Simple vs compound interest explained

Simple interest is principal × rate × time. The rate is the annual rate as a decimal, so 5% is 0.05. Time is in years. On a principal of 1,000 at 5% for 3 years, simple interest is 1,000 × 0.05 × 3 = 150. The simple amount is 1,150. Each year adds the same 50. Nothing in that product depends on how often a bank posts interest.

The interest calculator prints that simple amount next to a compound amount for the same principal, rate, and time. The compound-interest page does not. It only runs the compound formula. Neither page is financial, tax, or investment advice. Fees, a rate that changes, and taxes are not in the arithmetic.

Compound amount is principal × (1 + rate / n) raised to (n × time). n is how many times interest compounds each year. For the 1,000 at 5% for 3 years compounded monthly, n is 12. The amount is 1,000 × (1 + 0.05 / 12) raised to 36 = 1,161.472..., which the interest page shows as 1,161.47. Compound interest is 161.47. The extra over the simple 150 is 11.47.

The same inputs compounded once a year, n = 1, are 1,000 × 1.05 raised to 3 = 1,157.625. Monthly compounding is a little ahead of annual compounding because the interest posted during the year starts earning interest. The gap is smaller than the gap you get by changing the rate or the years.

The interest page also prints continuous compounding, principal × e raised to (rate × time). For 1,000 at 5% for 3 years that is 1,161.834..., shown as 1,161.83. It is slightly above the monthly figure and it is not a setting on the compound-interest page. Continuous is a mathematical limit as n grows, not a claim that a bank compounds infinitely often.

The effective annual rate for n compounds is ((1 + rate / n) raised to n − 1) × 100. For 5% compounded monthly it is 5.1162% when printed to four decimal places. That means one year of monthly compounding at a 5% nominal rate grows the balance as much as one year of simple growth at about 5.1162%. The nominal rate is still 5%. The effective rate is the comparison figure.

The compound page opens on 10,000 at 7% for 10 years with 12 compounds. The total is 20,096.613..., shown as $20,096.61. Interest is $10,096.61. The same boxes with 365 compounds total $20,136.18. The same boxes with 1 compound per year total $19,671.51. Daily is $39.57 ahead of monthly. Monthly is $425.10 ahead of annual. Those gaps are the frequency, not a change in the 7% rate.

A second case: 5,000 at 5% for 5 years compounded 4 times a year is $6,410.19. Compounded once a year it is $6,381.41. The quarterly schedule is ahead by $28.78. A compound count of zero is rejected and the totals go blank. Years of zero return the principal. There is no monthly deposit field. Adding a contribution each month is a different formula.

Open the interest calculator when you want simple, compound, continuous, and the effective rate side by side, and when you want a year-by-year table. n has to be a whole number from 1 to 365, the rate cannot be negative, and time is limited to 100 years. Open the compound page when you only want the compound total and you are comparing frequencies on one principal. Its check rejects a negative rate and a compound count that is not above zero.

Neither balance is a bank quote. A disclosure can use a different day count than n = 365. The salary calculator will not turn a future value into an hourly wage. The profit margin page will not grow a balance. It divides a sale’s profit by price or by cost. Nothing you type is invested, and nothing is uploaded.

Simple interest adds 50 a year on 1,000 at 5 percent. After one year the simple amount is 1,050. After two years it is 1,100. After three years it is 1,150. Monthly compound uses 1,000 × (1 + 0.05 / 12) raised to 12 for the first year, which is 1,051.161... and prints as 1,161.47 only after three years. The one-year figure prints near 1,051.16. The interest page’s table is that comparison, one whole year at a time.

The effective annual rate for 5 percent compounded monthly is 5.1162 percent to four decimals. It describes the first year. It does not replace the 5 percent you typed. The nominal rate stays 5. The effective rate is the one-year comparison with a rate that compounds once.

On the compound page, 10,000 at 7 percent for 10 years compounded 365 times ends at $20,136.18. Compounded monthly it ends at $20,096.61. The extra $39.57 is real and small next to $10,096.61 of interest. Raising the rate or the years moves the total more than switching from monthly to daily. A 360-day bank year is not a separate mode. The exponent is the integer n times the years you type.

Continuous compounding of the 1,000 at 5 percent for 3 years ends at 1,161.83, which is 0.36 above the monthly 1,161.47 when both are shown to the cent. The compound page has no continuous control. Use the interest calculator when you need that third number.

A rate of 0 returns the principal in every version. Simple interest is 0. The compound amount equals the principal. The continuous amount equals the principal. A rate of 0 is allowed. A negative rate is rejected on both pages. Time above 100 years is rejected on the interest calculator.

Neither formula adds a monthly deposit. The contribution series is a separate sum. Do not inflate the principal and pretend the deposits were included. Fees and taxes are not subtracted. This is a planning sketch, not financial, tax, or investment advice.

Simple interest is 2,500 × 0.08 × 1 = 200. The simple amount is 2,700. Compounded once, n = 1, the compound amount is also 2,700, because one compounding over one year matches simple interest. The effective rate for n = 1 is 8 percent. Continuous compounding is 2,500 × e raised to 0.08, about 2,708.22. The interest page prints that continuous figure. The compound page, set to 1 compound and 1 year, matches the 2,700.

Move the same 2,500 to monthly, n = 12, and the compound amount is 2,500 × (1 + 0.08 / 12) raised to 12, 2,707.50 when rounded to the cent. It sits between the annual 2,700 and the continuous 2,708.22. That ordering, annual then monthly then continuous, is what the formulas do at a positive rate. It is not a product recommendation, and it is not a promise that a bank posts interest twelve times.