Notes
SSC CHSL Exam
Qunatiative Aptitude
Interest

Interest

This summary breaks down the key concepts from the provided PDF on Interest for government exam preparation, with clear explanations, formulas, examples, cheat sheets, and practice questions.

Key Definitions

  • Interest: The extra amount paid for borrowing money or received for lending money.
  • Principal (P): The original amount borrowed or lent.
  • Amount (A): The total sum after adding interest to the principal ($A = P + \text{Interest}$).
  • Rate of Interest (r): The percentage charged or earned on the principal per year.
  • Time (t): The period for which money is borrowed or deposited (usually in years).

Types of Interest

Simple Interest (SI)

  • Definition: Interest calculated only on the principal for every year.
  • Formula:
$$ SI = \frac{P \times r \times t}{100} $$

Where: - $P$ = Principal - $r$ = Rate per annum - $t$ = Time in years

Cheatsheet:

  • If time is in months: $ t = \frac{months}{12} $
  • If time is in days: $ t = \frac{days}{365} $

Example:

Q: Rs.1080 invested for 3 months gave an interest of Rs.27. What is the annual rate?

Solution: $ t = \frac{3}{12} = 0.25 $ years $ 27 = \frac{1080 \times r \times 0.25}{100} $ $ r = 10% $

Compound Interest (CI)

  • Definition: Interest calculated on the principal plus accumulated interest.
  • Formula for Amount (A):
$$ A = P \left(1 + \frac{r}{100}\right)^t $$
  • Compound Interest:
$$ CI = A - P $$

Special Cases:

  • Compounded Half-Yearly: Use $ r/2 $ and $ 2t $
$$ A = P \left(1 + \frac{r}{2 \times 100}\right)^{2t} $$
  • Compounded Quarterly: Use $ r/4 $ and $ 4t $
$$ A = P \left(1 + \frac{r}{4 \times 100}\right)^{4t} $$
  • Variable Rates: For different rates each year:
$$ A = P \left(1 + \frac{r_1}{100}\right)\left(1 + \frac{r_2}{100}\right)\left(1 + \frac{r_3}{100}\right)... $$

Example 1:

Q: Find CI on Rs.50,000 at 12% per annum for 1 year, compounded half-yearly.

Solution: $ r = 6% $, $ t = 2 $ $ A = 50,000 \left(1 + \frac{6}{100}\right)^2 = 50,000 \times 1.1236 = 56,180 $ $ CI = 56,180 - 50,000 = 6,180 $

Example 2:

Q: Rs.25,000 at CI for 3 years at rates 4%, 8%, 10% for each year.

Solution: $ A = 25,000 \times 1.04 \times 1.08 \times 1.10 = 30,888 $

Comparison Table: Simple vs. Compound Interest

FeatureSimple Interest (SI)Compound Interest (CI)
Formula$ \frac{P \times r \times t}{100} $$ P \left(1 + \frac{r}{100}\right)^t - P $
Interest onPrincipal onlyPrincipal + Accumulated Interest
GrowthLinearExponential

Shortcuts and Effective Rate

  • Effective Rate for 2 years at r%: $ Effective Rate = r + r + \frac{r^2}{100} $
  • Difference between CI and SI for 2 years: $ D = \frac{P \times r^2}{100^2} $

Example:

Q: Difference between CI and SI on Rs.3,000 at 10% for 2 years?

Solution: $ D = \frac{3,000 \times 10^2}{100^2} = 30 $

Tree Method for CI

  • Assume a simple principal (like Rs.100 or Rs.1,000) to make calculations easier.
  • Calculate interest for each year, including interest on previous interest.

Example:

Q: CI for Rs.10,000 at 10% for 3 years.

Year 1: $ 1,000 $ Year 2: $ 1,100 $ (1,000 + 100) Year 3: $ 1,210 $ (1,000 + 100 + 100 + 10) Total: $ 1,000 + 1,100 + 1,210 = 3,310 $

Installments

  • Used for calculating the price when paying in equal installments with interest.

Example:

Q: Oven bought in 4 installments of Rs.2,500 at 10% SI. What is the price?

Solution: Assume installment is Rs.100, total payments = 460 Actual price = $ 2,500 \times \frac{460}{100} = 11,500 $

Practice Questions

  1. Calculate the SI on Rs.2,000 at 8% for 18 months.
  2. Find the CI on Rs.5,000 at 10% for 2 years, compounded annually.
  3. What is the difference between SI and CI on Rs.1,500 at 12% for 2 years?
  4. If Rs.1,200 becomes Rs.1,512 in 2 years at CI, what is the rate?
  5. A sum is invested at 5% SI and earns Rs.200 in 4 years. Find the principal.

Cheatsheet Summary

  • SI Formula: $ SI = \frac{P \times r \times t}{100} $
  • CI Formula: $ CI = P \left(1 + \frac{r}{100}\right)^t - P $
  • Effective Rate (2 years): $ r + r + \frac{r^2}{100} $
  • Time Conversion: Months $ \rightarrow $ Years: $ \frac{months}{12} $, Days $ \rightarrow $ Years: $ \frac{days}{365} $
  • Difference (CI - SI, 2 years): $ \frac{P \times r^2}{100^2} $

This structured approach, with formulas, worked examples, and practice, should make mastering interest calculations for exams much easier.

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