Introduction
You face constant financial choices about loans, investments, and daily purchases without knowing if the numbers work in your favor. Innumeracy leaves you vulnerable to hidden costs, predatory interest rates, and misleading commercial claims.
Robert L. Hershey demonstrates in All the Math You Need to Get Rich that pocket calculators and basic formulas make financial mathematics accessible to everyone. You do not need an advanced degree to master the quantitative skills that drive wealth accumulation.
The book provides a systematic guide to mastering financial arithmetic across four key areas. You will examine percentages and probabilities to avoid commercial traps, learn to handle big numbers, and master cash flows. You will calculate compound interest, evaluate mortgages, and analyze trade-off decisions to build long-term wealth.
- How do you calculate the true cost of a loan or credit card balance?
- What distinguishes a sound investment from a mathematical gamble?
- How does compound interest transform modest savings into a fortune?
1. Percentages and Probabilities
Failing to understand percentages leads to severe business and pricing errors. Markup percentage must be calculated using the wholesale price as the base, not the retail price, to avoid severe miscalculations.
Markup percentage must be calculated using the wholesale price as the base, not the retail price, to avoid severe miscalculations.
A clothing store owner thought his markup was 5 percent because he bought a bra for $1 and sold it for $5, whereas the actual markup was 400 percent. A clothing store also recorded a $3,222.50 wholesale price and a $5,985.00 retail price resulting in an 85.7% markup. You can calculate markups using a few straightforward steps:
- Find the actual dollar amount of the markup by subtracting the wholesale price from the retail price
- Turn the dollar markup into a decimal fraction by dividing it by the wholesale price
- Move the decimal point two places to the right to get the percentage
Changing the base number in percentage calculations drastically alters the resulting percentage, a tactic often used for misleading claims. Smaller bases yield larger percentages for the same absolute amount, which parties with an axe to grind exploit to make price hikes sound either massive or trivial.
A $0.25 price increase on a $0.75 candy bar is a 33.3% increase over the original price, but only 0.0224% of Juan Rodriguez's weekly salary of $1,117.50 and 20% of Junior's weekly allowance of $1.25. The Rule of 70 provides a rapid shortcut to estimate the time required for a quantity to double under a constant annual percentage growth rate. By dividing the percentage growth rate as a whole number into 70, you instantly get the approximate doubling period in years.
Linda Williams invests $500 at 6% interest; dividing 70 by 6 gives a doubling period of approximately 12 years. Charles Warren experienced a 7 percentage growth rate leading to a 10-year doubling period for his salary. A house value took an 8-year period to double from $80,000 to $160,000, yielding a 9% annual growth rate. You can use these two operations for growth planning:
- Divide the percentage of growth into 70 to find the doubling period
- Divide the doubling period into 70 to find the percentage of growth
Mathematical Expectations
Mathematical expectation measures the true long-term value of a chance-based decision by weighing winning payoffs against losing penalties. Commercial gambling and insurance options invariably feature negative expectations for the player.
Mathematical expectation measures the true long-term value of a chance-based decision by weighing winning payoffs against losing penalties.
Roger Young calculates that not buying collision insurance has a mathematical expectation of -$550 on a $30,000 car, whereas buying a $900 collision insurance policy with a $100 deductible results in -$926 due to premiums and deductibles. An unfair coin toss game yields a -$0.50 expectation. You follow these steps to compute expected value:
- Determine all possible outcomes and their respective probabilities so they add up to 1
- Multiply each outcome's financial gain or loss by its probability
- Sum the products to find the final mathematical expectation
The probability of multiple independent events all happening together is calculated by multiplying their individual probabilities. Multiplying probabilities of numbers less than one results in an exponentially smaller combined probability.
Bart London bets on four football parlay cards with a 50% chance on each; the joint probability of winning all four is 0.0625, leading to a negative mathematical expectation. Rolling snake eyes with two dice has a 1/36 probability. A crossword puzzle contest requiring 10 correct letter choices has a joint probability of winning of only 0.000003014. You evaluate combined chances with these actions:
- Verify that each event is completely independent of the others
- Multiply the individual probability of each event by the others to find the joint probability
2. Big Numbers and Cash Flows
Scientific notation simplifies cumbersome large and small numbers while tracking cash flows prevents first-cost blindness.
Scientific notation uses powers of 10 with exponents to easily write and compute cumbersome large numbers. When multiplying or dividing numbers in scientific notation, you multiply or divide the leading digits and add or subtract the exponents respectively.
Multiplying (2 * 10^2) by (4 * 10^3) yields 8 * 10^5, which is equal to 800,000.
You encounter various large quantities in commercial contexts:
- 1,000,000 = 10^6 (a million)
- 1,000,000,000,000 = 10^12 (a trillion)
- 1,000,000,000,000,000,000 = 10^18 (a quintillion)
- 10^24 (a septillion)
Negative exponents in scientific notation allow for writing and calculating numbers less than one by indicating decimal place shifts. A negative exponent represents the number of zeros after the 1 in the denominator of a fraction or the number of decimal places to the left.
A charity in New York City with a population of 7.9 million raising an average of 0.015 cents ($0.00015 or 1.5 * 10^-4) per person yields $1,185 total.
Small quantities follow similar exponential patterns:
- 1 tenth = 0.1 = 10^-1
- 1 millionth = 0.000001 = 10^-6
- 0.0000186 = 1.86 * 10^-5
First-Cost Blindness
Failing to project long-term operating costs leads to unexpected financial strain, zero cash flow, and potential repossession of assets when hidden expenses materialize.
First-cost blindness occurs when someone focuses entirely on the initial down payment while ignoring recurring operating expenses. Dick Ugson bought a gold and silver Fishtail 8 for $54,995 using all his $19,200 savings as a down payment, leaving him with monthly expenses of $2,100 that completely consumed his take-home pay and led to his car being repossessed.
Evaluating personal or business financial viability requires calculating cash flow by subtracting total monthly or annual expenses from total income. To determine true financial health, all irregular annual expenses must first be converted into monthly figures.
Tony Jackson calculated his family expenses with a boat would total $3,133.75 against an income of $2,700, resulting in a negative monthly cash flow of -$433.75, prompting him not to buy the boat.
You calculate cash flow with these actions:
- Restate all periodic or annual costs as monthly costs by dividing annual figures by 12.
- Add all monthly expenses together to find total expenses.
- Subtract total expenses from total monthly income to find the cash flow.
The foundational interest formula relates interest payment amount, principal, interest rate, and time as I = Prt. This can be algebraically rearranged to solve for unknown variables such as the interest rate r = I / Pt. Borrowing $5,000 at an 18% interest rate for 3 years generates $2,700 in total interest.
When a loan principal is repaid incrementally over time, interest must be calculated using the average unpaid balance rather than the initial principal. The average unpaid balance is found by summing the beginning and ending balances and dividing by 2.
Sam the loan shark offered Louis Stratton a $270 loan repaid via three monthly payments of $100, claiming an 11% interest rate, whereas calculating based on the average unpaid balance revealed a true annual interest rate of 64%.
You determine loan shark terms with these actions:
- Find the average unpaid balance by summing the beginning balance and ending balance and dividing by 2.
- Calculate the interest paid over the specific period divided by the average unpaid balance.
- Normalize the resulting periodic interest rate to an annual basis by multiplying appropriately.
3. Compound Interest and Mortgages
Compound interest tables and standard formulas allow you to project future worth, present values, and mortgage payments.
Compound interest means collecting interest on interest, where interest is added to the balance at the end of every period, dramatically accelerating growth over time. More frequent compounding leads to a slightly higher effective yield than the quoted annual rate due to compounding within the year.
If the Indians had invested the $24 proceeds from selling Manhattan in 1626 at an 8% annual compound interest rate, it would have grown to $82 trillion by 2001, which is enough to buy Manhattan back and pay every resident $55 million.
The future worth of an investment can be calculated efficiently using the formula F = P * (1 + r)^n or by utilizing factor tables found in the book. Alternatively, the U column in the interest factor tables provides ready-made multipliers for specific rates and periods.
Dorothy Worthington loans her son $2,500 at 6% interest compounded annually for 8 years, yielding $3,985 using the table factor of 1.594.
You calculate future worth with these actions:
- Identify the principal (P), periodic interest rate (r), and number of periods (n).
- Calculate using the formula F = P * (1 + r)^n or look up the U factor in the table corresponding to the rate and periods.
- Multiply the principal by the factor to get the future worth (F).
Inflation acts like negative compound interest, eroding the purchasing power of money over time if investment returns do not outpace it. Prices grow compound-wise according to the inflation rate, meaning money kept in low-yielding bank accounts loses real purchasing power.
A can of peaches costing $0.87 will cost $2.26 in 10 years at a 10% annual inflation rate, calculated using the 10% table factor of 2.594.
You evaluate inflation impacts with these actions:
- Identify the current price, the annual inflation rate, and the number of years.
- Locate the U factor in the corresponding rate table for the given number of years.
- Multiply the current price by the U factor to find the future price.
Using the V column in interest tables allows investors to calculate the present principal needed to reach a specific future worth. While the U column projects present money into the future, the V column reverses the process to discount future sums back to their present value.
Minnie Sanchez wants $18,000 in 6 years for a new car, using a 7% semi-annual certificate. Using the 3.5% table V factor of 0.6618 for 12 periods, she must deposit $11,912.40 today.
You find present principal with these actions:
- Determine the future worth, periodic interest rate, and total number of periods.
- Find the V factor in the table corresponding to the periodic rate and total periods.
- Multiply the future worth by the V factor to find the present principal (P = F * V).
Annuities and Mortgages
An annuity represents a series of repeated equal payments, and its future worth can be determined using the Y column in the interest tables. Regular periodic deposits accumulate compound interest over time.
Milton Edwards pays $35 monthly on his credit card at 1.5% per month for 2 years (24 months). Using the Y factor of 28.634, his payments are worth $1,002.19 to the credit card company.
You calculate annuity values with these actions:
- Identify the annuity payment amount (A), the periodic interest rate, and the total number of payment periods (n).
- Locate the Y factor in the table corresponding to the periodic rate and n.
- Multiply the annuity by the Y factor to get the future worth (F = AY).
Combining term insurance with separate high-yield investments typically yields a better return than buying an endowment or cash-value life insurance policy. Endowment policies bundle low-yielding savings with insurance, whereas buying cheap term insurance and investing the difference exploits higher interest rates.
Simon Walters chooses between a 20-year endowment policy giving just over 5% return and a term insurance policy plus investing $520 annually at 7%, which results in $21,317.40 after 20 years.
You compare insurance strategies with these actions:
- Calculate the annual savings achieved by buying lower-cost term insurance instead of an endowment policy.
- Use the formula F = AY with the appropriate table factor to calculate the future value of the annual savings at the investment interest rate.
- Compare the final accumulated savings against the endowment payout to choose the better option.
Mortgage payments, covering both principal and interest, are calculated using the X column of the interest tables and the mortgage principal. The X column factor converts a present principal loan into an annuity of equal periodic payments over the life of the mortgage.
Marvin Wolinski takes out a $63,000, 15-year mortgage with quarterly payments and 12% annual interest compounded quarterly. Using an X factor of 0.03613, his quarterly payment is $2,276.19.
You compute mortgage payments with these actions:
- Subtract the down payment from the property value to find the mortgage principal (P).
- Determine the periodic interest rate and the total number of payment periods (n).
- Find the X factor in the corresponding table for n and multiply it by the principal to get the periodic payment (A = PX).
Retirement fund accumulation and drawdown calculations use the W and Z columns of the interest tables respectively. The W column helps determine the periodic deposit needed to reach a specific future retirement sum, while the Z column calculates the principal needed to fund a series of periodic withdrawals.
Stephanie Swift wants to save $60,000 over 25 years at 5.5% interest. Using the W factor of 0.01955, she must deposit $1,173 at the end of each year.
You plan retirement funds with these actions:
- For accumulation goals, find the W factor for the target future sum and multiply F by W to find the required annual deposit (A = FW).
- For drawdown goals, find the Z factor for the desired withdrawal period and multiply the periodic withdrawal by Z to find the required starting principal (P = AZ).
4. Investing and Trade-Offs
Diverting funds from lotteries and credit card debt into diversified investments creates exponential wealth through trade-off analysis.
Over 50 percent of American households now own stocks, compared with 19 percent in 1983. Money market mutual funds keep their share price constant at one dollar while accumulating variable interest, acting as liquid cash reserves.
Harold West plans to put one thousand dollars at the end of each year for ten years into a money market fund paying six percent, resulting in a future worth of thirteen,one hundred eighty-one dollars.
Bonds are debt promises paying fixed interest and returning the principal at maturity, and their resale value moves inversely to market interest rates. Ben Bowman considers buying an eight percent coupon rate bond maturing in five years with a face value of one thousand dollars for nine hundred ninety dollars, yielding 5.9 percent.
Stock overall yield combines the annual increase in share price with annual dividends, divided by the initial share price. Avery Inverness owns shares of Amalgamated Thingees Corporation, where a quarterly dividend of sixty cents, a starting price of eighty-five dollars, and a current price of ninety dollars produce an overall yield of 8.71 percent. Mutual funds pool various securities to balance risks and rewards, but management fees and load charges reduce the overall yield.
Carrying credit card debt creates severe long-term financial drains, whereas investing those same amounts compounds into significant wealth. Patience Partridge calculates that maxing out her credit card at eighteen percent interest costs nine hundred dollars a year in finance charges, whereas investing that same nine hundred dollars annually in an S&P 500 index fund at fifteen percent growth yields eighteen thousand two hundred seventy-four dollars after ten years.
Routine spending on lottery tickets results in predictable long-term losses, whereas investing equivalent amounts accumulates into a million-dollar portfolio. Twin brothers Tim and Lucky illustrate this: Lucky bets ten dollars a day on the lottery for forty-five years and loses seventy thousand two hundred dollars to finish with zero, while Tim invests half that amount at a ten percent growth rate and reaches one million one hundred twenty-one thousand four hundred ninety-two dollars.
Trade-Offs and Scoring
When faced with competing priorities under a fixed limitation, you must first recognize the trade-off and then calculate a table of choices to make an intelligent decision.
- Realize that a trade-off exists.
- Calculate a table listing the choices.
- Examine the table of possible choices to make an intelligent decision.
Percival Ugmeister packed his suitcase for a trip to Hawaii and Alaska within a 70-lb limit. He brought too many summer clothes, suffered in Alaska, and had to buy heavy clothes for $1,388.92 and pay a $75 overweight baggage fee on return. Household financial decisions often require trading off immediate recreational spending against long-term savings goals.
The Frederick family had $410.62 left monthly after necessities. Saving it all at 6% interest yielded $27,775.98 in 5 years, whereas spending $4,500 yearly on vacations pushed the wait time out to 27 years. When a decision involves multiple competing objectives, a weighting and scoring system uses numerical values to rank choices objectively.
- Figure out how important each objective is on a scale of 1 to 10.
- Score each option from 1 to 10 for each objective.
- Multiply the weighting by the score for each category.
- Total up the ratings for each option to make the decision.
Don Gordon evaluated three cars across price, looks, and fuel economy, weighting price at 10, looks at 6, and fuel economy at 2. The Frugal-4 scored a total rating of 130 and was chosen over the Calamari Spyder and Provincial-6. Becoming a millionaire requires making more money than you spend through career progression and continuously investing the difference.
10 Key Ideas
- Markup percentage must be calculated using the wholesale price as the base, not the retail price, to avoid severe miscalculations.
- The Rule of 70 provides a rapid shortcut to estimate the time required for a quantity to double under a constant annual percentage growth rate.
- Mathematical expectation measures the true long-term value of a chance-based decision by weighing winning payoffs against losing penalties.
- Scientific notation simplifies cumbersome large and small numbers while tracking cash flows prevents first-cost blindness.
- Failing to project long-term operating costs leads to unexpected financial strain, zero cash flow, and potential repossession of assets when hidden expenses materialize.
- When a loan principal is repaid incrementally over time, interest must be calculated using the average unpaid balance rather than the initial principal.
- Compound interest tables and standard formulas allow you to project future worth, present values, and mortgage payments.
- Carrying credit card debt creates severe long-term financial drains, whereas investing those same amounts compounds into significant wealth.
- Routine spending on lottery tickets results in predictable long-term losses, whereas investing equivalent amounts accumulates into a million-dollar portfolio.
- When faced with competing priorities under a fixed limitation, you must first recognize the trade-off and then calculate a table of choices to make an intelligent decision.