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There are three main types of savings accounts, saving accounts, certificates of deposit (CD), and money market accounts. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
Savings account are very risk free, and so yield low interest rates. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
The differences in the three types of savings accounts relate to their convenience. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
Savings account typically have a lower interest rate that money market accounts, which typically have lower interest rates than CDs. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
Ordinary annuities more accurately reflect how we save, in that money is deposited repeatedly over time. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
Spreadsheet software, such as Google Sheets, have built in functions that can be used to quickly calculate both the future value of an ordinary annuity account, but also the payment necessary to reach a goal using an ordinary annuity. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
There are many different investments with different returns and risks. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
Bonds are loans form the purchaser to the entity selling the bond. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
Bonds have some tax benefits, low to no risk, and a low return. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
Stocks represent part ownership in a company. As such, stock holders share in the profits, and losses, of the company. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
Information, including price, P/E, yearly highs and lows, and dividend amount can be found in online stock tables available on many websites. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
Mutual funds represent collections of professionally administered investment vehicles. Have shares in a mutual fund has lower risk than ownership of stocks. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
Retirement accounts employ some of the same strategies as mutual funds, in that they spread the risk and are professionally managed. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
IRAs and Roth IRAs differ on when taxes are paid on the money, and who can use them. Roth IRAs have income limits while traditional IRAs do not. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
There are many reasons for a loan, but primarily it is taken out for a large expense when cash is not available. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
Each payment for an installment loan consists of an interest portion and a principal portion. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
There is a formula to calculate the payment necessary to pay off a loan in installments. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
Amortization schedules, or tables, show how each payment is applied to principal and interest. It also includes other details such as remaining balance and total interest paid. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
Loans often have other fees associated with them such as origination fees or application fees. The total of the interest paid and the fees is the cost of finance. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
The FAFSA must be filled out each year that a student wishes to borrow for. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
A studentâs funding gap determines how much they need in loans to pay for college. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
Federal subsidized student loans defer payments until after graduation and interest does not accrue on these loans. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
Unsubsidized student loans defer payment until after graduation but interest begins accruing as soon as the loan finds are disbursed. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
There are both yearly and aggregate limits for student loans to prevent over-borrowing, among other reasons. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
Federal direct loans have a low interest rate set by the government, but other student loans have varying rates of interest set by the banks. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
The standard repayment plan lasts 10 years and is made up of monthly payments. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
Consolidating or refinancing student loans merges many student loans into one loan. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
If only federal loans are consolidated, the interest rate is the same as the individual loans, currently set at 4.99%. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
If other loans are refinanced together, the interest rate may be lower with the new loan. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
Other repayment plans are available. Such a plan may have payment that start small and grow as the loan is paid off, or it may have a longer term, or may be based on the discretionary income of the student. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
Being delinquent on a student loan is a precursor to being in default. Making payments in a timely fashion allows the student to avoid this situation. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
Credit cards can be a flexible way to pay for almost anything, but can become a financial hazard if used unwisely. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
When deciding which credit card to apply for, evaluate the interest rate, fees (annual and late), reward programs and credit limit. Be sure they meet your criteria. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
Paying off the balance of your credit card every month will control your spending and will never result in paying interest. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
Credit card statements hold all important information about your credit card, including payment, balances, charges and billing cycle dates. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
Although the minimum payment is attractive precisely because it is so small, paying only the minimum results is a long payoff term and higher interest costs. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
There are many factors to consider when choosing to buy or lease a car. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
The cost of the car is increased by a number of fees and sales tax. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
There are advantages to buying a car and advantages to leasing a car. The decision between the two depends on the preference of the buyer. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
Insurance covers costs associate with accidents. It is made up of various components. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
The costs of owning a car, including insurance and maintenance, should be a part of the budgeting process. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
Budgeting for unexpected repairs can ease the stress of encountering large repair bill. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
There are many points of comparison between renting and buying a house. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
Before deciding to buy a house, you should carefully consider all the responsibilities that come with home ownership. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
Renting comes with more restrictions on the renter, but with fewer costs and is easier to move from. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
Owning a house has more costs but has more freedom, plus the owner creates equity. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
Mortgages are loans, and payments are calculated in the same way as any other loan. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
Amortization tables help a homeowner understand the mortgage and how the payments are applied to the principal and interest. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
In addition to paying the amount financed for a mortgage, the monthly payment will include an escrow payment, which covers insurance and taxes. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
Federal income tax is based on income after certain adjustments. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
Gross income is income from all sources, including gifts and winnings. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
Before taxes are calculated, the taxable income is found by subtracting deductions and exemptions from gross income. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
Income tax is progressive, increasing in rate as income increases. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
Being in the 32% tax bracket means some of your income is taxed at 10%, some at 12%, some at 22%, some at 24%, and the rest at 32%. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
Income in each tax bracket is taxed at that bracketâs rate, which means in 2022 the first $10,275 earned is taxed at 10% only. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
Tax credits are subtracted from the taxes that are owed. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
Some tax credits are refundable, which means they can make the amount you owe negative, which results in a refund. | https://openstax.org/books/contemporary-mathematics/pages/6-key-concepts |
Pnr=n!(nâr)!Pnr=n!(nâr)! | https://openstax.org/books/contemporary-mathematics/pages/7-formula-review |
The formula for counting combinations is:nCr=n!r!(nâr)!nCr=n!r!(nâr)! | https://openstax.org/books/contemporary-mathematics/pages/7-formula-review |
For an experiment whose sample spaceSSconsists of equally likely outcomes, thetheoretical probabilityof the eventEEis the ratioP(E)=n(E)n(S)P(E)=n(E)n(S)wheren(E)n(E)andn(S)n(S)denote the number of outcomes in the event and in the sample space, respectively. | https://openstax.org/books/contemporary-mathematics/pages/7-formula-review |
P(E)=1âP(Eâ²)P(E)=1âP(Eâ²) | https://openstax.org/books/contemporary-mathematics/pages/7-formula-review |
For an eventEE,odds forE=n(E):n(Eâ²)=P(E):P(Eâ²)=P(E):(1âP(E))odds againstE=n(Eâ²):n(E)=P(Eâ²):P(E)=(1âP(E)):P(E)odds forE=n(E):n(Eâ²)=P(E):P(Eâ²)=P(E):(1âP(E))odds againstE=n(Eâ²):n(E)=P(Eâ²):P(E)=(1âP(E)):P(E) | https://openstax.org/books/contemporary-mathematics/pages/7-formula-review |
If the odds in favor ofEEareA:BA:B, thenP(E)=AA+BP(E)=AA+B. | https://openstax.org/books/contemporary-mathematics/pages/7-formula-review |
IfEEandFFare mutually exclusive events, thenP(EorF)=P(E)+P(F)P(EorF)=P(E)+P(F). | https://openstax.org/books/contemporary-mathematics/pages/7-formula-review |
IfEEandFFare events that contain outcomes of a single experiment, thenP(EorF)=P(E)+P(F)âP(Eâ¢andâ¢F)P(EorF)=P(E)+P(F)âP(Eâ¢andâ¢F). | https://openstax.org/books/contemporary-mathematics/pages/7-formula-review |
IfEEandFFare events associated with the first and second stages of an experiment, thenP(Eandâ¢F)=P(E)ÃP(F|E)P(Eandâ¢F)=P(E)ÃP(F|E). | https://openstax.org/books/contemporary-mathematics/pages/7-formula-review |
Suppose we have a binomial experiment withnntrials and the probability of success in each trial ispp. Then:P(number of successes isa)=CnaÃpaÃ(1âp)nâaP(number of successes isa)=CnaÃpaÃ(1âp)nâa | https://openstax.org/books/contemporary-mathematics/pages/7-formula-review |
IfOOrepresents an outcome of an experiment andn(O)n(O)represents the value of that outcome, then the expected value of the experiment is:ân(O)ÃP(O)ân(O)ÃP(O)whereΣΣstands for the sum, meaning we add up the results of the formula that follows over all possible outcomes. | https://openstax.org/books/contemporary-mathematics/pages/7-formula-review |
whereΣΣstands for the sum, meaning we add up the results of the formula that follows over all possible outcomes. | https://openstax.org/books/contemporary-mathematics/pages/7-formula-review |
The Multiplication Rule for Counting is used to count large sets. | https://openstax.org/books/contemporary-mathematics/pages/7-key-concepts |
Using the Multiplication Rule for Counting to enumerate permutations. | https://openstax.org/books/contemporary-mathematics/pages/7-key-concepts |
Simplifying and computing expressions involving factorials. | https://openstax.org/books/contemporary-mathematics/pages/7-key-concepts |
Using factorials to count permutations. | https://openstax.org/books/contemporary-mathematics/pages/7-key-concepts |
Permutations are used to count subsets when order matters; combinations work when order doesn't matter. | https://openstax.org/books/contemporary-mathematics/pages/7-key-concepts |
Combinations can also be computed using factorials. | https://openstax.org/books/contemporary-mathematics/pages/7-key-concepts |
We identify the sample space of an experiment by identifying all of its possible outcomes. | https://openstax.org/books/contemporary-mathematics/pages/7-key-concepts |
Tables can help us find a sample space by keeping the possible outcomes organized. | https://openstax.org/books/contemporary-mathematics/pages/7-key-concepts |
Tree diagrams provide a visualization of the sample space of an experiment that involves multiple stages. | https://openstax.org/books/contemporary-mathematics/pages/7-key-concepts |
The theoretical probability of an event is the ratio of the number of equally likely outcomes in the event to the number of equally likely outcomes in the sample space. | https://openstax.org/books/contemporary-mathematics/pages/7-key-concepts |
Empirical probabilities are computed by repeating the experiment many times, and then dividing the number of replications that result in the event of interest by the total number of replications. | https://openstax.org/books/contemporary-mathematics/pages/7-key-concepts |
Subjective probabilities are assigned based on subjective criteria, usually because the experiment canât be repeated and the outcomes in the sample space are not equally likely. | https://openstax.org/books/contemporary-mathematics/pages/7-key-concepts |
The probability of the complement of an event is found by subtracting the probability of the event from one. | https://openstax.org/books/contemporary-mathematics/pages/7-key-concepts |
We use permutations and combinations to count the number of equally likely outcomes in an event and in a sample space, which allows us to compute theoretical probabilities. | https://openstax.org/books/contemporary-mathematics/pages/7-key-concepts |
Odds are computed as the ratio of the probability of an event to the probability of its compliment. | https://openstax.org/books/contemporary-mathematics/pages/7-key-concepts |
The Addition Rule is used to find the probability that one event or another will occur when those events are mutually exclusive. | https://openstax.org/books/contemporary-mathematics/pages/7-key-concepts |
The Inclusion/Exclusion Principle is used to find probabilities when events are not mutually exclusive. | https://openstax.org/books/contemporary-mathematics/pages/7-key-concepts |
Conditional probabilities are computed under the assumption that the condition has already occurred. | https://openstax.org/books/contemporary-mathematics/pages/7-key-concepts |
The Multiplication Rule for Probability is used to find the probability that two events occur in sequence. | https://openstax.org/books/contemporary-mathematics/pages/7-key-concepts |
Binomial experiments result when we count the number of successful outcomes in a fixed number of repeated, independent trials with a constant probability of success. | https://openstax.org/books/contemporary-mathematics/pages/7-key-concepts |
The binomial distribution is used to find probabilities associated with binomial experiments. | https://openstax.org/books/contemporary-mathematics/pages/7-key-concepts |
Probability density functions (PDFs) describe the probabilities of individual outcomes in an experiment; cumulative distribution functions (CDFs) give the probabilities of ranges of outcomes. | https://openstax.org/books/contemporary-mathematics/pages/7-key-concepts |
The expected value of an experiment is the sum of the products of the numerical outcomes of an experiment with their corresponding probabilities. | https://openstax.org/books/contemporary-mathematics/pages/7-key-concepts |
The expected value of an experiment is the most likely value of the average of a large number of replications of the experiment. | https://openstax.org/books/contemporary-mathematics/pages/7-key-concepts |
Suppose we have a set of data withnnvalues, ordered from smallest to largest. Ifnnis odd, then the median is the data value at positionn+12n+12. Ifnnis even, then we find the values at positionsn2n2andn2+1n2+1. If those values are namedaaandbb, then the median is defined to bea+b2a+b2. | https://openstax.org/books/contemporary-mathematics/pages/8-formula-review |
s=â(x-x¯)2n-1s=â(x-x¯)2n-1 | https://openstax.org/books/contemporary-mathematics/pages/8-formula-review |
Here,sis the standard deviation,xxrepresents each data value,x¯x¯is the mean of the data values,nnis the number of data values, and the capital sigma (ΣΣ) indicates that we take a sum. | https://openstax.org/books/contemporary-mathematics/pages/8-formula-review |
Ifxxis a member of a normally distributed dataset with meanµµand standard deviationÏÏ, then the standardized score forxxis | https://openstax.org/books/contemporary-mathematics/pages/8-formula-review |
If you know azz-score but not the original data valuexx, you can find it by solving the previous equation forxx: | https://openstax.org/books/contemporary-mathematics/pages/8-formula-review |
If a line has slopemmand passes through a point(x0,y0)(x0,y0), then the point-slope form of the equation of the line is: | https://openstax.org/books/contemporary-mathematics/pages/8-formula-review |
Supposexxandyyare explanatory and response datasets that have a linear relationship. If their means arex¯x¯andy¯y¯respectively, their standard deviations aresxsxandsysyrespectively, and their correlation coefficient isrr, then the equation of the regression line is: | https://openstax.org/books/contemporary-mathematics/pages/8-formula-review |
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