Simple and Compound Interest: AFCAT Numerical Ability

Formulas (3)

• Simple interest and amount · Simple interest
  $$SI = \dfrac{P \times R \times T}{100}, \quad A = P + SI$$
• Finding the principal, rate or time · Finding the rate or time
  $$R = \dfrac{100 \times SI}{P \times T}, \quad T = \dfrac{100 \times SI}{P \times R}$$
• Rates per month, quarter or half-year · Matching the units
  $$SI = \dfrac{P \times (\text{rate per period}) \times (\text{number of periods})}{100}$$

Watch out for (5)

• Turn months into years
  → Simple interest and amount
• Interest or amount
  → Simple interest and amount
• Take the interest out of the amount first
  → Finding the principal, rate or time
• A rate per period is not a yearly rate
  → Rates per month, quarter or half-year
• Compounding words at simple interest
  → Rates per month, quarter or half-year

Formulas (2)

• A sum given at two different times · One year's interest
  $$I_1 = \dfrac{A_2 - A_1}{t_2 - t_1}, \quad P = A_1 - t_1 I_1$$
• Find the rate in one case, use it in another · Rate from a complete case
  $$R = \dfrac{100\,(A - P)}{P \times T}$$

Watch out for (3)

• Take off the right number of years
  → A sum given at two different times
• 'A further' period
  → A sum given at two different times
• Change the rate, not the amount
  → Find the rate in one case, use it in another

Formulas (3)

• Interest scales with the sum, the rate and the time · Scaling the interest
  $$SI_2 = SI_1 \times \dfrac{P_2 R_2 T_2}{P_1 R_1 T_1}$$
• A difference of two interests gives the sum · Sum from a difference
  $$P = \dfrac{100 \times \text{difference}}{(R_2 - R_1)\,T}$$
• One sum split between two rates · Splitting a sum
  $$x \cdot \dfrac{R_1 T_1}{100} + (S - x) \cdot \dfrac{R_2 T_2}{100} = I$$

Watch out for (4)

• Scale, do not add
  → Interest scales with the sum, the rate and the time
• Multiply the rate difference by the time
  → A difference of two interests gives the sum
• Answer the part that is asked
  → One sum split between two rates
• Equal interest means the inverse ratio
  → One sum split between two rates

Formulas (3)

• The multiple and the rate · Multiple and rate
  $$R \times T = 100\,(n - 1)$$
• From one multiple to another · Time to a new multiple
  $$T_2 = T_1 \times \dfrac{n_2 - 1}{n_1 - 1}$$
• The multiple, amount or interest after another time · Multiple after t years
  $$1 + t \times \dfrac{n - 1}{T}$$

Watch out for (3)

• n times means n minus 1 as interest
  → The multiple and the rate
• Count interest, not the multiple
  → From one multiple to another
• Add the 1 back
  → The multiple, amount or interest after another time

Formulas (5)

• Amount and compound interest · Compound amount
  $$A = P\left(1 + \dfrac{R}{100}\right)^n, \quad CI = A - P$$
• Principal, rate or time from the factor · Present worth
  $$P = \dfrac{A}{\left(1 + \frac{R}{100}\right)^n}$$
• The same factor on another sum · Carrying the factor
  $$A_2 = P_2 \times \dfrac{A_1}{P_1}$$
• Two amounts in a row, and one year's interest · Rate from consecutive amounts
  $$1 + \dfrac{R}{100} = \dfrac{A_{n+1}}{A_n}$$
• Changing rates, part years, growth and depreciation · Depreciation
  $$V_n = V\left(1 - \dfrac{r}{100}\right)^n$$

Watch out for (6)

• Compound interest is not rate × years
  → Amount and compound interest
• Divide by the factor
  → Principal, rate or time from the factor
• A different time needs the yearly factor
  → The same factor on another sum
• Divide by the earlier amount
  → Two amounts in a row, and one year's interest
• A falling value multiplies by less than 1
  → Changing rates, part years, growth and depreciation
• The part year is simple
  → Changing rates, part years, growth and depreciation

Formulas (4)

• Whole powers: multiply the time · Multiples at compound interest
  $$\left(1 + \tfrac{R}{100}\right)^T = n \;\Rightarrow\; \left(1 + \tfrac{R}{100}\right)^{kT} = n^k$$
• Through a common base · Common base
  $$b^p \text{ in } T \text{ years} \;\Rightarrow\; b^q \text{ in } \dfrac{q}{p}\,T \text{ years}$$
• The rate from a multiple · Rate from a multiple
  $$R = 100\left(\sqrt[T]{n} - 1\right)$$
• The multiple after another time · Multiple after t years
  $$n^{t/T}$$

Watch out for (4)

• Multiples multiply, they do not add
  → Whole powers: multiply the time
• Scale the power, not the multiple
  → Through a common base
• The rate is the factor minus 1
  → The rate from a multiple
• Subtract the principal for the interest
  → The multiple after another time

Formulas (5)

• Two years: P(R/100)² · Two-year gap
  $$CI - SI = P\left(\dfrac{R}{100}\right)^2$$
• The rate from the two-year gap · Rate from the gap
  $$R = 100\sqrt{\dfrac{CI - SI}{P}}$$
• Three years · Three-year gap
  $$CI - SI = P\left(\dfrac{R}{100}\right)^2\left(3 + \dfrac{R}{100}\right)$$
• From one interest to the other · Sum from compound interest
  $$P = \dfrac{CI}{\left(1 + \frac{R}{100}\right)^n - 1}$$
• Both interests given: interest on interest · Rate from both interests
  $$\dfrac{R}{100} = \dfrac{CI - SI}{SI / 2}$$

Watch out for (5)

• Square the rate as a fraction
  → Two years: P(R/100)²
• R squared is not R
  → The rate from the two-year gap
• Three years is not three times two
  → Three years
• Divide by the compound share
  → From one interest to the other
• Halve the simple interest first
  → Both interests given: interest on interest

Formulas (4)

• Rate per period and number of periods · Compounded k times a year
  $$A = P\left(1 + \dfrac{R}{100k}\right)^{kn}$$
• Annual against half-yearly, and CI against SI · Gap between two compoundings
  $$\text{gap} = P \times (\text{factor}_1 - \text{factor}_2)$$
• Effective annual rate · Effective annual rate
  $$E = \left(1 + \dfrac{R}{100k}\right)^k - 1$$
• From an effective rate back to a period rate · Period rate from the effective rate
  $$\text{rate per period} = \sqrt[k]{1 + \dfrac{E}{100}} - 1$$

Watch out for (4)

• Change both the rate and the count
  → Rate per period and number of periods
• The two-year shortcut is annual only
  → Annual against half-yearly, and CI against SI
• Never below the quoted rate
  → Effective annual rate
• Take the root, do not divide
  → From an effective rate back to a period rate

Formulas (3)

• Equal instalments · Two equal instalments
  $$x = \text{loan} \times \dfrac{(1 + r)^2}{2 + r}$$
• The loan from its payments · Loan from its payments
  $$\text{loan} = \sum_k \dfrac{x_k}{(1 + r)^k}$$
• Carrying the balance forward · Balance after a year
  $$B_k = B_{k-1}(1 + r) - x_k$$

Watch out for (3)

• An instalment is not loan ÷ number
  → Equal instalments
• Each payment has its own factor
  → The loan from its payments
• Grow before you subtract
  → Carrying the balance forward