Introduction to Financial Math Study Guide Notes #5

Introduction to Financial Math NotesBasic Math ConceptsDevelop fluency in addition, subtraction, multiplication, and division of basic operations with and without calculators.addition: +subtraction: -multiplication: * or xdivision: ÷ or /Solve one- and two-step problems involving whole numbers, fractions, and decimals using addition, subtraction, multiplication, and division.Solve practical computation problems for business such as calculating wages after taxes, developing a budget, and balancing a checkbook.calculating wages after taxes: net income = gross income - total tax deductionsAnalyze problem statements for missing/irrelevant data; estimate/exact values, and inconsistent parameters.Identify business math terms.maker: the person authorized to write checks payee: a person or organization that is receiving a paymentcreditor: an individual or organization that is owed money by another partyendorser: an individual or entity that signs a negotiable instrument(check, promissory note, etc.) to transfer ownership or guarantee paymentliquidity: how quickly assets can be turned to cashassets: resources owned or controlled by a company that has economic value and can be used to generate future incomenet worth: total value of a company's assets minus its liabilitiesliabilities: a financial obligation that a company owes to another person or organizationcapital: the total amount of money or assets owned by a companyrestrictive endorsement: places a limitation on the use of a check or other negotiable financial instrumentblank endorsement: a signature on a financial instrument such as a checkspecial endorsement: allows you to make your business check payable to someone elseProve that results of computations using whole numbers, fractions, decimals, percents, and proportions are correct.Recognize patterns and relationships among numbers.Estimate the results of rational number computations and judge the reasonableness of the results.Identify and use relationships between operations, such as division as the inverse of multiplication to solve problems.Apply relational (equal, greater than, less than, etc.) and logical operations in a logical expression.equal: =greater than: >less than: <Select appropriate methods and tools for computing with whole numbers from mental computation, estimation, calculators, and paper and pencil.Convert within and between measurement systems (metric and customary) and monetary systems using technology where appropriate.Estimate conversions between the customary and metric systems.Use the appropriate type of unit to calculate measurement of length, area, weight, volume, angles, and perimeter.Construct or draw figures with given perimeters and areas.Interpret scale drawings and models using maps and blueprints.Use touch method on calculator to solve math problems that relate to business and industry.Consumer CreditDefine credit and credit terms.Credit - The ability to borrow money or access goods and services with the promise to pay for them laterCredit Limit - The maximum amount of credit availableInterest Rate - The percentage charged for borrowing, typically expressed as an annual rate (APR)Repayment Period - The length of time allowed for repaymentDiscounts for Early Payment - A discount offered if the borrower repays before the due date Late Fees or Penalties - Charges applied if the borrower fails to repay on timeIdentify the costs and benefits of various types of credit.Credit Cards:Benefits - Convenient, offers many rewards and perks (cash back, travel points, etc.), builds credit history/score, can provide quick access to funds in case of an emergencies, and some offer interest-free periods if the balance is paid in full.Costs - Interest payments, origination fees, potential collateral, and may impact credit scorePersonal Loans:Benefits - Fixed interest rates, larger loan amounts, lower interest rates (compared to credit cards), and predictable repayment termsCost - Interest payments, origination fees, potential collateral, and impact on credit scoreHome Equity Loans/Home Equity Lines of Credit (HELOCs):Benefits - Lower interest rates, large loan amounts, potential tax benefits, and flexible borrowing (HELOCs)Costs - Risk of losing your home, closing costs, variable interest rates (HELOCs), and debt accumulation Auto Loans:Benefits - Lower interest rates (compared to credit cards), predictable payments, and builds creditCosts - Interest payments, depreciation of the car, down payment, and risk of repossessionStore Credit:Benefits - Exclusive discounts and rewards, easier approval, and can help in building creditCosts - High interest rates, limited use, and risk of going into debtPayday Loans:Benefits - Quick access to cash and small loan amountsCosts - Extremely high interest rates, short repayment terms, and high debt trap riskCalculate sales tax.Sales Tax = Price of Item x (Sales Tax Rate/100)Compute total purchase price with interest added.Simple Interest = P * r * tTotal Purchase Price = P + Simple Price( P = principal amount (the original price or loan amount), r = Annual interest rate (expressed as a decimal), t = Time period the interest is charged for (in years))Compound Interest: A = P*(1+r/n)^n*t(A = Total amount (principle + interest), P = Principal amount (original price or loan amount), r = Annual interest rate (expressed as a decimal), t = Time period in years, n = Number of times the interest is compounded per year)Compute the costs involved in owning and buying an item such as an automobile.One-Time Costs (At the Time of Purchase)Sales Tax = Purchase Price * (Sales Tax Rate/100)Compute finance charges for single payment loans.Finance Charge = PrtP = The principal (the initial loan amount)r = annual interest rate (expressed as a decimal)t = time the loan is taken out for (typically in years)Compare installment and revolving credit costs.Loan Type: Installment Credit - Fixed loan amount, fixed team; Revolving Credit - Flexible borrowing up to credit limitRepayment Terms: Installment Credit - Fixed monthly payments; Revolving Credit - Flexible payments, minimum requiredInterest Rate: Installment Credit - Generally fixed, lower; Revolving Credit - Typically higher, can be variableCalculate installment loan costs such as amount financed the installment price, finance charge, and installment payment.Find the estimated annual percentage rate (APR) using a table.APR=2×Number of Payments per Year×Finance Charge/2×Number of Payments per Year×Finance ChargeLoan Amount6 Months12 Months24 Month$100020%18%16%$200018%16%14%$500015%14%12%Find the finance charge and new balance using the average daily balance method.Average Daily Balance = Sum of Daily Balances/Number of Days in Billing PeriodFinance Charge = Average Daily Balance * Daily Periodic Rate * Number of Days in Billing PeriodNew Balance = Previous Balance + Finance ChargeData Analysis and ProbabilityUsing a data set, determine mean, median, and mode.mean (average): the sum of all the data points divided by the number of data pointsmedian: the middle value when the dataset is ordered from least to greatest); if an even number of data points, the median the average of the two middle numbers.mode: the value that appears most frequently in the datasetDetermine the type of average that best represents the measure of central tendency.mean: symmetrical and no outliersmedian: skewed data or with outliersmode: categorical or multimodal dataDistinguish between a simple average and a weighted average and calculate each.simple average gives equal importance to all values in the data setweighted average adjusts the influence of each value according to its weightIdentify and construct various types of graphs and charts.bar chart (bar graph): categories are plotted along the x-axisdata values are plotted along the y-axiseach bar represents a category, and its height is proportional to the value of that categorypie chart:the entire pie represents the whole dataseteach slice of the pie represents a category or group, and the size of each slice corresponds to the percentage of the total that category represents line graph:the x-axis represents the time or other ordered variablethe y-axis represents the variable being measureddata points are plotted and connected by lineshistogram:the x-axis represents the bins(range of values)the y-axis represents the frequency or count of data points within each binbars are drawn to represent the frequency of each binscatter plot:the x-axis represents one variable, and the y-axis represents the other vaiableeach dot represents a pair of valuesbox plot (box-and-whisker plot):draw a rectangular box from Q1 to Q3, with a line at the medianextend "whiskers" from the box to the minimum and maximum values, excluding outliersoutliers are plotted as individual points outside the whiskersarea chart:the x-axis represents the variable (e.g., time)the y-axis represents the data valuesdata points are connected with lines, and the area under the lines is filled with colorbubble chart:the x-axis represents one variable, and the y-axis represents anotherthe size of each bubble is proportional to the third variableradar chart (spider chart)each axis represents a different variabledata points are plotted along each axis and connected to form a polygonstacked bar chart:each category is represented by a bar, and the bar is subdivided into smaller sections based on the data breakdownthe height of each section represents the proportion of the totalCompile and arrange facts in an organized manner for a table, chart, or figure.identify key variablesgather datacategorize the datalabel and format datadetermine the layoutcheck consistency and accuracyprovide contextExplain or prepare written summary of findings expressed in tables, charts, graphs, and figures.introduction to the datadescribe the trends or patternsmake use of charts and graphssummarize key findingscontextualize the findings ( if necessary)conclusion and recommendations (if applicable)Make predictions and decisions based on data and communicate their reasoning.review the data and trendsidentify the factors affecting the datamake predictionsmake decisions based on predictionscommunicate the reasoning summarize your conclusionDecimalsRound decimals to the nearest tenth and nearest hundredth.tenth: identify tenths and hundredths place -> if hundredths place =>5 round tenths place up 1 (leave the same otherwise) -> drop the any numbers after tenths placehundredth: identify hundredths and thousandths place -> if thousandths place =>5 round hundredths place up 1 (leave the same otherwise) -> drop the any numbers after hundredths placeAdd, subtract, multiply, and divide decimals.+: align decimal points -> add as normal-: align decimal points -> subtract as normal*: multiply normally -> count total decimal places from original numbers -> place the decimal so there is the same amount of decimal place/: move decimal point in both numbers to the right until they become whole numbers -> divide normally Convert a decimal to a fraction.count how many decimal places there is -> put that number underneath the decimal's original digits -> simplify ex: .25 -> 2 decimal places (hundredths) -> 100 -> 25/100 -> 1/4Convert a fraction to a decimal.numerator = dividend -> denominator = divisor -> divide as normalex: 1/4 -> 1 divided by 4 -> .25Read and write decimals using the place-value structure of the base-ten number system.read: read whole number normally if there is one (skip or say zero otherwise) -> say and or point for the decimal point -> read the fractional part as a whole number -> specify its place value.write: Identify the place value of each digit in the number. -> Write the digits in the appropriate positions relative to the decimal point. -> Include trailing zeros if a specific place value needs to be represented. Solve business problems with currency.Adding and Subtracting Money -Used for calculating totals or finding the remaining balance after a payment.Calculating Discounts - To find the price after a discount:Multiply the original price by the discount rate (discount=price×ratediscount=price×rate).Subtract the discount from the original price.Calculating Sales Tax - To include sales tax:Multiply the price by the tax rate (tax=price×ratetax=price×rate).Add the tax to the original price.Calculating Total Revenue - Revenue = Price × Quantity SoldFinding Profits - Profit = Revenue - CostsSplitting Bills or Payments -Divide the total by the number of people or payments.Currency Conversion - To convert currency:Multiply the amount by the exchange rate.DiscountsIdentify the various types of discounts.1. Trade Discount:Definition: A reduction in price given to a buyer by a seller, typically based on the buyer’s role in the distribution chain (e.g., wholesaler, retailer).2. Cash Discount:Definition: A discount offered to encourage prompt payment, usually within a specified time frame.3. Quantity Discount:Definition: A discount offered based on the quantity purchased, typically given to encourage larger orders.4. Seasonal Discount:Definition: A discount given for purchasing goods during the off-season or at a particular time of year.5. Promotional Discount:Definition: A discount used as part of a promotional campaign to increase sales or attract customers.6. Volume Discount:Definition: A discount given to customers who buy in large quantities, often to incentivize bulk buying.7. Rebate:Definition: A form of discount that involves offering a partial refund to the customer after the purchase.8. Cash-on-Delivery (COD) Discount:Definition: A discount offered when the buyer makes the payment in full at the time of delivery.9. Member/Employee Discount:Definition: A discount given to members of a club, employees, or affiliates of a company as a perk.Calculate the percentage of a discount of an item.Discount Percentage= Original Price/Amount of Discount ×100Calculate a chain discount and net selling price.Net Selling Price=List Price×(1−First Discount Rate)×(1−Second Discount Rate)×…Each discount is applied to the price that results from the previous discount. Calculate trade discounts and net selling price.Net Selling Price=List Price×(1−First Discount Rate)×(1−Second Discount Rate)×…Calculate cash discounts and net selling price.Net Selling Price=Original Price×(1−Cash Discount Rate)FractionsAdd, subtract, divide, and multiply fractions and mixed numbers with like (common) and unlike denominators.add -With like denominators:Simply add the numerators and keep the same denominator.With unlike denominators:Find a common denominator (least common multiple, LCM), adjust the fractions, and then add.mixed numbers:Convert to improper fractions, perform the operation, and convert back to mixed numbers if needed.subtract -With like denominators:Subtract the numerators and keep the denominator.With unlike denominators:Find a common denominator, adjust the fractions, and subtract.mixed numbers:Convert to improper fractions, perform the operation, and convert back to mixed numbers if needed.divide - Invert (flip) the second fraction and multiply.mixed numbers:Convert to improper fractions first.multiply - Multiply the numerators and denominators directly.mixed numbers:Convert to improper fractions first.Work flexibly with fractions to solve application problems.understand the problemchoose a strategy simplify before solving(Visualize, Estimate, Check Units, Practice)Convert fractions to decimals, decimals to fractions, and use properties of basic operations to simplify fractional arithmetic expressions.fractions -> decimals - Divide the numerator by the denominator.If the division doesn't terminate, identify if it's a repeating decimal.decimals -> fractions - Identify the decimal typeterminating:write the decimal as a fraction with a denominator of 10^n, where n is the number of decimal placessimplifyrepeating:let x equal the repeating decimalmultiply x by 10^n (where n aligns the repeating digits)subtract the equations to eliminate the repeating part, then solve for xuse properties of basic operations -addition and subtraction of fractions:find a common denominatorconvert fractions to have the same denominatoradd or subtract numerators, keeping the denominator the samemultiplication of fractions:multiply numerators togethermultiply denominators togethersimplifydivision of fractions:invert (reciprocal) the second fractionsmultiplyConvert an improper fraction to a whole or mixed number.divide the numerator by the denominatorthe quotient becomes the whole numberthe remainder becomes the numerator of the proper fractionswrite the mixed numbercombine the whole number and the proper fractionsimplify the fraction if possibleConvert a whole or mixed number to an improper fraction.whole number - write the whole number as a fraction with a denominator of 1mixed number - multiply the whole number by the denominator of the fraction partadd the numerator of the fraction part to the productwrite the result as the numerator, keeping the original denominatorReduce a fraction to lowest terms and raise a fraction to higher terms.reduce a fraction to lowest terms - find the GCD of the numerator and denominatordivide both the numerator and denominator by the GCDwrite the simplified fractionraising a fraction to higher terms - choose the number by which to multiplymultiply both the numerator and denominator by that numberApply appropriate methods for computing with fractions from among mental computation, estimation, calculators, or computers and paper and pencil.When to use ->mental computation:fractions with small numerators and denominatorscommon equivalent fractions or benchmarks like 1/2, 1/4, 3/4estimating results quickly(small numbers, easy relationships_estimation:complex problems where an exact answer isn't neededto check if an answer is logical(quick checks or approximations)calculators:large or complex fractionsreal-world applications requiring high accuracy (e.g., financial or scientific calculations)(precision or complexity)computers:repeated calculations or large datasetsproblems requiring graphs or models (e.g., excel for budgeting or analysis)(precision or complexity)paper and pencil:- solving problems manually to understand the processshowing work in exams or assignments(learning, showing work, or detailed manual solving)PercentagesDefine the terms base, rate, and percentage and identify them in word problems.base:def - The whole amount or the quantity that you're working with. It is the value from which a percentage is calculated. word prob - usually the total or initial amount given in the problem. rate:def - The percentage or proportion that is being applied to the base. It is often represented as a percentage, fraction, or decimal. word prob - often the percentage or ratio stated.percentage:def - The result of applying the rate to the base. It's the amount that represents the rate of the base. word prob - the actual value derived from applying the rate to the base.Solve for base, rate, or part in a percent problem.part = base * ratesolve for part (percentage value) -> base = part/ratesolve for base(total) -> base = part/ratesolve for rate(percentage) -> rate = part/basepart = the result or amount you're calculating (what you're trying to find)base = the whole or total amountrate = the percentage (in decimal form)Convert a whole number, fraction, or decimal to a percent.whole number -> whole number * 100 = percentfraction -> (numerator/denominator) * 100 = percentdecimal -> decimal *100 = percentConvert a percent to a whole number, fraction, or decimal.whole number -> percent/100 = whole numberfraction -> percent = percent number/100decimal -> percent/100 = decimalUse the percentage formula to find the unknown value when two values are known.refer to #2Find the rate or the base in the increase or decrease of an item.increase or decrease = the change in value (- or +)base = the original value (before increase or decrease)rate = the percentage change (as a decimal)amount of change = base * rateto find the rate (percentage change) -> rate = amount of change/baseto find the base (original value) -> amount of change/rateCalculate markup and markdown of original value of item.markup:formula - markup = original price * markup rateselling price after markup - selling price = original price + markupmarkdown:formula - markdown = original price * markdown ratenew price after markdown = new price = original price - markdownCalculate simple interest.simple interest (si) = pricipal (p) * rate (r) * time (t)principle (p) = initial amount of money invested/borrowedrate (r) = annual interest rate (decimal)time (t) = time period which the money is invested/borrowed, usually in yearstotal amount (a) after interest -> a = p + simple interest (si)