Interest
Two brothers win 50 000 € in a competition and share it out. One invests his share at 3 % compound and the other at 4 %. If after 5 years they have the same amount of money, how much did each one receive? To solve this you need to master both types of interest and know how to handle them as equations.
Simple interest: a quick recap
Calculate the simple interest generated by $4500$ € at $3,5\%$ per annum over $4$ years.
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…
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?
What initial principal needs to be invested at a simple annual interest rate of $4\%$ to obtain $600$ € in profit after $3$ years?
What initial principal needs to be invested at a simple annual interest rate of $4\%$ to obtain $600$ € in profit after $3$ years?
Compound interest: interest earns 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…
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 What will the final balance be after $5$ years if the interest earned is not withdrawn?
Calculate the final balance obtained by investing $2500$ € at a compound interest rate of $1,8\%$ per annum over $6$ years.
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…
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.
Finding the principal, the rate or the time
With the same formulae you can set up an equation when any quantity is missing, even with two unknowns shared between two investments.
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…
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?
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?
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 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 $4\%$…
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?
Non-annual compounding: a first idea of the APR (TAE)
The TIN (nominal interest rate) is the nominal rate. The TAE (annual equivalent rate, APR) lets you compare products by taking into account the frequency of payments, commissions and certain charges. In the simplified model below, with no charges or commissions and with reinvestment of interest, it coincides with the effective annual rate. It is not an adjustment for inflation.
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?
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 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 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.
Depreciation: compound interest in reverse
A van, a mobile phone or a computer loses value every year at a constant percentage rate: it is the same formula as compound interest, but subtracting instead of adding.
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…
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?
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 $819,20$…
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.
Compare investments with a spreadsheet
To compare the $8000$ € at $2{,}5\,\%$ over five years, type Year, Simple, Compound in A1:C1. In A2:A7 enter 0, 1, 2, 3, 4, 5. In B2 type =8000*(1+0,025*A2) and in C2 =8000*(1+0,025)^A2; copy both down to row 7. Adapt the decimal separator if your application requires it. Format the cells to two decimal places without rounding the internal values.
Plot the two series with years on the horizontal axis and the amount on the vertical axis. In year 0 both are worth $8000$ €; in year 1, $8200$ €; in year 5, $9000$ € and $9051{,}27$ €. Change the rate in both models and explain what happens. Always compare the same time period, and check commissions and conditions before interpreting the calculated amount as money available to you: no charges or taxes have been included here.
Source for distinguishing TIN and TAE: Banco de España.
Back to the start: If the elder receives $x$ €, we set up $x \cdot 1{,}03^5 = (50\,000 - x) \cdot 1{,}04^5$; on working it out, $1{,}1593 x = 60\,833 - 1{,}2167 x$ gives that the elder received approximately $25\,604$ € and the younger $24\,396$ €.