Simple Interest

ARITHMETIC HANDBOOK — SIMPLE INTEREST

1. What is Simple Interest?

Simple Interest (SI) is the interest calculated only on the original principal for the entire period. The interest does not get added to the principal for calculating interest in the next period.

Simple Interest is commonly used in questions involving loans, investments, deposits, borrowing, interest rates and amounts.

Important Terms

Principal (P): The original sum of money borrowed or invested.
Rate (R): Rate of interest per annum, usually expressed as a percentage.
Time (T): Period for which money is borrowed or invested.
Simple Interest (SI): Interest earned or paid on the principal.
Amount (A): Principal plus interest.

2. Main Formula

Simple Interest:

SI = (P × R × T) / 100

Amount:
A = P + SI

Where:

P = Principal
R = Rate of interest per annum
T = Time in years
SI = Simple Interest
A = Final Amount

3. Formula for Finding Principal, Rate and Time

P = (SI × 100)/(R × T)

R = (SI × 100)/(P × T)

T = (SI × 100)/(P × R)

4. Solved Examples

Example 1: Find the simple interest on ₹5,000 at 8% per annum for 3 years.

SI = (5000 × 8 × 3)/100
= ₹1,200

Amount = 5000 + 1200
= ₹6,200

Example 2: Find the principal if the simple interest is ₹900 at 10% per annum for 3 years.

P = (900 × 100)/(10 × 3)
= ₹3,000

Exam Tip: Before applying the formula, identify P, R and T. Then make sure the time is expressed in years when R is given per annum.

SIMPLE INTEREST — TIME, RATE & AMOUNT

5. Conversion of Time

Since the rate is generally given per annum, time should normally be converted into years.

1 year = 12 months
1 month = 1/12 year
3 months = 1/4 year
6 months = 1/2 year
9 months = 3/4 year
15 months = 5/4 years

Example: Find SI on ₹8,000 at 12% per annum for 9 months.

T = 9/12 = 3/4 year
SI = (8000 × 12 × 3/4)/100
= ₹720

6. Finding Rate

Example: A sum of ₹4,000 earns ₹960 as simple interest in 4 years. Find the rate.

R = (960 × 100)/(4000 × 4)
= 96000/16000
= 6% per annum

7. Finding Time

Example: A sum of ₹6,000 earns ₹1,800 as SI at 10% per annum. Find the time.

T = (1800 × 100)/(6000 × 10)
= 180000/60000
= 3 years

8. Finding Amount

Amount = Principal + Simple Interest
A = P + SI

Therefore:
A = P(1 + RT/100)

Example: Find the amount on ₹10,000 at 5% per annum for 4 years.

SI = (10000 × 5 × 4)/100
= ₹2,000
A = 10,000 + 2,000
= ₹12,000

9. Ratio Method

For the same principal, simple interest is directly proportional to rate and time.

SI₁ / SI₂ = (R₁T₁)/(R₂T₂)

Example: If ₹5,000 is invested at 8% for 3 years, compare its SI with the SI on the same principal at 12% for 2 years.

SI₁ : SI₂ = (8×3):(12×2)
= 24:24
= 1:1

Shortcut: When P is the same, you do not need to calculate the actual interest. Compare R × T directly.

SIMPLE INTEREST — SPECIAL PROBLEMS

10. Difference Between Amounts

When the principal and rate remain the same, the difference between two amounts is equal to the difference between the corresponding simple interests.

Difference in SI = P × R × Difference in Time / 100

Example: The difference between the simple interest for 5 years and 3 years on a certain sum at 8% per annum is ₹800. Find the principal.

Difference in time = 5 − 3 = 2 years
800 = (P × 8 × 2)/100
P = 800×100/16
= ₹5,000

11. Difference in Interest for Different Rates

If the principal and time are the same but the rates differ:

Difference in SI = P × (R₁ − R₂) × T / 100

Example: Find the difference in SI on ₹10,000 for 3 years at 8% and 6% per annum.

Difference = (10000 × 2 × 3)/100
= ₹600

12. When Amount and Principal are Given

SI = Amount − Principal

Example: A sum amounts to ₹7,200 after 4 years. If the principal was ₹6,000, find the SI and rate.

SI = 7200 − 6000
= ₹1,200

R = (1200×100)/(6000×4)
= 5% per annum

13. When Amount is a Multiple of Principal

If the amount is given as a fraction or multiple of the principal, the interest can be found quickly.

A = P + SI
Therefore:
SI = A − P

Example: A sum becomes 7/5 of itself in 4 years at simple interest. Find the rate.

SI = 7/5 P − P
= 2/5 P

Therefore:
(P×R×4)/100 = 2P/5
R×4/100 = 2/5
R = 10% per annum

14. Principal and Interest Ratio

SI : P = RT : 100

Example: Find the ratio of SI to principal when the rate is 8% per annum and time is 5 years.

SI : P = 8×5 : 100
= 40 : 100
= 2 : 5

15. Important Table

``` ```

Given	Formula
Simple Interest	PRT/100
Amount	P + SI
Principal	100SI/RT
Rate	100SI/PT
Time	100SI/PR
SI : P	RT : 100

Common Mistake: If time is given in months, do not directly put the number of months into the formula when the rate is per annum. Convert months into years first.

SIMPLE INTEREST — FORMULA & PRACTICE

16. Complete Formula Sheet

``` ```

Concept	Formula
Simple Interest	SI = PRT/100
Amount	A = P + SI
Amount Directly	A = P(1 + RT/100)
Principal	P = 100SI/RT
Rate	R = 100SI/PT
Time	T = 100SI/PR
SI : Principal	RT : 100
Difference in SI	P×R×ΔT/100
Difference due to Rate	P×ΔR×T/100
Same P	SI₁:SI₂ = R₁T₁:R₂T₂

17. Solved Revision Examples

Example 1: Find SI on ₹12,000 at 7.5% per annum for 2 years.

SI = (12000×7.5×2)/100
= ₹1,800

Example 2: A sum of ₹8,000 becomes ₹9,600 in 4 years. Find the rate.

SI = 9600 − 8000 = ₹1,600
R = (1600×100)/(8000×4)
= 5%

Example 3: Find the time required for ₹5,000 to earn ₹1,500 at 10% per annum.

T = (1500×100)/(5000×10)
= 3 years

18. Practice Questions

Find the simple interest on ₹6,000 at 5% per annum for 4 years.
Find the amount on ₹8,000 at 7.5% per annum for 2 years.
Find the principal if SI is ₹1,200, rate is 8% and time is 3 years.
A sum of ₹5,000 earns ₹1,500 as SI in 5 years. Find the rate.
Find the time required for ₹4,000 to earn ₹960 at 8% per annum.
Find the SI on ₹10,000 at 12% per annum for 9 months.
The difference between SI for 6 years and 4 years on a certain sum at 10% per annum is ₹1,000. Find the principal.
Find the difference between the SI on ₹12,000 for 3 years at 8% and 6% per annum.
A sum becomes ₹7,200 from ₹6,000 in 4 years. Find the rate.
At what rate will a sum double itself in 10 years under simple interest?

19. Answers

1. ₹1,200

2. ₹9,200

3. ₹5,000

4. 6% per annum

5. 3 years

6. ₹900

7. ₹5,000

8. ₹720

9. 5% per annum

10. 10% per annum

20. Quick Exam Shortcuts

``` Shortcut 1: SI is directly proportional to P, R and T.

Shortcut 2: If P is fixed:
SI ∝ R × T.

Shortcut 3: If R and T are fixed:
SI ∝ P.

Shortcut 4: If a sum doubles under simple interest:
SI = P.
Therefore RT = 100.

Shortcut 5: If a sum becomes three times:
SI = 2P.
Therefore RT = 200.

Shortcut 6: Always convert months into years when the rate is given per annum. ```

Final Rule: Simple Interest is always calculated on the original principal. The basic formula to remember is:

SI = P × R × T / 100