Episode 8 — Aptitude and Reasoning / 8.3 — Simple Interest
8.3.b Tips, Tricks, and Shortcuts -- Simple Interest
1. The Percentage-Per-Year Mental Model
Think of Simple Interest as a flat percentage of the principal applied each year.
At R% per annum, you earn/pay R% of P every single year.
10% on Rs. 5000 = Rs. 500 per year
For 3 years: 3 x 500 = Rs. 1500
This is faster than plugging into SI = PRT/100 every time.
Trick: Calculate SI for 1 year first, then multiply by T.
SI (1 year) = P x R / 100
SI (T years) = SI (1 year) x T
2. Breaking Down Complex Percentages
When R is not a "nice" number, break it down:
12.5% of P = 10% of P + 2.5% of P
= 10% of P + (1/4 of 10% of P)
Example: 12.5% of 4800
10% of 4800 = 480
2.5% of 4800 = 480/4 = 120
12.5% of 4800 = 480 + 120 = 600
Common Fraction Equivalents (Memorize These!)
Rate (%) | Fraction | Quick interpretation
─────────────────────────────────────────────────
5% | 1/20 | Divide by 20
6.25% | 1/16 | Divide by 16
8% | 2/25 | Multiply by 2, divide by 25
8.33% | 1/12 | Divide by 12
10% | 1/10 | Divide by 10
12.5% | 1/8 | Divide by 8
15% | 3/20 | Multiply by 3, divide by 20
16.67% | 1/6 | Divide by 6
20% | 1/5 | Divide by 5
25% | 1/4 | Divide by 4
33.33% | 1/3 | Divide by 3
50% | 1/2 | Divide by 2
Example using fractions:
SI on Rs. 7200 at 8.33% for 3 years
8.33% = 1/12
SI per year = 7200 / 12 = 600
SI for 3 years = 600 x 3 = Rs. 1800
3. The Doubling/Tripling/n-Times Shortcut
This is one of the most frequently tested shortcuts.
Doubling (Amount = 2P)
R x T = 100
Rate (%) | Time to Double
─────────────────────────────
5% | 20 years
8% | 12.5 years
10% | 10 years
12% | 8.33 years
12.5% | 8 years
15% | 6.67 years
20% | 5 years
25% | 4 years
50% | 2 years
Tripling (Amount = 3P)
R x T = 200
Rate (%) | Time to Triple
─────────────────────────────
5% | 40 years
10% | 20 years
12.5% | 16 years
20% | 10 years
25% | 8 years
General n-Times
R x T = (n - 1) x 100
Amount becomes | R x T =
────────────────────────────
2 times | 100
3 times | 200
4 times | 300
5 times | 400
n times | (n-1) x 100
Cross-Application Trick
If a sum doubles in T1 years, in how many years will it triple?
Doubling: R x T1 = 100 --> R = 100/T1
Tripling: R x T2 = 200 --> T2 = 200/R = 200/(100/T1) = 2 x T1
Answer: Time to triple = 2 x (Time to double)
Similarly:
Time to become n times = (n-1) x (Time to double)
Example: If money doubles in 6 years, when does it become 5 times?
T = (5-1) x 6 = 24 years
4. The Ratio Method for Split Investments
When a sum S is divided into two parts invested at rates R1 and R2, and you know the total SI:
Method 1: Alligation (When total SI for same time period is given)
If both parts are for the same time T and you want the
effective rate that gives the total SI:
Effective rate R_eff = (Total SI x 100) / (S x T)
Then use alligation:
Part at R1 R2 - R_eff
─────────── = ──────────────
Part at R2 R_eff - R1
Example: Rs. 10,000 is split into two parts at 8% and 12% for 2 years. Total SI = Rs. 2080.
R_eff = (2080 x 100) / (10000 x 2) = 10.4%
Part at 8% 12 - 10.4 1.6 2
────────── = ─────────── = ───── = ───
Part at 12% 10.4 - 8 2.4 3
Total parts = 2 + 3 = 5
Part at 8% = (2/5) x 10000 = Rs. 4000
Part at 12% = (3/5) x 10000 = Rs. 6000
Method 2: Direct Equation (Always works)
Let part at R1 = x
Then part at R2 = S - x
(x . R1 . T + (S-x) . R2 . T) / 100 = Total SI
Solve for x.
5. Shortcut When SI Equals Principal
When SI = P:
(P x R x T) / 100 = P
R x T = 100
This means: Rate x Time = 100
Quick check: 10% for 10 years, 20% for 5 years, 25% for 4 years, etc.
6. Shortcut for Finding Rate When Given Two Amounts at Different Times
If a principal becomes A1 in T1 years and A2 in T2 years at the same rate of SI:
R = [(A2 - A1) x 100] / [P x (T2 - T1)]
But often P is unknown. Use:
SI per year = (A2 - A1) / (T2 - T1)
Then: P = A1 - SI_per_year x T1
And: R = (SI_per_year x 100) / P
Example: A sum becomes Rs. 7000 in 2 years and Rs. 8500 in 5 years at SI. Find R and P.
SI per year = (8500 - 7000) / (5 - 2) = 1500 / 3 = 500
P = 7000 - (500 x 2) = 7000 - 1000 = 6000
R = (500 x 100) / 6000 = 8.33%
7. Shortcut for Comparing Two Investments
If you invest P1 at R1% and P2 at R2%, and both give the same SI for the same time:
P1 x R1 = P2 x R2
Or: P1 / P2 = R2 / R1
If the time periods also differ:
P1 x R1 x T1 = P2 x R2 x T2
Example: Rs. 600 at 5% gives the same SI as Rs. 500 at what rate (same time)?
600 x 5 = 500 x R
R = 3000 / 500 = 6%
8. Quick Calculation: The "1% Method"
For any problem, first find what 1% of the principal is, then scale.
P = 8400, R = 7.5%, T = 4
Step 1: 1% of 8400 = 84
Step 2: 7.5% of 8400 = 84 x 7.5 = 84 x 7 + 84 x 0.5 = 588 + 42 = 630
Step 3: SI for 4 years = 630 x 4 = Rs. 2520
9. Shortcut for Time in Months/Days
When time is given in months or days, simplify the fraction before computing.
T = 9 months = 9/12 = 3/4 years
T = 146 days = 146/365 = 2/5 years
T = 219 days = 219/365 = 3/5 years
T = 73 days = 73/365 = 1/5 years
T = 292 days = 292/365 = 4/5 years
Memorize: 365/5 = 73 days
So multiples of 73 give clean fractions.
10. The "Product Constant" Trick
If SI is fixed and you want to find how changes in one variable affect another:
SI is constant --> P x R x T = constant
If P doubles, either R must halve or T must halve (or some combination).
If R triples, then P x T must become 1/3.
11. Quick Mental Math for Common Scenarios
Scenario: Find SI on round numbers
Rs. 10,000 at 8% for 3 years
Think: 8% of 10,000 = 800 per year
For 3 years: 800 x 3 = Rs. 2400
(Took 3 seconds!)
Scenario: Find the rate
SI = 1500, P = 5000, T = 6 years
SI per year = 1500/6 = 250
Rate = (250/5000) x 100 = 5%
Shortcut thinking: 250 is what % of 5000?
250/5000 = 1/20 = 5%
Scenario: Mixed time period
P = 12,000 at 10% for 2 years and 6 months (= 2.5 years)
SI per year = 1200
SI for 2.5 years = 1200 x 2.5 = 1200 x 2 + 1200 x 0.5 = 2400 + 600 = 3000
12. Common Traps and How to Avoid Them
Trap 1: Time unit mismatch
Problem says "8 months" but you use T = 8.
ALWAYS convert months/days to years!
Trap 2: Rate vs. Amount confusion
"At what rate will Rs. 5000 amount to Rs. 6500 in 3 years?"
First find SI = 6500 - 5000 = 1500, THEN find R.
Don't use 6500 as SI.
Trap 3: Forgetting that A = P + SI
The question asks for Amount but you calculate only SI.
Or the question asks for SI but you give the Amount.
READ THE QUESTION CAREFULLY.
Trap 4: Per annum vs. total interest
"What is the annual interest?" --> Give SI for 1 year only.
"What is the total interest?" --> Give SI for all T years.
Trap 5: Half-yearly rate given as annual
"6% per half-year" means R = 12% per annum for SI calculations.
OR compute with R = 6% and T in half-year units.
13. Speed Drills (Practice These Mentally)
Try to solve each in under 15 seconds:
1. SI on Rs. 2000 at 10% for 3 years = ?
Answer: 200 x 3 = Rs. 600
2. SI on Rs. 5000 at 8% for 2 years = ?
Answer: 400 x 2 = Rs. 800
3. P = 4000, SI = 960, T = 3. Find R.
Answer: SI/year = 320. R = (320/4000) x 100 = 8%
4. P = 6000, R = 5%, SI = 1500. Find T.
Answer: SI/year = 300. T = 1500/300 = 5 years
5. Rs. 10000 doubles at 8%. Time = ?
Answer: T = 100/8 = 12.5 years
6. A sum triples in 10 years. Rate = ?
Answer: R = 200/10 = 20%
7. SI on Rs. 1200 at 12.5% for 4 years = ?
Answer: 12.5% = 1/8. SI/year = 150. Total = 600
8. Amount = 9000, P = 6000, T = 5. Rate = ?
Answer: SI = 3000, SI/year = 600. R = (600/6000) x 100 = 10%
Summary Table of Shortcuts
+──────────────────────────────────────────────────────────────+
| SHORTCUT | FORMULA / RULE |
+───────────────────────────────────+──────────────────────────+
| SI for 1 year then multiply | SI_total = (P.R/100) x T|
| Amount doubles | R x T = 100 |
| Amount triples | R x T = 200 |
| Amount becomes n times | R x T = (n-1) x 100 |
| Time to become n times | (n-1) x (time to double) |
| Rate from two amounts at diff T | SI/yr = (A2-A1)/(T2-T1) |
| Equal SI from two investments | P1.R1 = P2.R2 |
| Split investment (alligation) | Use effective rate method |
| Use fraction for R% | See fraction table above |
| Convert months to years | T = months / 12 |
| Convert days to years | T = days / 365 |
+──────────────────────────────────────────────────────────────+
Next: 8.3.c Solved Examples