// Numbas version: exam_results_page_options
{"name": "Patricia's copy of future value - ordinary annuity", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Patricia's copy of future value - ordinary annuity", "tags": [], "metadata": {"description": "Financial maths. Future value of an ordinary annuity.", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Suppose you are given $\\$\\var{C}$ to deposit into a savings account at the end of each&nbsp;{period[2]}&nbsp;for $\\var{years}$&nbsp;years. If the interest rate is $\\var{ipa}\\%$ per annum compounding {period[0]}, how much money is in the savings account after&nbsp;$\\var{years}$&nbsp;years?</p>", "advice": "<p>You are asked to find the future&nbsp;value of an ordinary annuity (since the payments are at the end of each&nbsp;period). Therefore we will use the future value of an ordinary annuity formula</p>\n<p>$\\displaystyle A=R(\\frac{(1+i)^n-1}{i})$</p>\n<p>where $A$ is the future&nbsp;value, $R$ is the&nbsp;amount deposited&nbsp;per period, $i$ is the interest rate per period, and $n$ is the number of periods.</p>\n<p>In our&nbsp;situation we have,</p>\n<p>$R=\\var{C}$,</p>\n<p><span data-jme-visible=\"period[1]&gt;1\">$i=\\frac{\\var{ipa}\\%}{\\var{period[1]}}=\\frac{\\var{ipadec}}{\\var{period[1]}}$,</span> <span data-jme-visible=\"period[1]=1\">$i=\\var{ipa}\\%=\\var{ipadec}$,</span></p>\n<p><span data-jme-visible=\"period[1]&gt;1\">$n=\\var{years}\\times \\var{period[1]}=\\var{n}$,</span> <span data-jme-visible=\"period[1]=1\">$n=\\var{n}$,</span></p>\n<p>and therefore we have</p>\n<p>$\\displaystyle A=\\var{C}(\\frac{(1+\\frac{\\var{ipadec}}{\\var{period[1]}})^\\var{n}-1}{\\frac{\\var{ipadec}}{\\var{period[1]}}})$</p>\n<p>Calculating this we find&nbsp;</p>\n<p>$\\begin{align}A&amp;\\approx \\var{F}\\\\&amp;=\\$\\var{Frounded}\\quad \\text{(to the nearest cent)}\\end{align}$</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"period": {"name": "period", "group": "Ungrouped variables", "definition": "random([random('yearly','annually'),1,'year'],[random('half-yearly','semiannually'),2,'half-year'],['quarterly', 4, 'quarter'],['monthly',12,'month'],['daily', 365,'day'])", "description": "", "templateType": "anything", "can_override": false}, "C": {"name": "C", "group": "Ungrouped variables", "definition": "if(period[1]=365,random(1..10),random(20..300#10))", "description": "", "templateType": "anything", "can_override": false}, "ipa": {"name": "ipa", "group": "Ungrouped variables", "definition": "ipadec*100", "description": "", "templateType": "anything", "can_override": false}, "F": {"name": "F", "group": "Ungrouped variables", "definition": "(C*period[1]/ipadec)*((1+ipadec/period[1])^(n)-1)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "years*period[1]", "description": "", "templateType": "anything", "can_override": false}, "years": {"name": "years", "group": "Ungrouped variables", "definition": "random(3..20)", "description": "", "templateType": "anything", "can_override": false}, "Frounded": {"name": "Frounded", "group": "Ungrouped variables", "definition": "precround(F,2)", "description": "", "templateType": "anything", "can_override": false}, "ipadec": {"name": "ipadec", "group": "Ungrouped variables", "definition": "random(0.02..0.10#0.001)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["years", "period", "ipadec", "ipa", "C", "n", "F", "Frounded"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Which formula should we use to calculate the future value of the annuity in $\\var{years}$ years?</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["$\\displaystyle A=R \\left [\\frac{(1+i)^n-1}{i} \\right]$", "$\\displaystyle P=R \\left [\\frac{1-(1+i)^{-n}}{i} \\right ]$", "$\\displaystyle P=R(1+i) \\left[ \\frac{1-(1+i)^{-n}}{i} \\right]$", "$\\displaystyle A=R(1+i) \\left[\\frac{(1+i)^n-1}{i} \\right]$"], "matrix": ["1", 0, 0, "0.5"], "distractors": ["", "This answer is incorrect because the question asks for the future value of the annuity (A). ", "This answer is incorrect because the question asks for the future value of the annuity (A). ", "This answer is incorrect because the periodic payments are deposited at the end of each compounding period."]}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Amount in the savings account after $\\var{years}$&nbsp;years:</p>\n<p></p>\n<p>$\\$$&nbsp;[[0]] (to the nearest cent)</p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>You are asked to find the future&nbsp;value of an ordinary annuity (since the payments are at the end of each&nbsp;period). Therefore we will use the future value of an ordinary annuity formula</p>\n<p>$\\displaystyle A=R\\left[\\frac{(1+i)^n-1}{i}\\right]$</p>\n<p>where $A$ is the future&nbsp;value, $R$ is the&nbsp;amount deposited&nbsp;per period, $i$ is the interest rate per period, and $n$ is the number of periods.</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "F-1", "maxValue": "F+1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the nearest cent (2 decimal places).", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Catherine Palmer", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/423/"}, {"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Patricia Cogan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3359/"}]}]}], "contributors": [{"name": "Catherine Palmer", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/423/"}, {"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Patricia Cogan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3359/"}]}