

1. Before doing any written calculations using the formula, make a mental estimate: round the annual interest to $3\%$, mentally calculate how much it would produce in a single year, and estimate between which two tens of euros the total profit will be after the three years.

2. Now apply the simple interest formula to accurately find the interest $I$ generated and the accumulated final capital $C_f$.

3. Compare the exact value obtained with your prior estimate and assess in a reasoned way whether your initial approximation was accurate.

Lucía has decided to save the money she received at her 15th birthday party to go on an Interrail trip around Europe when she finishes school. She has deposited $2500$ € in a youth savings account that offers a simple interest rate of $3,5$ % per annum. If she plans to leave the money in the bank for exactly $3$ years, what interest will she earn to fund her trip?

A group of 3rd ESO students wants to buy a 3D printer for their technology project. The model they have chosen costs $1200$ €. A financial institution offers them a loan at a simple interest rate of $4,2$ % per annum, to be repaid in $18$ months. Calculate how much money they will have to pay in interest at the end of that period.

To upgrade its e-sports equipment, a local club has applied for a microcredit of $3600$ €. The provider applies a simple interest rate of $12$ % per annum. If the club repays the money after $120$ days of competition, how much will the interest come to? (Use a $360$-day commercial year).

Jorge's grandfather gave him a sum of money for passing all his subjects. Jorge invested that principal in a deposit account paying a simple interest rate of $3$ % per annum. After $4$ years, the bank paid him $180$ € in interest alone. What was the initial principal given by his grandfather?

A bank account for young entrepreneurs offers a return such that depositing $8000$ € for $3$ years yields $480$ € in interest. What annual interest rate is the bank applying in this promotion?

Sara wants to buy an electric bicycle costing $1500$ €. She currently has $1200$ € and has deposited it in an account with a simple interest rate of $5$ % per annum. How long (expressed in years) must she keep the money in the account so that the interest generated is $300$ €, thus completing the amount she needs for the bicycle?

A young content creator has saved $5000$ € and decides to invest it in a financial product that guarantees a simple interest rate of $2,2$ % per annum for $5$ years. Calculate the final amount (the sum of the initial principal plus interest) that the content creator will have at the end of the investment period.

A dance academy has bought a sound system worth $2400$ €. They have agreed to pay the amount in $10$ months' time with a surcharge of $6$ % annual simple interest. What will be the total amount that the academy must pay in a single installment at the end of the $10$ months?

To calculate the interest generated by a principal of $4800\text{ €}$ invested at an annual simple interest rate of $3,6\%$ for $15$ months, three 3rd-year ESO students propose different methods:

• Sofía proposes first converting the $15$ months into years in decimal form by dividing by $12$ ($t = 1,25\text{ years}$) and applying the general formula $I = \frac{C \cdot r \cdot t}{100}$.

• Mateo prefers to directly apply the formula with the divisor adapted for months: $I = \frac{C \cdot r \cdot t}{1200}$ using $t = 15$.

• Elena calculates the profit for a full year by finding $3,6\%$ of $4800\text{ €}$, divides that amount by $12$ to know how much is generated in a single month, and multiplies it by $15$.

Solve the problem following each of the three methods and verify that they yield the same result. Next, explain with justification which of the three methods you would recommend if the term had been $7$ months instead of $15$ months and what drawbacks the other two would present.

A family decides to open a savings account at an annual simple interest rate of $4\%$ to save for the down payment on a hybrid vehicle. They know they need to withdraw the money exactly $2$ and a half years ($30$ months) after depositing it. Upon closing the account after that period, the bank pays them a total final capital (initial principal plus accrued interest) of exactly $13\,200\text{ €}$.

1. Set up a linear equation where the unknown is the initial principal $C$, expressing the interest in terms of $C$.

2. Solve the equation to determine what amount of money they originally deposited.

3. Determine what percentage of the final capital obtained corresponds exclusively to the interest generated and check the consistency of the result with the elapsed time.

A group of friends wants to buy a professional sound system for their rehearsal space costing $2450 \text{ €}$. They have saved part of the money and decide to deposit it in an account that offers $4,5\%$ annual simple interest. Knowing that after $3$ years and $4$ months they have achieved exactly $2450 \text{ €}$ (counting their initial principal plus interest), how much money did they deposit initially?

A young entrepreneur of your age needs a loan of $1200 \text{ €}$ to launch her online custom T-shirt shop. She has two options from financial institutions:

• Bank A: Offers her the money at a simple interest rate of $6\%$ per annum for $4$ years.

• Bank B: Offers her the same interest rate of $6\%$ per annum, but with a quarterly compounding of interest that she must withdraw each time (functioning as simple interest each quarter) over $48$ months. In addition, Bank B charges a fixed arrangement fee of $15 \text{ €}$.

Calculate how much total interest she will pay in each option and determine which one is the most economical for her.

In Spain, bank interest is subject to a tax withholding rate of $19\%$. This means the bank retains $19\%$ of the profit (interest) and pays you the rest. If we deposit $8500 \text{ €}$ for $540$ days at an interest rate of $3,2\%$ per annum, what will be the net profit (after tax) we receive at the end of the period? Use a $360$-day commercial year.

Daniela has split the $2000 \text{ €}$ she earned from her first summer job into two different financial products. The first offers a simple interest rate of $4\%$ per annum and the second a simple interest rate of $5\%$ per annum. If after one year she has obtained a total return of $92 \text{ €}$, how much money did she invest in each product?

Calculate what annual interest rate a principal must be deposited at so that, after $15$ years, the interest earned is exactly equal to the initial principal deposited. Does the result depend on the value of the principal chosen? Justify your answer mathematically.

A young athlete aged $18$ has just signed his first contract with a club, under which he will receive a bonus of $5000 \text{ €}$ upon completion. If the club decides to pay him that bonus today, but discounting the simple interest that this money would generate in an account at $3,5\%$ per annum over the remaining $6$ years of his contract, what amount will the athlete receive now?

Marta has $5000\text{ €}$ and is considering purchasing $9$-month Treasury bills issued by the Spanish Public Treasury at an annual simple interest rate of $3,2\%$ (considering a commercial year of $360$ days or $12$ months of $30$ days). To make an informed financial decision, she must consider two facts from the real socioeconomic context:

• The Tax Agency applies an immediate tax withholding of $19\%$ on the gross interest earned (personal income tax on returns on movable capital).

• The National Statistics Institute (INE) forecasts a cumulative inflation rate of $1,8\%$ over those same $9$ months on the shopping basket.

• Mathematically model the situation and calculate the gross interest generated in the $9$ months and the net amount Marta will receive after applying the tax withholding.

• Find the net final capital available ($C_f = 5000 + I_{\text{net}}$).

• Calculate how much a basket of goods and services that costs exactly $5000\text{ €}$ today will cost in $9$ months, taking into account the $1,8\%$ inflation. Based on that comparison, justify whether Marta will have gained or lost real purchasing power at the end of the investment.

A youth collective has a common fund of $18\,000\text{ €}$ from community activities and decides to distribute it entirely between two ethical finance projects at simple interest for a period of $3$ years:

• Part of the money ($x$) is deposited in an agricultural cooperative that offers an annual simple interest rate of $3\%$.

• The rest of the money is invested in a solar energy community that yields an annual simple interest rate of $5,5\%$.

After $3$ years, the sum of the interest generated by both investments amounts to exactly $2\,070\text{ €}$.

1. Algebraically express the interest generated by each of the two investments in terms of $x$.

2. Set up and solve a linear equation to find out how much money was allocated to each of the two projects.

3. Qualitative reasoning: If the collective had wanted both investments to generate exactly the same amount of profit in euros after those $3$ years, in which of the two projects would they have had to place a larger amount of money? Justify your answer by analyzing the proportional relationship between principal and interest rate in the simple interest formula, without needing to calculate numerical figures.