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How to calculate compound interest by hand and check the answer

How to calculate compound interest by hand and check the answer

A savings account or an investment quotes a yearly rate, but the balance grows faster than that rate suggests because interest earns interest. Work the number out yourself before you compare two offers.

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What rate times years gets wrong

Multiplying the rate by the years gives simple interest: $10,000 at 5% for 10 years would earn $5,000. Compound interest adds each period's interest to the balance, and the next period earns interest on it too. The same $10,000 at 5%, compounded monthly, ends at $16,470.09, which is $1,470.09 more than the simple sum.

The formula is A = P × (1 + r ÷ n)^(n × t). P is the starting amount, r the yearly rate as a decimal, n the number of times interest is added each year, and t the number of years. How often interest is added matters less than most people think. At 5% over 10 years, yearly compounding gives $16,288.95 and daily gives $16,486.65. The rate and the time the money stays invested matter far more, and without the math you can't tell whether two offers really differ or are only quoted differently.

Run the numbers three ways

Enter the starting amount, the yearly rate, and the years in the Compound Interest Calculator, then choose yearly, quarterly, monthly, daily, or continuous compounding. It shows the future value, the total interest, and the effective annual rate, which is the same as an APY: 5% compounded monthly is 5.1162%. Add a monthly contribution to see what regular saving does. With $200 a month, that $10,000 account reaches $47,526.55 after 10 years.

For a plan built on deposits, the Savings Calculator lets you choose whether each deposit goes in at the start or the end of the month, and works out how long it takes to reach a goal. $1,000 plus $200 a month at 5% for 10 years gives $32,703.47 with deposits at the end of each month and $32,832.87 with deposits at the start. Saving $500 a month at 4% from zero reaches $50,000 in 87 months.

For a check you can do in your head, use the Rule of 72 Calculator. Divide 72 by the yearly rate to estimate how many years money takes to double. At 8% that is 9 years, against an exact 9.01. It works backwards too: to double in 10 years you need about 7.2% a year. The shortcut drifts at high rates, so the calculator shows the exact figure beside it.

Calculate compound interest on a deposit

  1. Write the yearly rate as a decimal. 5% is 0.05.
  2. Divide it by the number of compounding periods in a year: 0.05 ÷ 12 for monthly.
  3. Add 1 and raise the result to the power of periods per year × years: (1 + 0.05 ÷ 12)^120 for 10 years.
  4. Multiply by the starting amount. For $10,000 the result is $16,470.09.
  5. Enter the same numbers in the Compound Interest Calculator to check the result, then add any monthly deposits.

What these figures leave out

  • The rate stays fixed and the deposits stay the same. Real returns and savings rates change.
  • Taxes on interest, account fees, and inflation are not included.
  • The rule of 72 assumes yearly compounding and drifts at high rates: at 30% it says 2.4 years against an exact 2.64.
  • The results are estimates for comparing scenarios. They are not financial advice.

Frequently asked questions

What is the formula for compound interest?

A = P × (1 + r ÷ n)^(n × t). P is the starting amount, r the yearly rate as a decimal, n the compounding periods per year, and t the years.

How much is $10,000 at 5% for 10 years with compound interest?

$16,470.09 with monthly compounding and $16,288.95 with yearly compounding.

How long does it take money to double at 8%?

About 9 years by the rule of 72. The exact answer is 9.01 years.

What is the effective annual rate?

The yearly growth once compounding is included. 5% compounded monthly is an effective 5.1162%, the same as an APY.

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