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{"name": "Present value - Annuity 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["perc", "int", "P", "n", "R", "Interest", "num", "perc2", "num2", "num3", "years"], "name": "Present value - Annuity 2", "tags": ["present value", "rebel", "REBEL", "rebelmaths"], "advice": "<p>The present value of an annuity, $P$ is given by:</p>\n<p></p>\n<p>$P=\\frac{R[1-(1+i)^{-n}]}{i}$</p>\n<p>where $R$ represents the periodic payment, $i$ represents the interest rate per period and $n$ represents the number of payments.&nbsp;</p>\n<p></p>\n<p>$P$ represents the present value of the annuity, this is what we are asked&nbsp;to calculate.</p>\n<p>$i$&nbsp;represents the rate of compound interest. the annual interest rate is $\\var{perc}$% so the monthly rate of interest is $\\frac {\\var{perc}} {12}=\\var{perc2}$% and therefore $i=\\frac{\\var{perc2}}{100}=\\var{int}$.</p>\n<p>$n$<b><i>&nbsp;</i></b>represents the number of payments, there are 12 payments over $\\var{years}$ year(s) so $n$ is $12 \\times \\var{years}=\\var{n}$.</p>\n<p>The value of each repayment, is &euro;$\\var{R}$&nbsp;</p>\n<p></p>\n<p>Using the formula:</p>\n<p><span>$P=\\frac{R[1-(1+i)^{-n}]}{i}$</span></p>\n<p>$P=\\frac{\\var{R}[1-(1+\\var{int})^{-\\var{n}}]}{\\var{int}}$</p>\n<p><span>$P=\\frac{\\var{R}[1-\\var{num}]}{\\var{int}}$</span></p>\n<p><span>$P=\\frac{\\var{R}[\\var{num2}]}{\\var{int}}$</span></p>\n<p><span><span><span><span>$P = \\var{R}&nbsp;\\times \\var{num3}$</span></span></span></span></p>\n<p>$P=\\var{P}$</p>\n<p></p>", "rulesets": {}, "parts": [{"prompt": "<p>Find the present value of these payments if the annual interest rate is $\\var{perc}$% compounded monthly. Give your answer to the nearest cent.</p>\n<p>&euro;[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "P+0.01", "minValue": "P-0.01", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": "5", "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "extensions": [], "statement": "<p>A $\\var{years}$-year lease for a company car requires a payment of &euro;$\\var{R}$&nbsp;at the end of each month.&nbsp;</p>\n<p></p>\n<p>The formula for calculating the present value ($P$) of an annuity is:</p>\n<p>$P=\\frac{R[1-(1+i)^{-n}]}{i}$</p>\n<p>where $R$ represents the value of each&nbsp;repayment, $i$ represents the interest rate and $n$ represents the number repayments.</p>\n<p></p>", "variable_groups": [{"variables": [], "name": "Unnamed group"}], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"perc": {"definition": "random(4.5..8.5#0.5)", "templateType": "anything", "group": "Ungrouped variables", "name": "perc", "description": ""}, "perc2": {"definition": "precround(perc/12, 5)", "templateType": "anything", "group": "Ungrouped variables", "name": "perc2", "description": ""}, "num2": {"definition": "1-num", "templateType": "anything", "group": "Ungrouped variables", "name": "num2", "description": ""}, "num3": {"definition": "num2/int", "templateType": "anything", "group": "Ungrouped variables", "name": "num3", "description": ""}, "int": {"definition": "perc2/100", "templateType": "anything", "group": "Ungrouped variables", "name": "int", "description": ""}, "n": {"definition": "years*12", "templateType": "anything", "group": "Ungrouped variables", "name": "n", "description": ""}, "P": {"definition": "precround(R*((1-(1+int)^(-n))/int),2)", "templateType": "anything", "group": "Ungrouped variables", "name": "P", "description": ""}, "num": {"definition": "(1+int)^-n", "templateType": "anything", "group": "Ungrouped variables", "name": "num", "description": ""}, "Interest": {"definition": "n*R-P", "templateType": "anything", "group": "Ungrouped variables", "name": "Interest", "description": ""}, "R": {"definition": "random(280..400 # 10)", "templateType": "anything", "group": "Ungrouped variables", "name": "R", "description": ""}, "years": {"definition": "random(1..3)", "templateType": "anything", "group": "Ungrouped variables", "name": "years", "description": ""}}, "metadata": {"description": "<p>A 1-year lease for a company car requires a payment of &euro;280 at the end of each month. Find the present value of these payments if the annual interest rate is 7% compounded monthly.</p>\n<p>rebelmaths</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}]}]}], "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}]}