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Financial Mathematics

rebel

rebelmaths

", "licence": "None specified"}, "type": "exam", "questions": [], "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": [{"name": "Financial Maths Formulae ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "functions": {}, "ungrouped_variables": ["a", "b", "ci", "ch1", "apr", "pv", "ch2", "ch3", "c", "ann", "ch4", "d"], "tags": ["financial", "rebel", "REBEL", "Rebel", "rebelmaths"], "advice": "

No solution given.

", "rulesets": {}, "parts": [{"maxAnswers": 0, "shuffleChoices": false, "matrix": [["1", 0, 0, 0], [0, "1", 0, 0], [0, 0, "1", 0], [0, 0, 0, "1"]], "shuffleAnswers": false, "minAnswers": 0, "marks": 0, "variableReplacements": [], "answers": ["

Compound Interest

", "

Present Value

", "

APR

", "

Annuity

"], "choices": ["{ch1}", "{ch2}", "{ch3}", "

{ch4}

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Match each of the following questions with the relevant formula.

", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"a": {"definition": "random(0..1)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "ci": {"definition": "\n [\"An individual intends to deposit \u20ac50,000 into a savings account. If no withdrawals are made from the account, it will have grown to \u20ac59,626 after four years. Calculate the annual rate of compound interest. \",\n \"A Lotto winner intends to deposit \u20ac125,000 into a savings account. If no withdrawals are made from the account, it will have grown to \u20ac170,061 after four years. Calculate the rate of compound interest per annum.\"\n ]\n ", "templateType": "anything", "group": "Ungrouped variables", "name": "ci", "description": ""}, "b": {"definition": "random(0..1)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "pv": {"definition": "[\"A payment of \u20ac2,000 is due to be made 4 years from today. If the rate of interest is 5% per annum, calculate the present value of the payment.\",\"Three years from today, on his twenty-first birthday, Tristan Snotte-Wragge will receive an inheritance of \u20ac50,000. If the rate of interest is 4%, what is the present value of Tristan\u2019s inheritance?\"]", "templateType": "anything", "group": "Ungrouped variables", "name": "pv", "description": ""}, "d": {"definition": "random(0..1)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "apr": {"definition": "[\"A savings account earns compound interest at 1.6% per month. Calculate the equivalent annual rate of interest.\", \"A savings account in Bank of Ruritania earns compound interest at 1.25% per month. Calculate the equivalent annual rate of interest.\"]\n ", "templateType": "anything", "group": "Ungrouped variables", "name": "apr", "description": ""}, "ann": {"definition": "[\"A company intends to make three equal deposits into a sinking fund, on January 1st 2016, January 1st 2017 and January 1st 2018. The fund will earn compound interest at 8% per annum. If no withdrawals are made from it, there will be \u20ac360,000 in the fund on December 31st 2018. Calculate the size of each deposit. \",\"On January 1st, 2015 a company borrowed \u20ac150,000 from a bank at 4.5% per annum compound interest. The loan must be repaid to the bank in three equal repayments, due on December 31st in 2015, 2016 and 2017. Calculate how large each repayment will need to be.\"]", "templateType": "anything", "group": "Ungrouped variables", "name": "ann", "description": ""}, "c": {"definition": "random(0..1)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "ch1": {"definition": "ci[a]", "templateType": "anything", "group": "Ungrouped variables", "name": "ch1", "description": ""}, "ch2": {"definition": "pv[b]", "templateType": "anything", "group": "Ungrouped variables", "name": "ch2", "description": ""}, "ch3": {"definition": "apr[c]", "templateType": "anything", "group": "Ungrouped variables", "name": "ch3", "description": ""}, "ch4": {"definition": "ann[d]", "templateType": "anything", "group": "Ungrouped variables", "name": "ch4", "description": ""}}, "metadata": {"description": "

Match the question with the correct formula.

rebelmaths

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Loan - compound interest 1 in euro", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "functions": {}, "ungrouped_variables": ["debt", "interest", "capital", "n"], "tags": ["compound interest", "financial", "REBEL", "rebel", "Rebel", "rebelmaths"], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"prompt": "

The principal, $P$, is the €[[0]] lent to the individual.

The final amount, $A$, is the €[[1]] paid back.

Since the individual has paid back €[[2]] more that s\\he borrowed, this is the interest.

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "{capital}", "minValue": "{capital}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "{interest}", "minValue": "{interest}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "{debt}", "minValue": "{debt}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "

A bank lends an individual €$\\var{capital}$. The individual has to pay back the bank €$\\var{interest}$ in $\\var{n}$ years time.

", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"n": {"definition": "random(2..5 #1)", "templateType": "anything", "group": "Ungrouped variables", "name": "n", "description": ""}, "debt": {"definition": "random(200..500#20)", "templateType": "randrange", "group": "Ungrouped variables", "name": "debt", "description": ""}, "interest": {"definition": "capital + debt", "templateType": "anything", "group": "Ungrouped variables", "name": "interest", "description": ""}, "capital": {"definition": "random(500..2000#20)", "templateType": "randrange", "group": "Ungrouped variables", "name": "capital", "description": ""}}, "metadata": {"description": "

Compound interest.

rebelmaths

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Once off payment Formulae (Financial Maths) ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "functions": {}, "ungrouped_variables": [], "tags": ["rebel", "Rebel", "REBEL", "rebelmaths"], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"maxAnswers": 0, "prompt": "

The Compound Interest Formula for savings is

$A=P(1+i)^n$

Match each letter to its meaning:

", "matrix": [["1", 0, 0, 0], [0, "1", 0, 0], [0, 0, "1", 0], [0, 0, 0, "1"]], "shuffleAnswers": true, "minAnswers": 0, "variableReplacements": [], "displayType": "radiogroup", "answers": ["

Amount in the deposit account after $n$ years

", "

Principal sum invested

", "

Rate of compound interest

", "

Number of compounding periods

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", "

"], "type": "m_n_x", "shuffleChoices": true, "minMarks": 0, "layout": {"expression": "", "type": "all"}}, {"maxAnswers": 0, "prompt": "

The Compound Interest Formula for loans is

$A=P(1+i)^n$

Match each letter to its meaning:

", "matrix": [["1", 0, 0, 0], [0, "1", 0, 0], [0, 0, "1", 0], [0, 0, 0, "1"]], "shuffleAnswers": true, "minAnswers": 0, "variableReplacements": [], "displayType": "radiogroup", "answers": ["

Amount owed to the bank after $n$ years

", "

Principal lent to the individual

", "

Rate of compound interest

", "

Number of compounding periods

"], "warningType": "none", "variableReplacementStrategy": "originalfirst", "maxMarks": 0, "showCorrectAnswer": true, "scripts": {}, "marks": 0, "choices": ["

", "

"], "type": "m_n_x", "shuffleChoices": true, "minMarks": 0, "layout": {"expression": "", "type": "all"}}, {"maxAnswers": 0, "prompt": "

The Present Value Formula is

$P=\\frac{A}{(1+i)^n}$

Match each letter to its meaning:

", "matrix": [["1", 0, 0, 0], [0, "1", 0, 0], [0, 0, "1", 0], [0, 0, 0, "1"]], "shuffleAnswers": true, "minAnswers": 0, "variableReplacements": [], "displayType": "radiogroup", "answers": ["

Future value

", "

Present value of investment

", "

Discounting rate

", "

Number of years

"], "warningType": "none", "variableReplacementStrategy": "originalfirst", "maxMarks": 0, "showCorrectAnswer": true, "scripts": {}, "marks": 0, "choices": ["

", "

"], "type": "m_n_x", "shuffleChoices": true, "minMarks": 0, "layout": {"expression": "", "type": "all"}}, {"maxAnswers": 0, "prompt": "

The APR Formula (Annual Percentage Rate) is

$APR=(1+i)^n-1$

Match each letter to its meaning:

", "matrix": [["1", 0, 0], [0, "1", 0], [0, 0, "1"]], "shuffleAnswers": true, "minAnswers": 0, "variableReplacements": [], "displayType": "radiogroup", "answers": ["

Annual Percentage Rate

", "

Interest rate per compounding period

", "

Number of compounding periods

"], "warningType": "none", "variableReplacementStrategy": "originalfirst", "maxMarks": 0, "showCorrectAnswer": true, "scripts": {}, "marks": 0, "choices": ["

APR

", "

"], "type": "m_n_x", "shuffleChoices": true, "minMarks": 0, "layout": {"expression": "", "type": "all"}}], "statement": "", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {}, "metadata": {"description": "

rebelmaths

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Regular Payment Formulae (Financial Maths) ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "functions": {}, "ungrouped_variables": [], "tags": ["rebel", "Rebel", "REBEL", "rebelmaths"], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"maxAnswers": 0, "prompt": "

Annuity Formulae:

$A=R\\left[\\frac{(1+i)^n-1}{i}\\right]$

and

$P=R\\left[\\frac{1-(1+i)^{-n}}{i}\\right]$

Match each letter to its meaning:

", "matrix": [["1", 0, 0, 0, 0], [0, "1", 0, 0, 0], [0, 0, "1", 0, 0], [0, 0, 0, "1", 0], [0, 0, 0, 0, "1"]], "shuffleAnswers": true, "minAnswers": 0, "variableReplacements": [], "displayType": "radiogroup", "answers": ["

Amount (Future Value)

", "

Regular payment

", "

Rate of interest per compounding period

", "

Number of compounding periods

", "

Present Value

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", "

"], "type": "m_n_x", "shuffleChoices": true, "minMarks": 0, "layout": {"expression": "", "type": "all"}}, {"maxAnswers": 0, "prompt": "

Annunity Due Formulae:

$P=R(1+i)\\left[\\frac{(1+i)^{n}-1}{i}\\right]$

and

$P=R(1+i)\\left[\\frac{1-(1+i)^{-n}}{i}\\right]$

Match each letter to its meaning:

Present Value

", "

Regular payment

", "

Rate of interest per compounding period

", "

Number of compounding periods

", "

Amount (Future Value)

"], "warningType": "none", "variableReplacementStrategy": "originalfirst", "maxMarks": 0, "showCorrectAnswer": true, "scripts": {}, "marks": 0, "choices": ["

", "

"], "type": "m_n_x", "shuffleChoices": true, "minMarks": 0, "layout": {"expression": "", "type": "all"}}], "statement": "

An annuity is a series of equal payments at regular intervals. Examples of annuities are regular deposits to a savings account, monthly home mortgage payments, monthly insurance payments and pension payments.

", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {}, "metadata": {"description": "

rebelmaths

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Savings compound interest 1 ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Catherine Palmer", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/423/"}, {"name": "Patricia Cogan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3359/"}], "tags": [], "metadata": {"description": "

Calculate the final amount in a savings account where compound interest is earned annually

", "licence": "None specified"}, "statement": "

A lump sum of €$\\var{P}$ is deposited into a savings account, that pays compound interest at a rate of $\\var{perc}$% per annum for $\\var{n}$ years. If no withdrawals are made from the account, then the amount that the lump sum will have grown to is given by the formula:

$\\ A = P(1+i)^n $

", "advice": "

The compound interest formula is: $\\ A = P(1+i)^n $

Part (a)

P represents the principal sum invested , so in this example it is €$\\var{P}$.

Part (b)

i represents the rate of compound interest, in this example, the annual interest rate is $\\var{perc}$% so i is $\\frac {\\var{perc}} {100}=\\var{int}$.

Part (c)

n represents the number of compounding periods , so in this example it is $\\var{n}$ years.

Part(d)

The amount in the deposit account after $\\var{n}$ years is denoted by A. Using the compound interest formula:

$A=P(1+i)^n$

$A=\\var{P} \\times(1+\\var{int})^\\var{n}=\\var{A}$

What is the value of P?

€[[0]]

What is the value of i written as a decimal?

[[0]]

What is the value of n?

[[0]]

How much will be in the deposit account after $\\var{n}$ years? Please give your answer to the nearest cent.

€[[0]]

The compound interest formula is: $\\ A = P(1+i)^n $

Part (a)

P represents the principal sum invested , so in this example it is €$\\var{P}$.

Part (b)

A represents the amount in the deposit account after $\\var{n}$ years, so in this example it is €$\\var{A}$.

Part (c)

n represents the number of compounding periods , so in this example it is $\\var{n}$ years.

Part(d)

Using the compound interest formula:

$A=P(1+i)^n$

$\\var{A}=\\var{P}(1+i)^\\var{n}$

We need to rearrange the equation to find the value of $i$.

$\\frac{\\var{A}}{\\var{P}}=(1+i)^\\var{n}$

$\\var{ratio}=(1+i)^\\var{n}$

$\\sqrt[\\var{n}]{\\var{ratio}}=1+i$

$\\var{intplus}=1+i$

$i=\\var{int}$ so the annual interest rate is $\\var{perc}$%.

", "rulesets": {}, "parts": [{"prompt": "

What is the value of P?

€[[0]]

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "P+0.0001", "minValue": "P-0.0001", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "

What is the value of A?

€[[0]]

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "A+0.0001", "minValue": "A-0.0001", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "

What is the value of n?

[[0]]

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "n+0.0001", "minValue": "n-0.0001", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "

What is the interest rate per annum?

Please give your answer as a percentage correct to 2 decimal places.

[[0]]%

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "perc+0.02", "minValue": "perc-0.02", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": "3", "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "

A lump sum of €$\\var{P}$ is deposited into a savings account that pays compound interest for $\\var{n}$ years. If no withdrawals are made from the account, then the amount that the lump sum will have grown to is €$\\var{A}$.

The compound interest formula is:

$\\ A = P(1+i)^n $

", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"A": {"definition": "precround(P*(1+int)^n,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "A", "description": ""}, "perc": {"definition": "random(1.5..5.5 #0.5)", "templateType": "anything", "group": "Ungrouped variables", "name": "perc", "description": ""}, "ratio": {"definition": "A/P", "templateType": "anything", "group": "Ungrouped variables", "name": "ratio", "description": ""}, "int": {"definition": "perc/100", "templateType": "anything", "group": "Ungrouped variables", "name": "int", "description": ""}, "intplus": {"definition": "ratio^(1/n)", "templateType": "anything", "group": "Ungrouped variables", "name": "intplus", "description": ""}, "n": {"definition": "random(2..6 #1)", "templateType": "anything", "group": "Ungrouped variables", "name": "n", "description": ""}, "P": {"definition": "random(1000..6000 #500)", "templateType": "anything", "group": "Ungrouped variables", "name": "P", "description": ""}}, "metadata": {"description": "

Calculate the annual interest rate for a savings account where A, P and n are given.

rebelmaths

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Savings compound interest 3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "functions": {}, "ungrouped_variables": ["perc1", "perc2", "int1", "int2", "n1", "n2", "P", "A1", "A2", "int3"], "tags": ["compound interest", "rebel", "REBEL", "Rebel", "rebelmaths"], "advice": "

The compound interest formula is: $\\ A = P(1+i)^n $

Part (a)

n represents the number of compounding periods , so for Bank A it is $\\var{n1}$ years.

i represents the rate of compound interest, for Bank A, the annual interest rate is $\\var{perc1}$% so i is $\\frac {\\var{perc1}} {100}=\\var{int1}$.

The total amount saved after $\\var{n1}$ years is denoted by A. Using the compound interest formula:

$A=P(1+i)^n$

$A=\\var{P} \\times(1+\\var{int1})^\\var{n1}=\\var{A1}$

Part(b)

n represents the number of compounding periods. For Bank B, interest is compounded daily for $\\var{n1}$ years so there are a total of $365 \\times \\var{n1} =\\var{n2}$ compounding periods.

i represents the rate of compound interest. For Bank B, the interest rate is ${\\var{perc2}}$% per annum compounded daily.

The interest rate per day is $\\frac{\\var{perc2}}{365}=\\var{int3}$%

Therefore $i=\\frac{\\var{int3}}{100}=\\var{int2}$

The amount saved after $\\var{n1}$ years is denoted by A. Using the compound interest formula:

$A=P(1+i)^n$

$A=\\var{P} \\times(1+\\var{int2})^\\var{n2}=\\var{A2}$

", "rulesets": {}, "parts": [{"prompt": "

Suppose that the potential customer chooses Bank A.

What is the value of $n$?

[[0]]

What is the value of $i$?

[[1]]

What is the value of $A$? Please give your answer to the nearest cent.

€[[2]]

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "n1+0.00001", "minValue": "n1-0.00001", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "int1+0.0001", "minValue": "int1-0.0001", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "A1+0.01", "minValue": "A1-0.01", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "

Suppose that the potential customer chooses Bank B.

What is the value of $n$?

[[0]]

What is the value of $i$? Please include all the decimal places in your answer.

[[1]]

What is the value of $A$? Please give your answer to the nearest cent.

€[[2]]

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Two rival high street banks offer customers a new deposit account.

Bank A offers an account that earns interest at a rate of $\\var{perc1}$% per annum where interest is compounded annually.

Bank B offers an account that earns interest at a nominal rate of $\\var{perc2}$% per annum where interest is compounded daily.

Suppose that a potential customer has €$\\var{P}$ to invest for $\\var{n1}$ years.

The compound interest formula is:

$\\ A = P(1+i)^n $

You may assume that there are 365 days per annum.

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Compare two savings accounts with different interest rates.

rebelmaths

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Loan compound interest 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Catherine Palmer", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/423/"}, {"name": "Patricia Cogan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3359/"}], "tags": [], "metadata": {"description": "

Compare two loans with different interest rates.

", "licence": "None specified"}, "statement": "

Two rival high street banks offer customers a new loan.

Bank A offers a loan at a rate of $\\var{perc1}$% per annum where interest is compounded annually.

Bank B offers a loan at a rate of $\\var{perc2}$% per annum where interest is compounded monthly.

Suppose that a potential customer wishes to borrow €$\\var{P}$ for $\\var{n1}$ years.

The compound interest formula is:

$\\ A = P(1+i)^n $

", "advice": "

The compound interest formula is: $\\ A = P(1+i)^n $

Part (a)

n represents the number of compounding periods , so for Bank A it is $\\var{n1}$ years.

i represents the rate of compound interest, for Bank A, the annual interest rate is $\\var{perc1}$% so i is $\\frac {\\var{perc1}} {100}=\\var{int1}$.

The amount owed to Bank A after $\\var{n1}$ years is denoted by A. Using the compound interest formula:

$A=P(1+i)^n$

$A=\\var{P} \\times(1+\\var{int1})^\\var{n1}=\\var{A1}$

The amount of interest accumulated over the $\\var{n1}$ years is the difference between the amount borrowed ($P$) and the amount payed back ($A$).

$A-P=\\var{A1}-\\var{P}=\\var{I1}$

Part(b)

n represents the number of compounding periods. For Bank B interest is compounded monthly for $\\var{n1}$ years so there are a total of $12 \\times \\var{n1} =\\var{n2}$ compounding periods.

i represents the rate of compound interest. For Bank B, the interest rate is ${\\var{perc2}}$% per annum compounded monthly.

The interest rate per month is $\\frac{\\var{perc2}}{12}=\\var{int3}$%

Therefore $i=\\frac{\\var{int3}}{100}=\\var{int2}$

The amount owed to Bank A after $\\var{n1}$ years is denoted by A. Using the compound interest formula:

$A=P(1+i)^n$

$A=\\var{P} \\times(1+\\var{int2})^\\var{n2}=\\var{A2}$

The amount of interest accumulated over the $\\var{n1}$ years is the difference between the amount borrowed ($P$) and the amount payed back ($A$).

$A-P=\\var{A2}-\\var{P}=\\var{I2}$

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Suppose that the potential customer chooses Bank A.

What is the value of $n$?

[[0]]

What is the value of $i$ as a decimal?

[[1]]

At the end of the $\\var{n1}$ years, how much must the customer pay to Bank A to clear the loan?

Please give your answer to the nearest cent.

€[[2]]

How much interest does the loan accumulate during the $\\var{n1}$ years?

Please give your answer to the nearest cent.

€[[3]]

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Suppose that the potential customer chooses Bank B.

What is the value of $n$?

[[0]]

What is the value of $i$? Please include all the decimal places in your answer.

[[1]]

At the end of the $\\var{n1}$ years, how much must the customer pay to Bank B to clear the loan?

Please give your answer to the nearest cent.

€[[2]]

How much interest does the loan accumulate during the $\\var{n1}$ years?

Please give your answer to the nearest cent.

€[[3]]

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Find the present value of an amount where interest is compounded annually

", "licence": "None specified"}, "statement": "

The formula for calculating the present value of an investment is:

$P=\\frac{A}{(1+i)^n}$

$\\var{n}$ years from today, on his twenty-first birthday, Sean MacLeinn will receive a gift of €$\\var{A}$ from his great-aunt Priscilla. The annual rate of interest is $\\var{perc}$%.

", "advice": "

The present value formula is: $ P = \\frac{A}{(1+i)^n} $

Part (a)

A represents the principal sum invested , so in this example it is €$\\var{A}$.

Part (b)

i represents the rate of compound interest, in this example, the annual interest rate is $\\var{perc}$% so i is $\\frac {\\var{perc}} {100}=\\var{int}$.

Part (c)

n represents the number of compounding periods , so in this example it is $\\var{n}$ years.

Part(d)

Using the present value formula:

$ P = \\frac{A}{(1+i)^n} $

$ P = \\frac{\\var{A}}{(1+\\var{int})^\\var{n}} $

$ P = €\\var{P}$

What is the value of $A$?

€[[0]]

What is the value of $i$ written as a decimal?

€[[0]]

What is the value of $n$?

[[0]]

What is the present value of this gift? Please give your answer to the nearest cent.

€[[0]]

Interest rate does not match compounding period.

Find much money should be invested at $r$% per annum so that after $y$ years the amount will be £$A$, interest compounded daily.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

A sum of money is be invested with a nominal interest rate of $\\var{r}$% compounded daily so that after $\\var{yrs}$ years the amount will have grown to €$\\var{amt}$. You may assume that there are 365 days per annum.

The formula for calculating the present value of an investment is:

$P=\\frac{A}{(1+i)^n}$

", "advice": "

Part(a)

Interest is compounded daily for $\\var{yrs}$ years so there are a total of $365 \\times \\var{yrs} =\\var{n}$ compounding periods.

Part(b)

It is important to remember that the interest rate must match the compounding period.

The interest rate given in the question is ${\\var{r}}$% per annum.

The interest rate per day is $\\frac{\\var{r}}{365}=\\var{r/365}$%

Therefore $i=\\frac{\\var{r/365}}{100}=\\var{int}$

Part(c)

We want to find the present value of the amount $A=\\var{amt}$ due at the end of $n=\\var{yrs}\\times 365 = \\var{365*yrs}$ interest periods (days).

Hence the present value i.e. the amount to be invested is:
\\[ \\begin{eqnarray*} P&=&\\frac{A}{(1+i)^n}\\\\ \\\\ &=& \\frac{\\var{amt}}{(1+\\var{int})^{\\var{n}}}\\\\ \\\\ &=&\\var{p} \\end{eqnarray*} \\]

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What is the value of $n$?

[[0]]

What is the value of $i$?

[[0]]

Please give your answer to 8 decimal places.

What is the value of the sum of money initially invested?

€[[0]]

Please give your answer to two decimal places.

A pharmaceutical company wishes to accumulate €$\\var{A}$ to upgrade its manufacturing plant in $3$ years time.

", "licence": "None specified"}, "statement": "

A pharmaceutical company wishes to accumulate €$\\var{A}$ to upgrade its manufacturing plant. The upgrade is due to be paid for on December 31st 2017 so the company makes three equal deposits into a savings account earning compound interest at a rate of $\\var{perc}$% per annum. The payments are made on 31st December 2015, 31st December 2016 and 31st December 2017.

", "advice": "

Since the payments are made at the end of the year, the formula for calculating the future value ($A$) of an annuity is given by:

$A=\\frac{R[(1+i)^{n}-1]}{i}$

where $R$ represents the periodic payment, $i$ represents the interest rate per period and $n$ represents the number of payments.

Part (a)

$A$ represents the future value of the annuity, this is the amount to be saved, therefore $A=€\\var{A}$

Part (b)

$i$ represents the rate of compound interest, the annual interest rate is $\\var{perc}$% so $i$ is $\\frac {\\var{perc}} {100}=\\var{int}$.

Part(c)

$n$ represents the number of payments , so $n$ is $\\var{n}$.

Part (d)

The value of each repayment, €$R$ can be calculated using the future value formula or it can be calculated directly by calculating the amount of interest that each deposit earns.Using the formula:

$A=\\frac{R[(1+i)^{n}-1]}{i}$

$\\var{A}=\\frac{R[(1+\\var{int})^{\\var{n}}-1]}{\\var{int}}$

$\\var{A}=\\frac{R[\\var{num}-1]}{\\var{int}}$

$\\var{A}=\\frac{R[\\var{num2}]}{\\var{int}}$

$\\var{A} = R \\times \\var{num3}$

$\\frac{\\var{A}}{\\var{num3}}=R$

$\\var{R}=R$

Alternatively, we can calculate $R$ directly:

The amount to be saved is €$\\var{A}$ so $A=\\var{A}$. The interest rate per annum is $\\var{int}$ and there are $n$ repayments so $n=\\var{n}$.

The future value of the deposit made on 31st December 2015 is: $R \\times (1+\\var{int})^2$
The future value of the deposit made on 31st December 2016 is: $R\\times(1+\\var{int})$
Thevalue of the deposit made on 31st December 2016 is: $R$

Thus the future value is given by:

$A=R\\times(1+\\var{int})^2$+$R\\times(1+\\var{int})$+$R$

$\\var{A}=R\\times(1+\\var{int})^2$+$R\\times(1+\\var{int})$+$R$

$\\var{A}=R[(1+\\var{int})^2$+$(1+\\var{int})+1$

$\\var{A}=R[\\var{frac2}+\\var{frac1}+1]$

$\\var{A} = R \\times \\var{num4}$

$\\frac{\\var{A}}{\\var{num4}}=R$

$\\var{R}=R$

Part (e)

The amount of interest paid is the difference between the amount accumulated ($A$) and the three deposits of €$\\var{R}$

Interest = $\\var{A}- 3 \\times \\var{R}=\\var{Interest}$

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Which formula should we use to calculate how large each deposit should be?

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What is the value of $A$?

€[[0]]

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What is the value of $i$?

[[0]]

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What is the value of $n$?

[[0]]

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Calculate how large each deposit will need to be.

Please give your answer to the nearest cent.

€[[0]]

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Since the payments are made at the end of the year, the formula for calculating the future value ($A$) of an annuity is:

$A=R\\left[\\frac{(1+i)^{n}-1}{i}\\right]$

where $R$ represents the value of the regular deposit, $i$ represents the interest rate and $n$ represents the number of payments.

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Calculate the total amount of interest that the company will earn over the three years.

Please give your answer to the nearest cent.

€[[0]]

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The loan must be repaid to the bank in three equal repayments
Calculate:
(i) how large each repayment will need to be
(ii) the total amount of interest that the company will pay to the bank over the four years.

", "licence": "None specified"}, "statement": "

On January 1, 2015 a company borrowed €$\\var{P}$ from a bank at $\\var{perc}$% per annum compound interest. The loan must be repaid to the bank in three equal repayments, due on December 31 in 2015, 2016 and 2017.

", "advice": "

The present value of an annuity, $P$ is given by:

$P=\\frac{R[1-(1+i)^{-n}]}{i}$

where $R$ represents the periodic payment, $i$ represents the interest rate per period and $n$ represents the number of payments.

Part (a)

$P$ represents the present value of the annuity, this is the amount borrowed, therefore $P=€\\var{P}$

Part (b)

$i$ represents the rate of compound interest, the annual interest rate is $\\var{perc}$% so $i$ is $\\frac {\\var{perc}} {100}=\\var{int}$.

Part(c)

$n$ represents the number of payments , so $n$ is $\\var{n}$.

Part (d)

The value of each repayment, €$R$ can be calculated using the present value formula or it can be calculated directly by discounting each payment according to how far into the future it lies.

Using the formula:

$P=\\frac{R[1-(1+i)^{-n}]}{i}$

$\\var{P}=\\frac{R[1-(1+\\var{int})^{-\\var{n}}]}{\\var{int}}$

$\\var{P}=\\frac{R[1-\\var{num}]}{\\var{int}}$

$\\var{P}=\\frac{R[\\var{num2}]}{\\var{int}}$

$\\var{P} = R \\times \\var{num3}$

$\\frac{\\var{P}}{\\var{num3}}=R$

$\\var{R}=R$

Alternatively, we can calculate $R$ directly:

The principle borrowed from the bank is €$\\var{P}$ so $P=\\var{P}$. The interest rate per annum is $\\var{int}$ and there are $n$ repayments so $n=\\var{n}$.

The present value of the first repayment is: $\\frac{R}{(1+\\var{int})}$
The present value of the second repayment is: $\\frac{R}{(1+\\var{int})^2}$
The present value of the third repayment is: $\\frac{R}{(1+\\var{int})^3}$

Thus the present value is given by:

$P=\\frac{R}{(1+\\var{int})}$+$\\frac{R}{(1+\\var{int})^2}$+$\\frac{R}{(1+\\var{int})^3}$

$\\var{P}=\\frac{R}{(1+\\var{int})}$+$\\frac{R}{(1+\\var{int})^2}$+$\\frac{R}{(1+\\var{int})^3}$

$\\var{P}=R[\\frac{1}{(1+\\var{int})}$+$\\frac{1}{(1+\\var{int})^2}$+$\\frac{1}{(1+\\var{int})^3}]$

$\\var{P}=R[\\var{frac1}+\\var{frac2}+\\var{frac3}]$

$\\var{P} = R \\times \\var{num4}$

$\\frac{\\var{P}}{\\var{num4}}=R$

$\\var{R}=R$

Part (e)

The amount of interest paid is the difference between the amount borrowed ($P$) and the amount paid back (three payments of €$\\var{R}$)

Interest = $ 3 \\times \\var{R} - \\var{P}=\\var{Interest}$

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Which formula should we use to calculate how large each repayment will need to be?

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What is the value of $P$?

€[[0]]

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What is the value of $i$?

[[0]]

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What is the value of $n$?

[[0]]

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Calculate how large each repayment will need to be.

Please give your answer to the nearest cent.

€[[0]]

The formula for calculating the present value ($P$) of an annuity is:

$P=R\\left[\\frac{1-(1+i)^{-n}}{i}\\right]$

where $R$ represents the value of each repayment, $i$ represents the interest rate and $n$ represents the number of repayments.

Calculate the total amount of interest that the company will pay to the bank over the three years.

Please give your answer to the nearest cent.

€[[0]]

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The present value of an annuity, $P$ is given by:

$P=\\frac{R[1-(1+i)^{-n}]}{i}$

where $R$ represents the periodic payment, $i$ represents the interest rate per period and $n$ represents the number of payments.

$P$ represents the present value of the annuity, this is what we are asked to calculate.

$i$ 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 $i=\\frac{\\var{perc2}}{100}=\\var{int}$.

$n$ represents the number of payments, there are 12 payments over $\\var{years}$ year(s) so $n$ is $12 \\times \\var{years}=\\var{n}$.

The value of each repayment, is €$\\var{R}$

Using the formula:

$P=\\frac{R[1-(1+i)^{-n}]}{i}$

$P=\\frac{\\var{R}[1-(1+\\var{int})^{-\\var{n}}]}{\\var{int}}$

$P=\\frac{\\var{R}[1-\\var{num}]}{\\var{int}}$

$P=\\frac{\\var{R}[\\var{num2}]}{\\var{int}}$

$P = \\var{R} \\times \\var{num3}$

$P=\\var{P}$

", "rulesets": {}, "parts": [{"prompt": "

Find the present value of these payments if the annual interest rate is $\\var{perc}$% compounded monthly. Give your answer to the nearest cent.

€[[0]]

", "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"}], "statement": "

A $\\var{years}$-year lease for a company car requires a payment of €$\\var{R}$ at the end of each month.

The formula for calculating the present value ($P$) of an annuity is:

$P=\\frac{R[1-(1+i)^{-n}]}{i}$

where $R$ represents the value of each repayment, $i$ represents the interest rate and $n$ represents the number repayments.

", "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": "

A 1-year lease for a company car requires a payment of €280 at the end of each month. Find the present value of these payments if the annual interest rate is 7% compounded monthly.

rebelmaths

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "AER", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "functions": {}, "ungrouped_variables": ["perc", "apr2", "int", "perc2", "apr"], "tags": ["AER", "Annual Equivalent Rate", "REBEL", "rebel", "Rebel", "rebelmaths"], "preamble": {"css": "", "js": ""}, "advice": "

The quarterly interest rate is $\\var{perc}$% per quarter. There are 4 quarters in one year so the equivalent annual rate of interest can be calculated as follows:

$i=(1+\\frac{\\var{perc}}{100})^{4}-1=\\var{apr2}-1=\\var{apr}$

Therefore the equivalent annual rate is $\\var{perc2}$%

", "rulesets": {}, "parts": [{"prompt": "

Please give your answer as a percentage to two decimal places.

[[0]]%

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "perc2+0.01", "minValue": "perc2-0.01", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": "5", "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "

A savings account earns compound interest at a rate of $\\var{perc}$% per quarter.
Calculate the equivalent annual rate of interest.

", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"perc": {"definition": "random(1.4..2.5#0.1)", "templateType": "anything", "group": "Ungrouped variables", "name": "perc", "description": ""}, "int": {"definition": "perc/100", "templateType": "anything", "group": "Ungrouped variables", "name": "int", "description": ""}, "perc2": {"definition": "precround(apr*100,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "perc2", "description": ""}, "apr": {"definition": "precround(apr2-1,4)", "templateType": "anything", "group": "Ungrouped variables", "name": "apr", "description": ""}, "apr2": {"definition": "precround((1+int)^4,4)", "templateType": "anything", "group": "Ungrouped variables", "name": "apr2", "description": ""}}, "metadata": {"description": "

A savings account earns compound interest at 1.5% per month.
Calculate the equivalent annual rate of interest

rebelmaths

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "APR", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "functions": {}, "ungrouped_variables": ["perc", "apr2", "int", "perc2", "apr"], "tags": ["Annual Percentage Rate", "APR", "REBEL", "rebel", "Rebel", "rebelmaths"], "preamble": {"css": "", "js": ""}, "advice": "

The monthly interest rate is $\\var{perc}$% per month. There are 12 months in one year so the APR can be calculated as follows:

$i=(1+\\frac{\\var{perc}}{100})^{12}-1=\\var{apr2}-1=\\var{apr}$

Therefore the APR is $\\var{perc2}$%

", "rulesets": {}, "parts": [{"prompt": "

Calculate the Annual Percentage Rate (APR) for this credit card.

Please give your answer as a percentage to two decimal places.

", "allowFractions": false, "variableReplacements": [], "maxValue": "perc2+0.01", "minValue": "perc2-0.01", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": "5", "type": "numberentry", "showPrecisionHint": false}], "statement": "

A credit card company charges compound interest at a rate of $\\var{perc}$% per month on unpaid balances.

", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"perc": {"definition": "random(0.8..1.6#0.1)", "templateType": "anything", "group": "Ungrouped variables", "name": "perc", "description": ""}, "int": {"definition": "perc/100", "templateType": "anything", "group": "Ungrouped variables", "name": "int", "description": ""}, "perc2": {"definition": "precround(apr*100,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "perc2", "description": ""}, "apr": {"definition": "precround(apr2-1,4)", "templateType": "anything", "group": "Ungrouped variables", "name": "apr", "description": ""}, "apr2": {"definition": "precround((1+int)^12,4)", "templateType": "anything", "group": "Ungrouped variables", "name": "apr2", "description": ""}}, "metadata": {"description": "

A credit card company charges compound interest at x% per month
on unpaid balances.

rebelmaths

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Compound Interest - find interest rate", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "functions": {}, "ungrouped_variables": ["A", "A0", "n", "a_p", "a_p_rt", "int", "ip"], "tags": ["rebel", "Rebel", "REBEL", "rebelmaths"], "advice": "

We use the Compound Interest Formula $A=P(1+i)^n$ where

$A=\\var{A}$

$P=\\var{A0}$

$i=?$

$n=\\var{n}$

So, we get

$\\var{A}=\\var{A0}(1+i)^\\var{n}$

Divide both sides by $\\var{A0}$

$\\frac{\\var{A}}{\\var{A0}}=(1+i)^\\var{n}$

$\\var{a_p}=(1+i)^\\var{n}$

Next,

$\\var{a_p}^{\\frac{1}{\\var{n}}}=1+i$

$\\var{a_p_rt}=1+i$

So, $\\var{a_p_rt}-1=\\var{int}=i$

And so the answer as a percentage is $\\var{ip}$%.

Then, just round to the correct number of decimal places.

", "rulesets": {}, "parts": [{"prompt": "

Calculate the annual interest rate.

Please give your answer as a percentage to 3 decimal places.

[[0]]%

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "3", "maxValue": "100*(({A}/{A0})^(1/{n}) - 1)", "minValue": "100*(({A}/{A0})^(1/{n}) - 1)", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": "5", "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "

An investor puts €$\\var{A0}$ in a banks saving account with a fixed interest rate earning compound interest. In return they receive €$\\var{A}$ in $\\var{n}$ years time.

", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"A": {"definition": "random(115000..125000#100)", "templateType": "randrange", "group": "Ungrouped variables", "name": "A", "description": ""}, "a_p": {"definition": "A/A0", "templateType": "anything", "group": "Ungrouped variables", "name": "a_p", "description": ""}, "int": {"definition": "a_p_rt-1", "templateType": "anything", "group": "Ungrouped variables", "name": "int", "description": ""}, "ip": {"definition": "int*100\n", "templateType": "anything", "group": "Ungrouped variables", "name": "ip", "description": ""}, "n": {"definition": "random(4..10#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "n", "description": ""}, "A0": {"definition": "random(100000..111000#100)", "templateType": "randrange", "group": "Ungrouped variables", "name": "A0", "description": ""}, "a_p_rt": {"definition": "(A/A0)^(1/n)", "templateType": "anything", "group": "Ungrouped variables", "name": "a_p_rt", "description": ""}}, "metadata": {"description": "

Find i using compound interest formula A=P(1+i)^n

rebelmaths

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Future Value - Annuity 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "functions": {}, "ungrouped_variables": ["R", "perc", "perc2", "int", "num", "years", "n", "num2", "num3", "A"], "tags": ["future value", "rebel", "REBEL", "Rebel", "rebelmaths"], "advice": "

The future value of an annuity, $A$ is given by:

$A=\\frac{R[(1+i)^n-1]}{i}$

where $R$ represents the periodic payment, $i$ represents the interest rate per period and $n$ represents the number of payments.

$A$ represents the future value of the annuity, this is what we are asked to calculate.

$n$ represents the number of payments, there are 12 payments over $\\var{years}$ year(s) so $n$ is $12 \\times \\var{years}=\\var{n}$.

The value of each repayment, is €$\\var{R}$

Using the formula:

$A=\\frac{R[(1+i)^n-1]}{i}$

$A=\\frac{\\var{R}[(1+\\var{int})^{\\var{n}}-1]}{\\var{int}}$

$A=\\frac{\\var{R}[\\var{num}-1]}{\\var{int}}$

$A=\\frac{\\var{R}[\\var{num2}]}{\\var{int}}$

$A = \\var{R} \\times \\var{num3}$

$A=\\var{A}$

", "rulesets": {}, "parts": [{"prompt": "

What is the future value of the annuity in $\\var{years}$ years?

€[[0]]

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "A+0.10", "minValue": "A-0.10", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": "5", "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "

To provide for retirement an employee deposits €$\\var{R}$ at the end of each month into an account that earns $\\var{perc}$% annual interest compounded monthly.

", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"perc": {"definition": "random(4..9#0.5)", "templateType": "anything", "group": "Ungrouped variables", "name": "perc", "description": ""}, "A": {"definition": "precround(R*num3,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "A", "description": ""}, "perc2": {"definition": "precround(perc/12,5)", "templateType": "anything", "group": "Ungrouped variables", "name": "perc2", "description": ""}, "num2": {"definition": "num-1", "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": ""}, "years": {"definition": "random(5..10)", "templateType": "anything", "group": "Ungrouped variables", "name": "years", "description": ""}, "num": {"definition": "(1+int)^n", "templateType": "anything", "group": "Ungrouped variables", "name": "num", "description": ""}, "R": {"definition": "random(50..100#10)", "templateType": "anything", "group": "Ungrouped variables", "name": "R", "description": ""}, "n": {"definition": "years*12", "templateType": "anything", "group": "Ungrouped variables", "name": "n", "description": ""}}, "metadata": {"description": "

To provide for retirement an employee deposits R at the end of each month in an account that earns $perc% annual interest compound monthly. What is the future value of the annuity in nyears?

rebelmaths

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Net Present Value", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "functions": {}, "ungrouped_variables": ["Invest", "Return1", "perc", "int", "NPV", "n1", "num"], "tags": ["present value", "rebel", "Rebel", "REBEL", "rebelmaths"], "advice": "

We wish to calculate the present value of an investment that will be worth €$\\var{Return1}$ in $\\var{n1}$ years time. Using the present value formula:

$P=\\frac{A}{(1+i)^{n}}$

where $A$ is €$\\var{Return1}$, $n$ is $\\var{n1}$ and $i$ is $\\frac{\\var{perc}}{100}=\\var{int}$ gives:

$P=\\frac{\\var{Return1}}{(1+\\var{int})^{\\var{n}}}$

$P=\\frac{\\var{Return1}}{\\var{num}}$

$P=\\var{NPV}$

", "rulesets": {}, "parts": [{"prompt": "

Calculate the present value of the investment given that the discount rate is $\\var{perc}$% per annum.

€[[0]]

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "NPV+0.1", "minValue": "NPV-0.1", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": "5", "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "

An investment will give a return of €$\\var{Return1}$ in $\\var{n1}$ years time.

", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"perc": {"definition": "random(2..7#0.5)", "templateType": "anything", "group": "Ungrouped variables", "name": "perc", "description": ""}, "int": {"definition": "perc/100", "templateType": "anything", "group": "Ungrouped variables", "name": "int", "description": ""}, "Invest": {"definition": "random(50000..150000#20000)", "templateType": "anything", "group": "Ungrouped variables", "name": "Invest", "description": ""}, "Return1": {"definition": "siground(Invest*(1+int+0.01)^n1,4)", "templateType": "anything", "group": "Ungrouped variables", "name": "Return1", "description": ""}, "num": {"definition": "(1+int)^n1", "templateType": "anything", "group": "Ungrouped variables", "name": "num", "description": ""}, "n1": {"definition": "random(3..5#1)", "templateType": "anything", "group": "Ungrouped variables", "name": "n1", "description": ""}, "NPV": {"definition": "precround(Return1/(1+int)^n1,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "NPV", "description": ""}}, "metadata": {"description": "

An investment of €x invested today will give a return of €y in n years time Calculate the net present value (NPV) of the investment given that the discount rate is 3.5% per annum.

rebelmaths

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Net Present Value 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "functions": {}, "ungrouped_variables": ["Invest", "Return1", "perc", "int", "PV1", "n1", "num", "Return2", "n2", "PV2", "NPV", "num2"], "tags": ["present value", "rebel", "Rebel", "REBEL", "rebelmaths"], "advice": "

We wish to calculate the net present value (NPV) of an investment that will be worth €$\\var{Return1}$ in $\\var{n1}$ years time plus an additional €$\\var{Return2}$ in $\\var{n2}$ years time. Using the present value formula, the present value of the €$\\var{Return1}$ is:

$P=\\frac{A}{(1+i)^{n}}$

where $A$ is €$\\var{Return1}$, $n$ is $\\var{n1}$ and $i$ is $\\frac{\\var{perc}}{100}=\\var{int}$

$P1=\\frac{\\var{Return1}}{(1+\\var{int})^{\\var{n}}}$

$P1=\\frac{\\var{Return1}}{\\var{num}}$

$P1=\\var{PV1}$

Using the present value formula, the present value of the €$\\var{Return2}$ is:

$P=\\frac{A}{(1+i)^{n}}$

where $A$ is €$\\var{Return2}$, $n$ is $\\var{n2}$ and $i$ is $\\frac{\\var{perc}}{100}=\\var{int}$

$P2=\\frac{\\var{Return2}}{(1+\\var{int})^{\\var{n}}}$

$P2=\\frac{\\var{Return2}}{\\var{num}}$

$P2=\\var{PV2}$

The NPV of the total amount is $P1+P2-Investment=\\var{PV1}+\\var{PV2}-\\var{INVEST}=€\\var{NPV}$

", "rulesets": {}, "parts": [{"prompt": "

Calculate the net present value (NPV) of the investment given that the discount rate is $\\var{perc}$% per annum.

€[[0]]

An investment of €$\\var{Invest}$ invested today will give a return of €$\\var{Return1}$ in $\\var{n1}$ years' time and a further return of €$\\var{Return2}$ in $\\var{n2}$ years' time.

", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"perc": {"definition": "random(2..6#0.5)", "templateType": "anything", "group": "Ungrouped variables", "name": "perc", "description": ""}, "PV1": {"definition": "precround(Return1/(1+int)^n1,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "PV1", "description": ""}, "PV2": {"definition": "precround(Return2/(1+int)^n2,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "PV2", "description": ""}, "num": {"definition": "(1+int)^n1", "templateType": "anything", "group": "Ungrouped variables", "name": "num", "description": ""}, "num2": {"definition": "(1+int)^n2", "templateType": "anything", "group": "Ungrouped variables", "name": "num2", "description": ""}, "int": {"definition": "perc/100", "templateType": "anything", "group": "Ungrouped variables", "name": "int", "description": ""}, "Invest": {"definition": "siground(Return1/(1+int+0.01)^n1, 3)+siground(Return2/(1+int+0.01)^n2, 3)", "templateType": "anything", "group": "Ungrouped variables", "name": "Invest", "description": ""}, "Return1": {"definition": "random(20000..30000#5000)", "templateType": "anything", "group": "Ungrouped variables", "name": "Return1", "description": ""}, "Return2": {"definition": "random(40000..70000#5000)", "templateType": "anything", "group": "Ungrouped variables", "name": "Return2", "description": ""}, "n1": {"definition": "random(2..4#1)", "templateType": "anything", "group": "Ungrouped variables", "name": "n1", "description": ""}, "n2": {"definition": "n1+3", "templateType": "anything", "group": "Ungrouped variables", "name": "n2", "description": ""}, "NPV": {"definition": "(PV1+PV2)- INVEST", "templateType": "anything", "group": "Ungrouped variables", "name": "NPV", "description": ""}}, "metadata": {"description": "

An investment of x invested today will give a return of €y in n1 years time and a further €z n2 years from today. Calculate the net present value (NPV) of the investment given that the discount rate is perc% per annum.

rebelmaths

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}]}], "contributors": [{"name": "Deirdre Casey", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/681/"}], "extensions": [], "custom_part_types": [], "resources": []}