Money Math Center
Learn the math behind everyday money decisions. Work with budgets, percentages, interest, loans, savings, prices, wages, profit, markup, margin, taxes, tips, inflation and more—with formulas and plain-language explanations.
Budget & Cash Flow Math
See where money goes and turn totals into useful percentages.
Monthly Budget
Leftover = income − expenses. Savings rate = leftover ÷ income × 100.
What is this for?
Check whether your monthly take-home income covers expenses and what remains for goals.
How it works · worked example
Leftover = income − expenses. Leftover share = leftover ÷ income × 100.
- $4,000 income − $3,100 expenses = $900 remaining.
- $900 ÷ $4,000 = 0.225. Multiply by 100 to get 22.5%.
- That $900 is available to allocate; it only becomes savings if you actually save it.
Check your understanding: Use the same monthly period and take-home income. Include irregular expenses. A negative result means a shortfall; a savings percentage is undefined when income is zero.
50/30/20 Guide
A planning rule, not a requirement: 50% needs, 30% wants, 20% saving/debt goals.
What is this for?
Make a starting plan for dividing take-home pay among needs, wants, and savings or extra debt payments.
How it works · worked example
Needs = income × 0.50; wants = income × 0.30; goals = income × 0.20.
- With $4,000 take-home pay, needs receive $2,000.
- Wants receive $1,200 and saving/debt goals receive $800.
- Check: $2,000 + $1,200 + $800 = $4,000.
Check your understanding: This is a flexible planning framework. High housing costs, irregular income or urgent debt may require different proportions. Keep categories consistent and avoid counting an expense twice.
Percentages in Money
Percent math appears in sales, taxes, tips, raises, interest, investment returns and price changes.
Discount + Sales Tax
Discount first, then tax the discounted price.
What is this for?
Estimate a checkout total when a percentage discount is applied before sales tax.
How it works · worked example
Discount = price × discount% ÷ 100. Tax = discounted price × tax% ÷ 100.
- A $100 item at 20% off saves $20, leaving $80.
- Tax at 7% is $80 × 0.07 = $5.60.
- Add tax: $80 + $5.60 = $85.60.
Check your understanding: Tax treatment depends on the location and type of discount. This model taxes the discounted amount and excludes shipping, other fees and taxable-item exceptions.
Tip + Split Bill
What is this for?
Work out a tip and divide the full bill equally among a group.
How it works · worked example
Tip = bill × tip% ÷ 100; share = (bill + tip) ÷ people.
- For an $80 bill and 20% tip, $80 × 0.20 = $16.
- The combined total is $96.
- Two people split $96 ÷ 2 = $48 each.
Check your understanding: Enter the bill amount you want to base the tip on. This assumes equal shares and a whole number of people; individual purchases may need a different split.
Savings & Interest Math
Simple Interest
I = P × r × t.
What is this for?
Understand interest charged only on the original amount, without interest earning more interest.
How it works · worked example
I = P × r × t; P is principal, r is annual rate as a decimal, t is years.
- Convert 5% to 0.05.
- For $1,000 over 3 years: 1,000 × 0.05 × 3 = $150 interest.
- Add principal: $1,000 + $150 = $1,150 total.
Check your understanding: Match rate and time units. Simple interest is different from compound interest; many real products use daily balances, fees or payments.
Compound Growth
A = P(1 + r/n)^(nt).
What is this for?
See how a starting balance grows when earned interest is reinvested.
How it works · worked example
A = P × (1 + r/n)^(n×t). n is compounding periods per year.
- For $1,000 at 5% nominal annual interest compounded monthly, r = 0.05 and n = 12.
- The monthly rate is 0.05 ÷ 12; 10 years contains 120 periods.
- Calculate 1,000 × (1 + 0.05/12)^120 ≈ $1,647.01. Growth is about $647.01.
Check your understanding: No deposits, withdrawals, taxes or fees are included. This field is a nominal annual rate, not an APY to divide by 12. A market return does not stay constant.
Savings Goal
What is this for?
Decide how much to set aside each month to reach a target by a chosen deadline.
How it works · worked example
Monthly saving = max(goal − already saved, 0) ÷ months.
- A $10,000 target minus $2,500 already saved leaves $7,500.
- Divide by 18 months: $7,500 ÷ 18 ≈ $416.67 monthly.
- Round up slightly if necessary so cent rounding does not leave you short.
Check your understanding: This assumes no interest, withdrawals or changing target. Use a positive whole number of months; if you have already reached the goal, the required contribution is zero.
Rule of 72 Estimate
Approximation: 72 ÷ annual percentage rate.
What is this for?
Quickly compare how long a lump sum could take to double at a steady positive compound growth rate. Useful for mental estimates and learning why time matters.
How it works · worked example
Estimated years to double = 72 ÷ annual rate expressed as a percentage.
- At 8%, enter 8, not 0.08. Calculate 72 ÷ 8 = 9 years.
- A $1,000 balance would be roughly $2,000 near that time, if growth is steady and reinvested.
- For an approximate 12-year doubling target, reverse the rule: 72 ÷ 12 = about 6% annually.
Check your understanding: This is an estimate, not a promised return. It assumes reinvestment, a steady rate, and no contributions, withdrawals, taxes or fees. Zero/negative rates do not produce a doubling time.
Rule of 72: understand the shortcut
What is it for? Use it for a quick comparison of steady compound growth rates. It estimates how many years a starting amount takes to become twice as large. It does not tell you which investment to choose or predict a market return.
Why does money double?
Compound growth adds the return to the balance, so later returns apply to a larger amount. With $1,000 at a steady 8%, year one earns $80, leaving $1,080. Year two earns 8% of $1,080, or $86.40, leaving $1,166.40. Keeping the growth invested is what makes the increase accelerate.
Use the right form of the rate
For the shortcut, an 8% annual rate is entered as 8: 72 ÷ 8 = about 9 years. For the compound-growth formula, the same rate is written as 0.08. Dividing 72 by 0.08 would incorrectly give 900 years.
Why 72?
The exact annual-growth model solves (1 + r)t = 2, giving t = ln(2) ÷ ln(1 + r). Here ln is the natural logarithm and r is a decimal annual rate. At small rates, ln(1 + r) is close to r, which leads to about 69.3 divided by the percentage rate. The number 72 is easy to divide mentally and adjusts the estimate for typical annual compounding rates.
| Annual rate | Rule of 72 | Model solution |
|---|---|---|
| 2% | 36 years | 35.003 years |
| 4% | 18 years | 17.673 years |
| 6% | 12 years | 11.896 years |
| 8% | 9 years | 9.006 years |
| 12% | 6 years | 6.116 years |
| 20% | 3.6 years | 3.802 years |
When the shortcut is useful—and what changes the answer
The initial amount does not change the doubling time in this model: $100 and $10,000 double at the same rate over the same time. Contributions, withdrawals, taxes, fees, changing returns and different compounding schedules change the result. At higher rates or when you need precision, use the full formula rather than the shortcut.
The formula’s fractional year is a mathematical interpolation. If interest is actually credited only once a year, the balance first reaches the doubling target at a whole-year credit date. At 8%, the model solution is about 9.006 years, so the first annual credit meeting the target is year 10.
Reverse the rule
To estimate the rate needed for a chosen doubling time, divide 72 by the years. Doubling in 12 years suggests about 6% annually. This is a mathematical target, not evidence that a suitable product can earn that rate.
What about debt and inflation?
Unpaid debt can also compound, but payments, new charges and fees mean the shortcut cannot replace a payoff calculation. At a steady 3% inflation rate, prices would roughly double in 24 years; an unchanged cash balance would then buy roughly half as much. This is different from the cash balance itself shrinking.
Try it: Enter 6%, then 12%. The shortcut changes from 12 years to 6 years. Compare both with the model solution and yearly balances above.
Further reading: Investor.gov: compound interest · Stanford: the mathematics of the Rule of 72. Educational examples assume constant rates.
Loans & Debt Math
Loan Payment
Uses the standard amortizing loan payment formula.
What is this for?
Estimate the fixed monthly payment on a loan repaid in equal monthly installments.
How it works · worked example
Payment = P×i ÷ [1 − (1+i)^(−m)], where i = annual rate ÷ 12 and m = months.
- For $20,000 at 7% for 60 months, the monthly decimal rate is 0.07/12.
- Insert those values in the formula to get about $396.02 per month.
- Before rounding, multiply the payment by 60 to estimate total payments; subtract $20,000 to find modeled interest.
Check your understanding: This uses the APR input as a simple nominal annual rate. Actual APR may include fees; variable rates, insurance, extra payments and payment rounding change real loan costs.
Debt Payoff — No New Interest
Simple learning model: balance ÷ payment. Real interest-bearing debt takes longer.
What is this for?
Get a simple lower-complexity payoff estimate for a balance that stops accruing interest.
How it works · worked example
Months = ceiling(balance ÷ monthly payment).
- $5,000 ÷ $300 = 16.67 months.
- Round up to 17 payment months.
- Sixteen $300 payments total $4,800; the final payment is $200 if no interest or fees are added.
Check your understanding: This is not a credit-card payoff model with continuing interest. A payment of zero cannot pay off a positive balance.
Business Money Math
Markup & Margin
Markup uses cost as the base. Margin uses selling price as the base.
What is this for?
Understand pricing and compare the gross profit share with the percentage added to cost.
How it works · worked example
Profit = sale − cost; markup% = profit/cost × 100; margin% = profit/sale × 100.
- Buying for $60 and selling for $100 gives $40 gross profit.
- Markup is $40 ÷ $60 × 100 = 66.67%.
- Margin is $40 ÷ $100 × 100 = 40%.
Check your understanding: Markup and margin have different denominators. These are gross figures before overhead, tax and other expenses; a percentage with a zero denominator is undefined.
Break-Even Units
Fixed costs ÷ (price − variable cost).
What is this for?
Estimate how many sales are needed to cover fixed costs in a simplified business model.
How it works · worked example
Contribution/unit = price − variable cost. Break-even units = ceiling(fixed costs ÷ contribution/unit).
- Selling at $50 with $30 variable cost leaves $20 toward fixed costs per unit.
- With $5,000 fixed costs, 5,000 ÷ 20 = 250 units.
- At 250 units, revenue is $12,500 and modeled costs are $5,000 + $7,500 = $12,500.
Check your understanding: All units are assumed to sell at the same price and cost. If price is no greater than variable cost, selling more will not cover positive fixed costs.
Pay, Wages & Time Math
Hourly ↔ Annual Pay
What is this for?
Compare an hourly rate with its estimated yearly gross pay based on the hours and paid weeks you enter.
How it works · worked example
Weekly pay = hourly rate × hours/week. Annual pay = weekly pay × paid weeks/year.
- At $25/hour and 40 hours/week, weekly gross pay is $1,000.
- Multiply by 52 paid weeks for $52,000 a year.
- To reverse it, divide $52,000 by 40 × 52 to get $25/hour.
Check your understanding: This excludes overtime premiums, taxes and deductions. Unpaid leave reduces paid weeks; variable schedules need a realistic average.
Raise Calculator
What is this for?
Calculate the added pay and new total after a percentage raise.
How it works · worked example
Increase = current pay × raise% ÷ 100; new pay = current pay + increase.
- 4% of $50,000 is 50,000 × 0.04 = $2,000.
- Add $2,000 to get $52,000 new pay.
- If both entries are annual figures, the gross monthly increase is about $166.67.
Check your understanding: Keep the pay period consistent: hourly input gives an hourly increase. A gross raise is not the same as the take-home increase after deductions.
Shopping & Comparison Math
Unit Price
What is this for?
Compare package sizes fairly, such as cost per ounce, item or liter.
How it works · worked example
Unit price = total price ÷ quantity.
- A $8.99 package with 12 units costs 8.99 ÷ 12 ≈ $0.7492 per unit.
- The display rounds that to $0.75.
- Compare another package using the same unit; convert pounds to ounces first if needed.
Check your understanding: Quantity must be positive. Do not compare a price per ounce with a price per pound or ignore quality and unusable waste.
Percent Change
What is this for?
Measure how much a price, income or other value increased or decreased relative to its original size.
How it works · worked example
Change% = (new − old) ÷ |old| × 100.
- From $80 to $100, the difference is $20.
- Divide $20 by the original $80 and multiply by 100 = 25% increase.
- Going back from $100 to $80 is a 20% decrease, because the starting value changed.
Check your understanding: A zero starting value makes percentage change undefined. For negative starting values, this tool uses the absolute old value; explain that convention when reporting the result.
Inflation & Purchasing Power
Future Cost Estimate
Future cost = present cost × (1 + inflation rate)^years.
What is this for?
Explore what an item might cost later if prices grow at a constant annual inflation rate.
How it works · worked example
Future cost = cost today × (1 + inflation rate)^years.
- Convert 3% to 0.03.
- Over 10 years, multiply today’s $100 by 1.03^10.
- The result is about $134.39, meaning roughly $34.39 more for the same item.
Check your understanding: This is a scenario, not a forecast. Individual prices may move differently from an overall inflation measure.
Purchasing Power
What is this for?
See how much today’s money could buy in future prices if the cash amount stays unchanged.
How it works · worked example
Buying power in today’s dollars = fixed money amount ÷ (1 + inflation rate)^years.
- For a fixed $1,000 and 3% inflation for 10 years, calculate 1,000 ÷ 1.03^10.
- The result is about $744.09 of today’s buying power.
- The account still contains $1,000; the lower figure describes what that amount can purchase.
Check your understanding: This assumes the cash earns no interest and inflation remains constant. Do not subtract inflation once when modeling several compounded years.
Money Math Lessons
Core ideas that connect practical money decisions to arithmetic, percentages, algebra and exponential growth.
Quick Money Math Formula Reference
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