Exercises: Interest

Calculate the simple interest generated by $4500$ € at $3,5\%$ per annum over $4$ years.

A farmer takes out a loan of $12000$ € at a simple annual interest rate of $5\%$ to buy new machinery. If he repays the money in a single payment after $18$ months, how much will he pay in interest?

An amount of $8000$ € is deposited into a bank product at a compound interest rate of $2,5\%$ per annum. What will the final balance be after $5$ years if the interest earned is not withdrawn?

What initial principal needs to be invested at a simple annual interest rate of $4\%$ to obtain $600$ € in profit after $3$ years?

A language school invests $6500$ € at simple interest for $2$ years and makes a total profit of $455$ €. What was the annual interest rate of the investment?

A family finances the installation of solar panels costing $15000$ € at a compound interest rate of $4\%$ per annum over $3$ years. What total amount will they have to pay back at the end of the period?

Calculate the final balance obtained by investing $2500$ € at a compound interest rate of $1,8\%$ per annum over $6$ years.

If you invest $1000$ € at $5\%$ per annum for exactly $1$ year, is there any difference in the final balance obtained depending on whether the interest applied is simple or compound? Justify your answer mathematically.

Mateo wants to calculate the interest produced by a fixed-term deposit of $2400$ € at a $3\%$ annual simple interest rate over a period of $8$ months. On his exam, he writes the following:

And concludes that he will earn $576$ € in those $8$ months.

1. a) Explain with reasons why Mateo's result makes no financial or logical sense, estimating how much he would earn in a full year.
2. b) Point out the specific mathematical error he made when applying the formula.
3. c) Correct the procedure and calculate the exact amount of interest Mateo will obtain at the end of the $8$ months.

According to estimates from the automotive sector in Spain (GANVAM and DGT associations), a passenger car loses an average of $16\%$ of its value each year due to depreciation and wear and tear. A company purchases a new delivery van for a price of $20\,000$ €.

1. a) Before calculating with the formula, make a reasoned estimate of whether, after $3$ years, the vehicle will be worth more or less than half of its original price ($10\,000$ €).
2. b) Model the evolution of the vehicle's value using the annual depreciation formula and calculate its exact value after $3$ years.
3. c) Compare the obtained value with your initial estimate and explain why the accumulated depreciation over $3$ years is not equivalent to subtracting $16\%$ three times (that is, $48\%$ of $20\,000$ €).

Compare the growth of an investment of $5000$ € at $6\%$ simple interest versus $6\%$ compound interest over $10$ years. Look at the attached graph showing the progress of both investments and determine, by calculating the numerical values, what the exact difference in final balance will be between the two at the end of the period.

A transport company deposited $48000$ € at a simple interest rate of $3,2\%$ per annum and earned total interest of $5376$ €. How many months was the money invested in the bank?

A university student wants to have $15000$ € in $4$ years' time to do a master's degree abroad. If their bank offers a compound interest deposit of $2,2\%$ per annum, how much money must they deposit today to achieve their goal?

An amount of $10000$ € is invested at a compound interest rate of $4,8\%$ per annum, but compounded monthly. What will the final balance be after $3$ years?

A new delivery van costs $24000$ €. Due to wear and tear and time, its value is estimated to depreciate at a rate of $12\%$ compounded annually. What will its estimated value be after $5$ years?

When investing $6000$ € at compound interest for $4$ years, a final balance of $6753,05$ € was obtained. What annual interest rate was applied to the transaction?

A bank offers a deposit at a nominal annual interest rate of $3\%$ with interest paid quarterly. Calculate the Annual Equivalent Rate (AER) corresponding to this deposit.

A student decides to invest the $2000$ € she saved from her first summer job and splits it into two equal parts. She deposits the first half into a youth account at $4\%$ simple annual interest and the second half into a deposit at $3,5\%$ compound annual interest. Which half will have yielded greater returns after $8$ years?

A family has $6000$ € in savings that they will not need for the next $3$ years and compares two options offered in the Spanish banking market with no maintenance fees:

• Option A (Simple interest deposit): Fixed rate of $3,2\%$ per year, paying out all interest at the end of the $3$ years.

• Option B (Quarterly compounding account): Nominal interest rate (NIR) of $3\%$ per year, whose interest is compounded and reinvested every quarter.

1. a) Model the final accumulated capital that would be obtained with each option at the end of the $3$ years.
2. b) Compare the two results and choose with justification the most financially advantageous method.
3. c) It is often assumed that compound interest always beats simple interest. Explain why the opposite happens in this specific situation, analyzing the difference in interest rates compared to quarterly compounding.

An investor placed an initial capital $C$ in an annual compound interest fund for $4$ years. At the end of that time, the bank statement shows that the final capital represents a total cumulative increase of $21,55\%$ relative to the initial amount invested.

1. a) Write the equation that relates the annual interest rate $r$ to this cumulative increase after the $4$ years and justify why it is not necessary to know the numerical value of the initial capital $C$ to state it.
2. b) Check with justification which of the following integer annual interest rates is the exact solution to your equation: $r = 3\%$, $r = 4\%$, or $r = 5\%$.
3. c) If the investment had been made at simple interest over the same $4$ years obtaining exactly the same $21,55\%$ total gain, would the annual interest rate have had to be greater or lower than the one found in the previous section? Justify it conceptually without doing long calculations.

Two siblings win a prize of $50000$ € in an innovation competition and split it between them. The older sibling invests their share at $3\%$ compound annual interest, and the younger sibling invests theirs at $4\%$ compound annual interest. After $5$ years, both have exactly the same final balance. Set up the corresponding system of equations and calculate how much each initially received.

A young woman has $12000$ € saved from her first jobs. She puts part of it into a deposit at $5\%$ simple annual interest and the rest into a savings account at $4\%$ simple annual interest. After $3$ years, the total interest generated by both products sums to $1680$ €. How much did she deposit into each?

The value of computer equipment depreciates continuously at a fixed percentage rate. If $2$ years after purchase its value is $1280$ € and after $4$ years it is $819,20$ €, assuming a constant annual rate of depreciation, calculate the original price of the equipment and the percentage depreciation it undergoes each year.

Bank A offers a deposit with a nominal annual interest rate of $4,5\%$ with monthly compounding. Bank B offers a deposit with annual compounding. What nominal annual interest rate must Bank B offer so that the Annual Equivalent Rate (AER / TAE) of both offers is exactly the same?

A person invests $5000$ € at a compound interest rate of $3\%$ per annum. However, the average annual inflation during that period is $2\%$, which means that the purchasing power of money decreases cumulatively. Calculate the nominal final capital after $10$ years and then determine its real value (purchasing power equivalent to that of the initial year).

A saver has $18\,000$ € and analyzes two investment alternatives in Spain for a fixed term of $5$ years:

• Alternative 1 (Parking space): Buys a parking space for $18\,000$ € and rents it out for $95$ € per month for the full $5$ years (monthly rents are saved without generating further interest). In addition, it is estimated that the property's value will increase at an annual compound appreciation rate of $2\%$.

• Alternative 2 (Government bonds): Invests the $18\,000$ € in government debt at an annual compound interest rate of $3,4\%$ for $5$ years, reinvesting the returns each year.

1. a) Model and calculate the total accumulated capital after the $5$ years with Alternative 1 (adding the estimated final sale value of the parking space and all the rents collected).
2. b) Model and calculate the final capital obtained after the $5$ years with Alternative 2.
3. c) Determine which option is financially more profitable and interpret which real-world factors (such as unrented months, community fees, or taxes) could alter the final decision.

A person splits a capital of $30\,000$ € between two different bank products, $C_1$ and $C_2$, for a period of $4$ years:

• The amount $C_1$ is deposited at $4\%$ annual simple interest.

• The remaining amount, $C_2$, is placed at $3,5\%$ annual compound interest with annual compounding periods.

After the $4$ years, both deposits have produced exactly the same amount of interest earned ($I_1 = I_2$).

1. a) Before doing algebraic calculations, make a reasoned estimate of which of the two initial capitals ($C_1$ or $C_2$) must be greater for the final gains to be equal, comparing the relative return of each model.
2. b) Set up an equation with a single unknown (in terms of $C_1$) that models the equality of the interest earned by both investments and solve it to find the exact amount allocated to each product.
3. c) Check whether your initial estimate was correct and calculate the total combined gain that the $30\,000$ € have generated at the end of the $4$ years.