Simple and Compound Interest: Formulas, Accumulation and Discount Factors

Understand Simple and Compound Interest

Key rules to remember

Simple interest accumulation

A(t) = 1 + it; amount = P(1 + it)

Interest is on the original capital only. t in years, i an annual rate.

Compound interest accumulation

A(t) = (1 + i)^t; amount = P(1 + i)^t

i is the effective annual rate. t can be fractional.

Discount factor (compound)

v = 1 ÷ (1 + i); v^t = (1 + i)^(−t)

Present value of ₹1 due in t years.

Discount factor (simple)

1 ÷ (1 + it)

Reciprocal of the simple accumulation factor.

Effective rate in year n under simple interest

i_n = i ÷ (1 + i(n − 1))

Falls as n rises. Under compound interest the effective rate stays at i.

Effective rate from accumulation function

i_n = [A(n) − A(n − 1)] ÷ A(n − 1)

Works for any accumulation function.

Present value and accumulation

PV = S × v^t; S = PV × (1 + i)^t

Move money forward by multiplying by the accumulation factor, back by the discount factor.

How to solve Simple and Compound Interest questions

1. 1Read the question and note which basis applies: simple, compound or not stated. If not stated, use compound.
2. 2Write down the rate, its type (effective annual or other) and the time in years.
3. 3Draw a time line. Mark the known amount and the date you need the value at.
4. 4Decide the direction. Moving forward in time means multiplying by the accumulation factor. Moving back means multiplying by the discount factor.
5. 5Write the factor in notation, such as (1 + i)^t or 1 + it, before putting in numbers.
6. 6Calculate and keep full calculator precision until the final step.
7. 7If the question asks for a rate or time, rearrange: i = (S ÷ P)^(1/t) − 1 for compound, or i = (S ÷ P − 1) ÷ t for simple.
8. 8Check that the answer is sensible: accumulated values exceed the starting amount when i > 0.

Quickest way: Factor shortcut

Common mistakes in Simple and Compound Interest

• Using the compound formula when the question says simple interest, or the reverse.

  Compound is the default in most of the course, so students apply it automatically.

  Fix: Underline the word simple or compound before writing any formula. If neither appears, use compound.

• Treating the effective rate in each year of a simple interest investment as constant.

  The interest amount is constant, so students assume the rate is too.

  Fix: Use i_n = i ÷ (1 + i(n − 1)). The rate falls because the same interest is earned on a growing balance.

• Discounting simple interest with v^t.

  Students remember v = 1 ÷ (1 + i) and apply it everywhere.

  Fix: Under simple interest the discount factor is 1 ÷ (1 + it). Take the reciprocal of the accumulation factor.

• Entering the rate as 8 instead of 0.08 in the formula.

  Rates are quoted in percent but formulas need decimals.

  Fix: Convert to decimal before substituting. Write 1 + 0.08, not 1 + 8.

• Rounding the factor early, for example using 1.26 for 1.08^3.

  Students want tidy numbers.

  Fix: Keep the full value, 1.259712, until the final answer.

• Assuming compound always exceeds simple.

  Students remember that compound grows faster in the long run.

  Fix: This holds for t > 1. For 0 < t < 1 and i > 0, simple interest gives a larger accumulation.

Worked examples