A $340,000 mortgage at 6.5% interest over 30 years will cost you $433,651 in interest alone. You'll pay back $773,651 total for a home that cost $340,000. That gap between what you borrow and what you repay is the single most important number in personal finance, and most people never calculate it before signing. The math behind mortgages, compound interest, loans, and even tips follows patterns that are straightforward once you see them. This guide breaks down how each calculation works so you can run the numbers yourself and know what the results mean.
How Mortgage Payments Work
A mortgage payment stays the same every month for the life of the loan. On a $340,000 loan at 6.5% over 30 years, that payment is $2,149 per month. But what happens inside that fixed payment changes dramatically over time.
In your first month, the lender calculates interest on the full $340,000 balance. The monthly interest rate is 6.5% divided by 12, which is 0.5417%. Multiply $340,000 by 0.005417 and you get $1,842 in interest. Out of your $2,149 payment, only $307 goes toward the actual loan balance. You just paid $2,149 and your debt dropped by $307.
This is what lenders call amortization. Each month, interest is calculated on the remaining balance. As that balance slowly drops, the interest portion shrinks and the principal portion grows. But the shift is gradual. On a 30-year mortgage, the tipping point where more than half your payment goes to principal instead of interest typically does not arrive until around year 18 or 19. For the first 17 years, the majority of every payment is interest.
A 15-year mortgage on the same $340,000 at 6.5% has a higher monthly payment ($2,963) but reaches that tipping point in year 3 or 4. Total interest paid over the life of the loan: $193,118 instead of $433,651. You save $240,533 by choosing the shorter term. Use the mortgage calculator to compare different terms and rates side by side.
The Amortization Formula
The standard formula for a fixed monthly payment is:
M = P × [r(1+r)^n] / [(1+r)^n - 1]
Where M is the monthly payment, P is the principal (loan amount), r is the monthly interest rate (annual rate divided by 12), and n is the total number of payments (years times 12).
You do not need to memorize this. But understanding what it does helps explain why extra payments are so powerful. If you add $200 per month to that $340,000 mortgage, you pay off the loan 6 years and 4 months early and save roughly $107,000 in interest. That extra $200 goes entirely to principal, which reduces the balance that future interest is calculated on.
Where Rates Stand Now
Context matters when thinking about mortgage math. In the early 1980s, 30-year fixed rates exceeded 18%. Through the 2010s, they hovered between 3% and 5%. Rates climbed sharply in 2022-2023, briefly touching 7.8% in October 2023. As of early 2026, the 30-year fixed rate averages around 6.5%, with fluctuations between 6.3% and 6.8% depending on the week. At 6.5%, a $400,000 mortgage costs $510,617 in total interest. At 3.5% (which many 2020-2021 borrowers locked in), that same loan costs only $246,612 in interest. Same house, same price, $264,005 difference based entirely on timing.
Compound Interest and the Rule of 72
Compound interest is interest calculated on both the initial amount and on all previously accumulated interest. Simple interest calculates only on the original amount. The difference between the two grows slowly at first, then accelerates.
Start with $10,000 at 7% annual interest. After one year, both simple and compound interest give you $10,700. No difference yet. After 10 years, simple interest gives you $17,000 ($700 per year, flat). Compound interest gives you $19,672. After 20 years: simple interest yields $24,000, compound interest yields $38,697. After 30 years: $31,000 versus $76,123. The gap more than doubles every decade.
The compound interest calculator lets you plug in any starting amount, rate, time period, and monthly contribution to see exactly how your money grows.
The Rule of 72
Want a quick estimate of how long it takes money to double? Divide 72 by the annual interest rate. At 6% interest, your money doubles in about 12 years (72 / 6 = 12). At 8%, it doubles in 9 years. At 3%, it takes 24 years.
This shortcut first appeared in print in 1494, in "Summa de Arithmetica" by the Italian mathematician Luca Pacioli. Pacioli was a Franciscan friar and collaborator of Leonardo da Vinci. He presented the rule without derivation or proof, which means it likely circulated among merchants and bankers before he wrote it down. The underlying math relies on natural logarithms, which were not formally developed until more than a century later.
The number 72 works because it is close to 100 times the natural log of 2 (which is 69.3), and because 72 divides evenly by 1, 2, 3, 4, 6, 8, 9, and 12, making the mental math easy for common interest rates. The rule is most accurate between 5% and 10%. Outside that range, it still gives a reasonable ballpark.
Despite persistent internet claims, Albert Einstein never called compound interest "the eighth wonder of the world." No biographer, letter, or contemporary source contains that quote. It appears to have been attributed to him sometime in the 1980s and spread from there.
How Compounding Frequency Matters
The same annual rate produces different results depending on how often interest compounds. A $10,000 investment at 7% for 20 years:
| Compounding | Final Value | Interest Earned |
|---|---|---|
| Annually | $38,697 | $28,697 |
| Quarterly | $40,064 | $30,064 |
| Monthly | $40,387 | $30,387 |
| Daily | $40,547 | $30,547 |
The jump from annual to quarterly compounding adds $1,367. Going from quarterly to daily adds another $483. The total difference between annual and daily compounding is $1,850 on $10,000 over 20 years. That is real money, but not transformative. The far bigger lever is time. Starting five years earlier at any compounding frequency will beat daily compounding with a five-year late start every time.
Savings accounts and CDs typically compound daily. Bonds usually compound semi-annually. Most investment returns effectively compound annually through reinvested gains. When comparing rates, look for the APY (Annual Percentage Yield), which already accounts for compounding frequency, rather than the APR (Annual Percentage Rate), which does not.
Loan Calculations: How They Differ from Mortgages
The core math is identical. Auto loans, personal loans, and student loans all use the same amortization formula as mortgages. What changes are the terms, the rates, and the collateral.
Auto loans typically run 3 to 7 years. Because the term is shorter, less total interest accumulates. A $35,000 car loan at 5.9% for 5 years has a monthly payment of $675 and total interest of $5,501. The same amount borrowed over 7 years drops the payment to $510 but increases total interest to $7,808. The loan calculator handles these comparisons instantly.
Personal loans are unsecured, meaning no asset backs them. Rates are higher as a result, typically ranging from 7% to 20% depending on credit score. Terms are usually 2 to 7 years. A $15,000 personal loan at 11% over 4 years costs $3,639 in interest.
Student loans have their own ecosystem. Federal student loan rates are set by Congress and have historically been lower than personal loan rates. Federal loans also offer income-driven repayment plans that adjust your monthly payment based on earnings. Private student loans behave more like personal loans, with rates that vary by creditworthiness.
The Key Difference: Secured vs. Unsecured
Mortgages and auto loans are secured by the thing you bought. If you stop paying, the lender takes the house or the car. This collateral reduces the lender's risk, which is why mortgage rates (currently around 6.5%) and auto loan rates (around 5-7%) are lower than personal loan rates (7-20%) and credit card rates (18-28%). The percentage calculator can help you figure out rate differences and percentage changes between loan options.
Tip Calculations
Tipping math is simpler than mortgage math, but the cultural rules around it are surprisingly complex.
United States
A standard restaurant tip in the US ranges from 15% to 20% of the pre-tax bill. Service workers in many states earn a sub-minimum wage (as low as $2.13 per hour federally for tipped employees), making tips a primary income source rather than a bonus. On a $86 dinner bill, a 15% tip is $12.90 and a 20% tip is $17.20.
Should you tip on the pre-tax or post-tax amount? Etiquette guides consistently say pre-tax, but in practice, many people tip on the total including tax because it is the number right in front of them. On that $86 dinner in a state with 8% sales tax, the post-tax total is $92.88. Tipping 18% on the pre-tax amount gives $15.48. Tipping 18% on the post-tax amount gives $16.72. The difference is $1.24. The tip calculator lets you calculate both ways and split the result across multiple people.
Tipping Around the World
| Region | Expected Tip | Notes |
|---|---|---|
| United States | 15-20% | Expected at restaurants, bars, salons, taxis |
| Canada | 15-20% | Similar to US customs |
| United Kingdom | 10-15% | Only if no service charge included |
| France | Not expected | Service included by law ("service compris") |
| Germany | 5-10% | Round up or small percentage |
| Scandinavia | Not expected | High wages; rounding up is fine |
| Japan | Do not tip | Considered rude or confusing |
| South Korea | Not expected | Service charge sometimes included |
| China | Not expected | Uncommon outside tourist hotels |
| Thailand | 10% | Increasingly expected in tourist areas |
| Australia | Not expected | Fair wage system; 10% appreciated for great service |
| Brazil | 10% | Often included as "servico" on the bill |
Travelers from the US tend to over-tip abroad, which in countries like Japan can actually cause offense. The baseline expectation in most of the world is zero tip, with service costs built into menu prices or added as a fixed service charge.
When to Use Each Calculator
Different financial questions call for different tools. Here is a quick decision guide.
Use the mortgage calculator when you need to:
- Compare monthly payments at different interest rates
- See how much you will pay in total interest over the life of a loan
- Determine how extra monthly payments reduce your payoff timeline
- Figure out how much house you can afford based on your target monthly payment
Use the compound interest calculator when you need to:
- Project how much a savings account or investment will grow
- Compare the impact of different contribution amounts
- See how starting earlier affects your ending balance
- Understand the effect of different annual return rates
Use the loan calculator when you need to:
- Calculate auto loan payments and total cost
- Compare personal loan offers at different rates and terms
- Determine total interest on any fixed-rate installment loan
- Decide between a shorter term with higher payments or a longer term with lower payments
Use the tip calculator when you need to:
- Split a restaurant bill among multiple people
- Calculate pre-tax vs. post-tax tip amounts
- Determine per-person cost including tip
- Quickly figure out 15%, 18%, or 20% of a bill
Use the percentage calculator when you need to:
- Calculate percentage increases or decreases between two values
- Find what percentage one number is of another
- Convert between fractions, decimals, and percentages
- Work out discount amounts or markup rates
Use the salary calculator when you need to:
- Convert between hourly, weekly, biweekly, monthly, and annual pay
- Compare job offers with different pay structures
- Estimate take-home amounts across different pay periods
Common Financial Math Mistakes
Confusing APR and APY. APR (Annual Percentage Rate) is the stated rate before compounding. APY (Annual Percentage Yield) is the effective rate after compounding. A credit card with 24% APR compounded daily has an APY of about 27.1%. Banks advertise APY on savings accounts (because it is higher) and APR on loans (because it is lower). Always compare like with like.
Ignoring the total cost of a loan. A $500/month car payment over 7 years sounds manageable. Multiply it out: that is $42,000 in payments for a car that might be worth $12,000 when you finish paying. Total interest on a $35,000 loan at 6.5% for 7 years is about $8,700. The monthly payment is not the cost of the loan. The total outflow is the cost.
Assuming longer terms save money. Lower monthly payments feel like savings, but longer terms always increase total interest paid. A $25,000 auto loan at 6%: over 4 years you pay $3,182 in interest. Over 6 years, $4,831. The monthly payment drops by $173, but you pay $1,649 more overall.
Forgetting that percentages work both directions asymmetrically. A 50% loss requires a 100% gain to recover. If your $10,000 investment drops 50% to $5,000, it needs to double (gain 100%) to get back to $10,000. This asymmetry is why protecting against large losses matters more than chasing large gains.
Tipping on the wrong base. If a coupon reduced your $100 dinner to $60, tip on the original $100. The server did the same work regardless of your discount.
Rounding in the wrong direction on mortgage quotes. A difference of 0.25% on a $300,000 mortgage adds up to about $16,000 over 30 years. When comparing loan offers, round to the exact numbers, not to the nearest whole percent. The mortgage calculator handles this precision for you.
Quick Reference Formulas
Monthly mortgage/loan payment: M = P × [r(1+r)^n] / [(1+r)^n - 1] (P = principal, r = monthly rate, n = total payments)
Compound interest (future value): A = P(1 + r/n)^(nt) (P = principal, r = annual rate, n = compounds per year, t = years)
Rule of 72 (doubling time): Years to double = 72 / annual interest rate
Simple interest: I = P × r × t (P = principal, r = annual rate, t = years)
Tip amount: Tip = Bill × (tip percentage / 100)
APY from APR: APY = (1 + APR/n)^n - 1 (n = compounding periods per year)
These formulas cover the math behind every financial calculator on Calcflux. For quick answers without the algebra, use the mortgage calculator, compound interest calculator, loan calculator, tip calculator, or percentage calculator.