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We are asked to find the interest rate per annum using compound interest. Therefore we will use the compound interest equation
\n$S=P(1+i)^n$,
\nwhere $S$ is the future value, $P$ is the present value, $i$ is the interest rate per time period and $n$ is the number of time periods.
\nIn our situation we have,
\n$S=\\var{S}$
\n$P=\\var{P}$,
\n$n=\\var{years}\\times\\var{period[1]}=\\var{n}$,
\nand therefore we have
\n$\\var{S}=\\var{P}\\left(1+r\\right)^\\var{n}$,
\nwhich we need to rearrange to solve for $i$.
\n\nWe want to get $i$ by itself. We start by dividing both sides by $\\var{P}$ (to remove the multiplication by $\\var{P}$)
\n$\\displaystyle \\frac{\\var{S}}{\\var{P}}=\\left(1+i\\right)^\\var{n}$.
\n\nNext we remove the power on the right hand side by raising both sides to the power of $\\frac{1}{\\var{n}}$ (this is equivalent to applying $\\sqrt[\\var{n}]{\\phantom{XX}}$ to both sides)
\n$\\displaystyle\\left(\\frac{\\var{S}}{\\var{P}}\\right)^{\\frac{1}{\\var{n}}}=\\left(\\left(1+i\\right)^\\var{n}\\right)^{\\frac{1}{\\var{n}}}$.
\nWhich simplifies (by an index law) to
\n$\\displaystyle\\left(\\frac{\\var{S}}{\\var{P}}\\right)^{\\frac{1}{\\var{n}}}=1+i$.
\nNext to get $i$ by itself we subtract $1$ from both sides to get
\n$\\displaystyle\\left(\\frac{\\var{S}}{\\var{P}}\\right)^{\\frac{1}{\\var{n}}}-1=i$.
\n\nRecall that $i$ is the interest rate per time period but we were asked for the interest rate per annum so we multiply by $\\var{period[1]}$ to get
\n$\\begin{array}\\displaystyle \\text{interest rate pa}&=\\var{period[1]}\\left(\\left(\\frac{\\var{S}}{\\var{P}}\\right)^{\\frac{1}{\\var{n}}}-1\\right)\\\\&\\approx\\var{ipadec}\\\\&=\\var{iparounded}\\%\\quad \\text{(2 decimal places)}\\end{array}$
", "rulesets": {}, "extensions": [], "functions": {}, "ungrouped_variables": ["P", "S", "years", "period", "n", "ippdec", "ipadec", "ipa", "iparounded"], "name": "interest rate - compound interest", "tags": [], "variablesTest": {"condition": "", "maxRuns": 100}, "variable_groups": [], "variables": {"years": {"name": "years", "group": "Ungrouped variables", "definition": "random(3..15)", "description": "", "templateType": "anything"}, "S": {"name": "S", "group": "Ungrouped variables", "definition": "P+random(3000..10000#500)", "description": "", "templateType": "anything"}, "ippdec": {"name": "ippdec", "group": "Ungrouped variables", "definition": "(S/P)^(1/n)-1", "description": "", "templateType": "anything"}, "ipadec": {"name": "ipadec", "group": "Ungrouped variables", "definition": "period[1]*((S/P)^(1/n)-1)", "description": "", "templateType": "anything"}, "iparounded": {"name": "iparounded", "group": "Ungrouped variables", "definition": "precround(ipa,2)", "description": "\\var{period[2]}\\left(\\left(\\frac{\\var{S}}{\\var{P}}\\right)^{\\frac{1}{\\var{n}}}-1\\right)
", "templateType": "anything"}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "years*period[1]", "description": "", "templateType": "anything"}, "period": {"name": "period", "group": "Ungrouped variables", "definition": "random([random('half-yearly','semiannually'),2],['quarterly', 4],['monthly',12],['daily', 365])", "description": "", "templateType": "anything"}, "ipa": {"name": "ipa", "group": "Ungrouped variables", "definition": "ipadec*100", "description": "", "templateType": "anything"}, "P": {"name": "P", "group": "Ungrouped variables", "definition": "random(1000..100000#1000)", "description": "present value
", "templateType": "anything"}}, "statement": "If you are unsure of how to do a question, click on Reveal answers to see the full working. Then, once you understand how to do the question, click on Try another question like this one to start again. Do each question repeatedly to ensure you have mastered it.
", "parts": [{"customName": "", "type": "gapfill", "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "marks": 0, "unitTests": [], "gaps": [{"precision": "2", "maxValue": "ipa", "mustBeReducedPC": 0, "type": "numberentry", "correctAnswerStyle": "plain", "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "minValue": "ipa", "showCorrectAnswer": true, "customName": "", "customMarkingAlgorithm": "", "showPrecisionHint": false, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "marks": 1, "precisionPartialCredit": 0, "precisionType": "dp", "precisionMessage": "You have not given your answer to two decimal places.", "strictPrecision": false, "scripts": {}, "extendBaseMarkingAlgorithm": true, "useCustomName": false, "mustBeReduced": false, "unitTests": [], "variableReplacements": [], "correctAnswerFraction": false}], "sortAnswers": false, "scripts": {}, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "variableReplacements": [], "showCorrectAnswer": true, "prompt": "Suppose $\\var{years}$ years ago you deposited $\\$\\var{P}$ into a bank account and today the balance accumulated is $\\$\\var{S}$. If the interest was compounded {period[0]} what was the rate of compound interest per annum?
\n\n[[0]] $\\%$ (to two decimal places)
", "useCustomName": false}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}