Simple and Compound Interest: SSC CGL Quantitative Aptitude

Formulas (3)

• Simple interest formula and the amount · Simple interest and amount
  $$SI = \frac{P \times R \times T}{100} \qquad A = P + SI = P\left(1 + \frac{R T}{100}\right)$$
• Simple interest for months and days · Time in years
  $$T = \frac{\text{months}}{12} \qquad T = \frac{\text{days}}{365}$$
• Principal from the amount (present worth) · Principal from amount
  $$P = \frac{100 \times A}{100 + R T}$$

Watch out for (5)

• Giving the amount when the interest is asked
  → Simple interest formula and the amount
• Taking a percentage earning on the interest
  → Simple interest formula and the amount
• Putting months in place of years
  → Simple interest for months and days
• Counting both the first and the last day
  → Simple interest for months and days
• Taking the percent off the amount
  → Principal from the amount (present worth)

Formulas (4)

• Finding the rate or the time from the interest · Rate, time and principal
  $$R = \frac{100 \times SI}{P \times T} \qquad T = \frac{100 \times SI}{P \times R} \qquad P = \frac{100 \times SI}{R \times T}$$
• Two amounts after two different times · Two amounts at two times
  $$\text{Interest per year} = \frac{A_2 - A_1}{t_2 - t_1} \qquad P = \frac{t_2 A_1 - t_1 A_2}{t_2 - t_1}$$
• Equal simple interest on two sums · Equal simple interest
  $$P_1 R_1 T_1 = P_2 R_2 T_2$$
• Rate tied to the number of years · Rate equal to the years
  $$R^2 = \frac{100 \times SI}{P}$$

Watch out for (5)

• Using the amount as the interest
  → Finding the rate or the time from the interest
• Rounding the rate too early
  → Finding the rate or the time from the interest
• Dividing the rise by the later time
  → Two amounts after two different times
• Turning the ratio of sums upside down
  → Equal simple interest on two sums
• Answering the time when the rate is asked
  → Rate tied to the number of years

Formulas (3)

• Simple interest as a fraction of the principal · Interest as a share of the principal
  $$\frac{SI}{P} = \frac{R \times T}{100} \qquad R = \frac{100}{T} \times \frac{SI}{P}$$
• A sum that becomes n times itself at simple interest · Becoming n times at simple interest
  $$R \times T = 100(n - 1) \qquad T_2 = T_1 \times \frac{n_2 - 1}{n_1 - 1}$$
• Simple interest scales with principal, rate and time · Scaling simple interest
  $$\frac{SI_2}{SI_1} = \frac{P_2 R_2 T_2}{P_1 R_1 T_1} \qquad B^2 = AC$$

Watch out for (3)

• Reading the amount's fraction as the interest's fraction
  → Simple interest as a fraction of the principal
• Scaling the time by n instead of n minus 1
  → A sum that becomes n times itself at simple interest
• Comparing the rates alone
  → Simple interest scales with principal, rate and time

Formulas (3)

• One sum split between two rates · Splitting a sum between two rates
  $$\text{Higher part} = \frac{100 I - r_1 P}{r_2 - r_1} \qquad \frac{\text{Lower part}}{\text{Higher part}} = \frac{r_2 - \bar{r}}{\bar{r} - r_1}$$
• Several loans at one rate · Several loans at one rate
  $$SI_{\text{total}} = \frac{R}{100}\left(P_1 T_1 + P_2 T_2 + \dots\right)$$
• Parts linked by a ratio or by equal interest · Equal interest from every part
  $$a : b : c = \frac{1}{r_1} : \frac{1}{r_2} : \frac{1}{r_3}$$

Watch out for (4)

• Running alligation on a several-year interest
  → One sum split between two rates
• Reversing the alligation ratio
  → One sum split between two rates
• Charging a later loan for the whole year
  → Several loans at one rate
• Splitting in the ratio of the rates
  → Parts linked by a ratio or by equal interest

Formulas (3)

• Extra interest from a change in rate or time · Extra interest from a change
  $$\Delta SI = \frac{P \times \Delta R \times T}{100} \qquad \Delta SI = \frac{P \times R \times \Delta T}{100}$$
• Different rates in different years · Rates that change over the years
  $$SI = \frac{P}{100}\left(R_1 T_1 + R_2 T_2 + R_3 T_3\right)$$
• Part repayments and interest added to the principal · Final payment after a part repayment
  $$\text{Final payment} = (P - x) + \frac{R}{100}\left(P + (P - x)\right)$$

Watch out for (4)

• Applying a rate rise for one year only
  → Extra interest from a change in rate or time
• Averaging the rates without the years
  → Different rates in different years
• Charging the second year on the original sum
  → Part repayments and interest added to the principal
• Leaving out the unpaid first-year interest
  → Part repayments and interest added to the principal

Formulas (4)

• Compound interest amount and principal · Compound interest, yearly
  $$A = P\left(1 + \frac{r}{100}\right)^n \qquad CI = A - P \qquad \text{2 years: } \left(2r + \frac{r^2}{100}\right)\%$$
• Rate or time from the growth factor · Growth factor
  $$\left(1 + \frac{r}{100}\right)^n = \frac{A}{P}$$
• Interest year by year and consecutive amounts · Year-by-year compound interest
  $$r = \frac{A_{n+1} - A_n}{A_n} \times 100 \qquad I_k = P\left(1 + \frac{r}{100}\right)^{k-1} \times \frac{r}{100}$$
• Compound and simple interest on the same sum · Gap between compound and simple interest
  $$CI - SI = P\left(\frac{r}{100}\right)^2 \ (2 \text{ years}) \qquad CI - SI = P\left(\frac{r}{100}\right)^2\left(3 + \frac{r}{100}\right) \ (3 \text{ years})$$

Watch out for (6)

• Doubling the first year's interest
  → Compound interest amount and principal
• Giving the amount when compound interest is asked
  → Compound interest amount and principal
• Dividing the total growth by the years
  → Rate or time from the growth factor
• Reading constant growth as a constant addition
  → Rate or time from the growth factor
• Using the simple interest method on compound amounts
  → Interest year by year and consecutive amounts
• Expecting a gap in the first year
  → Compound and simple interest on the same sum

Formulas (3)

• A sum that becomes n times itself at compound interest · Repeated multiples at compound interest
  $$\left(1 + \frac{r}{100}\right)^{t} = n \quad \Rightarrow \quad \left(1 + \frac{r}{100}\right)^{kt} = n^{k}$$
• Part years and compounding more often than yearly · Part years and shorter periods
  $$A = P\left(1 + \frac{r}{100}\right)^{n}\left(1 + \frac{r m}{1200}\right) \qquad \text{every } k \text{ months: } \frac{rk}{12}\% \text{ per period, } \frac{12n}{k} \text{ periods}$$
• Equal annual instalments · Equal annual instalments
  $$\text{CI: } P = \frac{x}{1 + \frac{r}{100}} + \frac{x}{\left(1 + \frac{r}{100}\right)^2} + \dots \qquad \text{SI: } A = nx + \frac{x r}{100} \cdot \frac{n(n - 1)}{2}$$

Watch out for (5)

• Using the simple interest rule under compound interest
  → A sum that becomes n times itself at compound interest
• Compounding the part year at the full rate
  → Part years and compounding more often than yearly
• Halving the rate but not doubling the periods
  → Part years and compounding more often than yearly
• Charging every instalment interest for the full term
  → Equal annual instalments
• Forgetting the down payment
  → Equal annual instalments