// Numbas version: exam_results_page_options
{"name": "cash flow amount - ordinary annuity", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "functions": {}, "statement": "<p>If you are unsure of how to do&nbsp;a question, click on&nbsp;<em>Reveal answers</em>&nbsp;to see the full working. Then, once you understand how to do the question, click on&nbsp;<em>Try another question like this one</em>&nbsp;to start again. Do each question repeatedly to ensure you have mastered it.</p>", "extensions": [], "tags": [], "rulesets": {}, "parts": [{"scripts": {}, "prompt": "<p>Suppose you borrow&nbsp;$\\$\\var{P}$ to buy a house. The term of the loan is $\\var{years}$ years, and you will need to make {period[0]} repayments on the loan at the end of each&nbsp;{period[2]}. Interest is $\\var{ipa}\\%$ per annum compounding {period[0]}. Calculate the size of your&nbsp;monthly repayment.</p>\n<p></p>\n<p>$\\$$&nbsp;[[0]] (to the nearest cent)</p>", "gaps": [{"scripts": {}, "useCustomName": false, "mustBeReducedPC": 0, "customMarkingAlgorithm": "", "unitTests": [], "precisionMessage": "You have not given your answer to the nearest cent.", "precision": "2", "correctAnswerStyle": "plain", "customName": "", "minValue": "C", "showCorrectAnswer": true, "precisionType": "dp", "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "showPrecisionHint": false, "showFeedbackIcon": true, "mustBeReduced": false, "strictPrecision": true, "notationStyles": ["plain", "en", "si-en"], "marks": 1, "precisionPartialCredit": 0, "type": "numberentry", "correctAnswerFraction": false, "maxValue": "C", "variableReplacementStrategy": "originalfirst", "allowFractions": false}], "unitTests": [], "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "showFeedbackIcon": true, "type": "gapfill", "marks": 0, "useCustomName": false, "showCorrectAnswer": true, "customName": "", "sortAnswers": false, "variableReplacementStrategy": "originalfirst"}], "variables": {"n": {"group": "Ungrouped variables", "definition": "years*period[1]", "name": "n", "description": "", "templateType": "anything"}, "period": {"group": "Ungrouped variables", "definition": "['monthly',12,'month']", "name": "period", "description": "", "templateType": "anything"}, "ipa": {"group": "Ungrouped variables", "definition": "ipadec*100", "name": "ipa", "description": "", "templateType": "anything"}, "ipadec": {"group": "Ungrouped variables", "definition": "random(0.01..0.06#0.0001)", "name": "ipadec", "description": "", "templateType": "anything"}, "years": {"group": "Ungrouped variables", "definition": "random(20,25,30)", "name": "years", "description": "", "templateType": "anything"}, "C": {"group": "Ungrouped variables", "definition": "P*(ipadec/period[1])/(1-1/(1+ipadec/period[1])^n)", "name": "C", "description": "", "templateType": "anything"}, "Crounded": {"group": "Ungrouped variables", "definition": "precround(C,2)", "name": "Crounded", "description": "", "templateType": "anything"}, "P": {"group": "Ungrouped variables", "definition": "random(100000..1000000#1000)", "name": "P", "description": "", "templateType": "anything"}}, "preamble": {"js": "", "css": ""}, "name": "cash flow amount - ordinary annuity", "ungrouped_variables": ["years", "period", "ipadec", "ipa", "n", "P", "C", "Crounded"], "metadata": {"description": "Financial maths. Repayments on a home loan.<br>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>You are asked to find the cash flow amount&nbsp;of an ordinary annuity (since the payments are at the end of each&nbsp;period) and we are given the present value of the annuity. Therefore we will use the present value of an ordinary annuity formula&nbsp;</p>\n<p>$\\displaystyle P=\\frac{C}{i}\\left(1-\\frac{1}{(1+i)^n}\\right)$</p>\n<p>where $P$ is the present value, $C$ is the cash flow 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>$P=\\var{P}$,</p>\n<p>$i=\\frac{\\var{ipa}\\%}{\\var{period[1]}}=\\frac{\\var{ipadec}}{\\var{period[1]}}$,&nbsp;</p>\n<p>$n=\\var{years}\\times \\var{period[1]}=\\var{n}$,&nbsp;</p>\n<p>and therefore we have</p>\n<p>$\\displaystyle \\var{P}=\\frac{C}{\\left(\\simplify[unitDenominator]{{ipadec}/{period[1]}}\\right)}\\left(1-\\frac{1}{\\left(1+\\simplify[unitDenominator]{{ipadec}/{period[1]}}\\right)^\\var{n}}\\right)$</p>\n<p>which we need to rearrange to solve for $C$.</p>\n<p></p>\n<p>We want to get $C$ by itself.&nbsp;We start by multiplying&nbsp;both sides by $\\left(\\simplify[unitDenominator]{{ipadec}/{period[1]}}\\right)$ (to remove the&nbsp;division&nbsp;of $C$ by it&nbsp;on the right hand side)</p>\n<p>$\\displaystyle \\var{P}\\left(\\simplify[unitDenominator]{{ipadec}/{period[1]}}\\right)=C\\left(1-\\frac{1}{\\left(1+\\simplify[unitDenominator]{{ipadec}/{period[1]}}\\right)^\\var{n}}\\right)$</p>\n<p>Now we will divide both sides by $\\left(1-\\frac{1}{\\left(1+\\simplify[unitDenominator]{{ipadec}/{period[1]}}\\right)^\\var{n}}\\right)$ (to remove the multiplication by it on $C$)</p>\n<p>$\\displaystyle \\frac{\\var{P}\\left(\\simplify[unitDenominator]{{ipadec}/{period[1]}}\\right)}{\\left(1-\\frac{1}{\\left(1+\\simplify[unitDenominator]{{ipadec}/{period[1]}}\\right)^\\var{n}}\\right)}=C$</p>\n<p>Calculating this we find&nbsp;</p>\n<p>$\\begin{align}C&amp;\\approx \\var{C}\\\\&amp;=\\$\\var{Crounded}\\quad \\text{(to the nearest cent)}\\end{align}$</p>", "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}