Episode 8 — Aptitude and Reasoning / 8.3 — Simple Interest
8.3.a Concepts and Formulas -- Simple Interest
1. What is Simple Interest?
When you borrow money or deposit money in a bank, the borrower pays (or the bank pays you) an extra amount called interest for the use of that money. Simple Interest is the interest calculated only on the original principal for each time period. It does not compound -- that is, interest from previous periods does not itself earn interest.
Key Property: Simple Interest grows linearly with time. If you plot SI against time on a graph, you get a straight line.
2. Core Definitions
Principal (P)
The original sum of money that is borrowed or invested before any interest is applied.
Example: If you deposit Rs. 5000 in a bank, then P = 5000.
Rate of Interest (R)
The percentage of the principal charged or earned per time period (usually per annum/year).
Example: If the bank offers 8% per annum, then R = 8.
Time (T)
The duration for which the money is borrowed or invested. Usually expressed in years.
Important conversions:
Months to Years: T = Number of months / 12
Days to Years: T = Number of days / 365
Examples:
6 months = 6/12 = 0.5 years
146 days = 146/365 = 2/5 years
18 months = 18/12 = 1.5 years
73 days = 73/365 = 1/5 years
Simple Interest (SI)
The extra amount earned or paid over the principal.
Amount (A)
The total money at the end of the time period. It is always the sum of Principal and Interest.
Amount = Principal + Simple Interest
A = P + SI
3. The Fundamental Formula
P x R x T
SI = ─────────────────
100
Where:
SI = Simple Interest
P = Principal
R = Rate of interest per annum (in %)
T = Time period (in years)
Why divide by 100?
Because R is given as a percentage. Dividing by 100 converts it to a decimal multiplier.
Equivalent form: SI = P x (R/100) x T
4. The Amount Formula
A = P + SI
A = P + (P x R x T) / 100
A = P [1 + (R x T) / 100]
A = P (100 + R x T) / 100
This last form is particularly useful when you are given the Amount and need to find the Principal.
5. Deriving Each Unknown
5.1 Finding Simple Interest (SI)
Given: P, R, T
SI = (P x R x T) / 100
Worked Example 1: Find the Simple Interest on Rs. 4000 at 5% per annum for 3 years.
P = 4000, R = 5, T = 3
SI = (4000 x 5 x 3) / 100
SI = 60000 / 100
SI = Rs. 600
Worked Example 2: Find SI on Rs. 7500 at 12% per annum for 8 months.
P = 7500, R = 12, T = 8/12 = 2/3 years
SI = (7500 x 12 x 2/3) / 100
SI = (7500 x 8) / 100
SI = 60000 / 100
SI = Rs. 600
5.2 Finding Principal (P)
Given: SI, R, T
P = (SI x 100) / (R x T)
Derivation:
SI = (P x R x T) / 100
SI x 100 = P x R x T
P = (SI x 100) / (R x T)
Worked Example 3: A sum of money earns Rs. 1200 as simple interest in 4 years at 6% per annum. Find the principal.
SI = 1200, R = 6, T = 4
P = (1200 x 100) / (6 x 4)
P = 120000 / 24
P = Rs. 5000
Finding P from Amount:
A = P (100 + RT) / 100
P = (A x 100) / (100 + RT)
Worked Example 4: What principal will amount to Rs. 14000 in 5 years at 8% per annum simple interest?
A = 14000, R = 8, T = 5
P = (14000 x 100) / (100 + 8 x 5)
P = 1400000 / (100 + 40)
P = 1400000 / 140
P = Rs. 10000
Verification: SI = (10000 x 8 x 5)/100 = 4000
A = 10000 + 4000 = 14000 [Correct]
5.3 Finding Rate (R)
Given: SI (or A), P, T
R = (SI x 100) / (P x T)
Derivation:
SI = (P x R x T) / 100
R = (SI x 100) / (P x T)
Worked Example 5: Rs. 8000 becomes Rs. 10400 in 4 years at simple interest. Find the rate per annum.
P = 8000, A = 10400, T = 4
SI = A - P = 10400 - 8000 = 2400
R = (2400 x 100) / (8000 x 4)
R = 240000 / 32000
R = 7.5%
5.4 Finding Time (T)
Given: SI (or A), P, R
T = (SI x 100) / (P x R)
Worked Example 6: In what time will Rs. 6000 earn Rs. 900 as simple interest at 5% per annum?
P = 6000, SI = 900, R = 5
T = (900 x 100) / (6000 x 5)
T = 90000 / 30000
T = 3 years
Worked Example 7: In what time will Rs. 2000 amount to Rs. 2600 at 10% per annum simple interest?
P = 2000, A = 2600, R = 10
SI = A - P = 2600 - 2000 = 600
T = (600 x 100) / (2000 x 10)
T = 60000 / 20000
T = 3 years
6. Important Relationships and Observations
6.1 SI is Directly Proportional to P, R, and T
SI = (P x R x T) / 100
If P doubles (R, T constant) --> SI doubles
If R doubles (P, T constant) --> SI doubles
If T doubles (P, R constant) --> SI doubles
If all three double --> SI becomes 8 times
6.2 When Does the Amount Double?
If A = 2P, then SI = P.
SI = P
(P x R x T) / 100 = P
R x T = 100
Therefore: T = 100 / R (time for money to double)
R = 100 / T (rate at which money doubles)
Worked Example 8: At what rate of simple interest will a sum double itself in 8 years?
R x T = 100
R x 8 = 100
R = 12.5%
6.3 When Does the Amount Triple?
If A = 3P, then SI = 2P.
(P x R x T) / 100 = 2P
R x T = 200
Therefore: T = 200 / R
R = 200 / T
6.4 General n-Times Formula
If the amount becomes n times the principal:
SI = (n - 1) x P
R x T = (n - 1) x 100
T = (n - 1) x 100 / R
R = (n - 1) x 100 / T
Worked Example 9: In how many years will a sum become 4 times itself at 15% per annum SI?
n = 4
R x T = (4 - 1) x 100 = 300
T = 300 / 15 = 20 years
7. Difference Between Simple Interest and Compound Interest (Introduction)
| Feature | Simple Interest | Compound Interest |
|---|---|---|
| Interest calculated on | Original principal only | Principal + accumulated interest |
| Growth pattern | Linear | Exponential |
| Formula | SI = PRT/100 | CI = P(1 + R/100)^T - P |
| Interest each year | Same (constant) | Increases each year |
| Total interest earned | Less (for T > 1) | More (for T > 1) |
Example comparison: P = 1000, R = 10%, T = 3 years
Simple Interest:
Year 1: Interest = 100, Total = 1100
Year 2: Interest = 100, Total = 1200
Year 3: Interest = 100, Total = 1300
Total SI = 300
Compound Interest:
Year 1: Interest = 100, Total = 1100
Year 2: Interest = 110, Total = 1210
Year 3: Interest = 121, Total = 1331
Total CI = 331
Key insight for exams:
For T = 1 year: SI = CI (always equal for same P and R)
For T = 2 years: CI - SI = P x (R/100)^2
For T > 1 year: CI > SI (always)
This topic is covered in depth in 8.4 Compound Interest.
8. Real-World Applications
8.1 Bank Fixed Deposits (FDs)
Many small savings schemes and short-term FDs use simple interest calculation.
Example:
You deposit Rs. 50,000 in a 1-year FD at 7% SI.
SI = (50000 x 7 x 1) / 100 = Rs. 3500
You receive Rs. 53,500 at maturity.
8.2 Personal Loans and Flat-Rate EMIs
Some personal loans quote a "flat rate" which is essentially simple interest on the original loan amount.
Example:
Loan = Rs. 2,00,000 at flat rate of 10% for 3 years.
Total Interest = (200000 x 10 x 3) / 100 = Rs. 60,000
Total repayment = 200000 + 60000 = Rs. 2,60,000
Monthly EMI = 260000 / 36 = Rs. 7,222 (approx.)
8.3 Lending and Borrowing (Profit on Interest)
A common exam scenario: a person borrows at one rate and lends at a higher rate.
Example:
Ramesh borrows Rs. 10,000 at 8% SI and lends it at 12% SI for 2 years.
Interest paid (borrowing): (10000 x 8 x 2) / 100 = Rs. 1600
Interest earned (lending): (10000 x 12 x 2) / 100 = Rs. 2400
Profit = 2400 - 1600 = Rs. 800
8.4 Government Savings Schemes
Post office savings, National Savings Certificates (certain types), and Kisan Vikas Patra (older versions) used simple interest calculations.
8.5 Split Investment Problems
When a total sum is split and invested at different rates:
Example:
Rs. 15,000 is split into two parts. One part is invested at 8% and
the other at 12%. Total SI after 2 years is Rs. 2880. Find the split.
Let part at 8% = x, then part at 12% = (15000 - x)
SI from first part: (x x 8 x 2) / 100 = 16x / 100
SI from second part: ((15000 - x) x 12 x 2) / 100 = (360000 - 24x) / 100
Total SI = (16x + 360000 - 24x) / 100 = 2880
360000 - 8x = 288000
8x = 72000
x = 9000
Part at 8% = Rs. 9000
Part at 12% = Rs. 6000
Verification:
SI1 = (9000 x 8 x 2)/100 = 1440
SI2 = (6000 x 12 x 2)/100 = 1440
Total = 2880 [Correct]
9. Special Cases and Edge Scenarios
9.1 Half-Yearly and Quarterly Interest
When interest is calculated half-yearly or quarterly but still as simple interest:
Half-yearly:
Effective rate per half-year = R / 2
Number of half-year periods = 2T
SI = P x (R/2) x (2T) / 100 = P x R x T / 100
(Same as annual -- no difference in SI!)
Important: For Simple Interest, the frequency of calculation
does NOT matter. SI is the same whether calculated annually,
half-yearly, or quarterly. This is different from Compound Interest.
9.2 Different Rates for Different Years
If rate is R1% for first T1 years, R2% for next T2 years, and R3% for next T3 years:
Total SI = P x (R1 x T1 + R2 x T2 + R3 x T3) / 100
Worked Example 10: Find SI on Rs. 5000 for 2 years at 6%, then 3 years at 8%.
SI = 5000 x (6 x 2 + 8 x 3) / 100
SI = 5000 x (12 + 24) / 100
SI = 5000 x 36 / 100
SI = Rs. 1800
9.3 Equal Installments under Simple Interest
If a sum P is to be repaid in n equal annual installments at R% SI:
Each installment = (P x 100) / [n x 100 + R x n(n-1)/2]
(This is an advanced formula -- memorize it only if you frequently
encounter installment problems.)
10. Summary of All Formulas
+---------------------------------------------------------+
| SIMPLE INTEREST FORMULAS |
+---------------------------------------------------------+
| SI = (P x R x T) / 100 |
| A = P + SI = P(100 + RT) / 100 |
| P = (SI x 100) / (R x T) |
| P = (A x 100) / (100 + RT) |
| R = (SI x 100) / (P x T) |
| T = (SI x 100) / (P x R) |
+---------------------------------------------------------+
| Amount = n times Principal: R x T = (n-1) x 100 |
| Doubling: R x T = 100 |
| Tripling: R x T = 200 |
+---------------------------------------------------------+
| Varying rates: |
| SI = P x (R1.T1 + R2.T2 + ... + Rn.Tn) / 100 |
+---------------------------------------------------------+
| Profit on lending: |
| Gain = P x (R_lend - R_borrow) x T / 100 |
+---------------------------------------------------------+