Principal \(P\)
The original sum borrowed, lent or invested. Under compounding, the amount at the end of one period becomes the base for the next period.
7 SSC CGL · Mathematics
Simple and Compound Interest
Handwritten-style notes for SSC CGL Tier I and Tier II.
01MAP IT!
The Big Picture — Graphic Mind Map
Interest is the price of using money; the growth rule decides whether old interest also earns interest.
INTEREST
GROWTH
SYSTEM
GROWTH
SYSTEM
1. Vocabularyprincipal, rate, time, interest, amount
2. Simple Interestconstant interest on original principal
3. Compound Interestinterest on the updated amount
4. Compounding Clockannual, half-yearly and quarterly
5. Change Modelsgrowth, depreciation and variable rates
6. Exam Speeddifference formulas, factors and reverse work
Two roads begin with the same principal
Simple interest walks on a straight road because each period adds the same interest. Compound interest bends upward because every new amount becomes the next principal.
\[\text{SI: constant addition}\qquad\text{CI: repeated multiplication}\]
\[\boxed{A=P+I}\qquad\text{and}\qquad\boxed{\text{growth factor}=1\pm\frac{R}{100}}\]
02ZERO TO EXPERT
The Foundation (Basics) — Complete Concept Build
Learn the language first, then choose the correct time clock and growth model.
AThe five quantities
Rate \(R\)
Interest charged per hundred of principal per stated time unit, normally per annum.
\[R\%=\frac{R}{100}\]
Time \(T\)
The duration must use the same unit as the rate. Convert months to years when the rate is per annum.
\[T=\frac{m}{12}\text{ years}\]
Interest \(I\)
The extra money paid or earned for the use of principal.
\[I=A-P\]
Amount \(A\)
The final total after adding interest to principal.
\[A=P+I\]
Language decoder
- “Sum” usually means \(P\).
- “Worth” or “amount” means \(A\).
- “Per annum” means per year.
- “Interest accrued” means \(I\).
The unit check prevents half the errors
Rate and time must speak the same language. A yearly rate needs time in years; a half-yearly rate needs the number of half-years.
\[18\text{ months}=\frac{18}{12}=\frac32\text{ years}\]
BSimple Interest — one fixed base
Master formula
\[\boxed{\mathrm{SI}=\frac{PRT}{100}}\]
The formula assumes \(R\) percent per year and \(T\) years.
Amount under SI
\[A=P+\frac{PRT}{100}=P\left(1+\frac{RT}{100}\right)\]
Rearrangements
\[P=\frac{100I}{RT},\quad R=\frac{100I}{PT},\quad T=\frac{100I}{PR}\]
Interest per year
\[\text{yearly SI}=\frac{PR}{100}\]
It stays constant because the original \(P\) never changes.
Months and days
\[I=\frac{PRm}{1200}\]
\[I=\frac{PRd}{100\times365}\]
Use \(365\) days unless the question explicitly states another convention.
Ratio property
With other quantities fixed:
\[I\propto P,\qquad I\propto R,\qquad I\propto T\]
Simple interest climbs equal steps
If annual interest is \(j\), the amounts are \(P, P+j, P+2j,\ldots\). This arithmetic progression is the visual signature of SI.
\[A_n=P+nj\quad\text{where}\quad j=\frac{PR}{100}\]
CSpecial SI relationships
Amount becomes \(k\) times
If \(A=kP\), then \(I=(k-1)P\).
\[T=\frac{100(k-1)}{R}\]
Doubling and tripling
\[T_{2P}=\frac{100}{R},\qquad T_{3P}=\frac{200}{R}\]
This shortcut is for SI only.
Amounts at two times
If amounts at \(T_1,T_2\) are \(A_1,A_2\):
\[\text{annual SI}=\frac{A_2-A_1}{T_2-T_1}\]
\[P=A_1-T_1\left(\frac{A_2-A_1}{T_2-T_1}\right)\]
DCompound Interest — the amount becomes the next principal
Annual compounding
\[\boxed{A=P\left(1+\frac{R}{100}\right)^T}\]
\[\mathrm{CI}=A-P\]
Growth factor
Write \(r=R/100\). One period multiplies money by \(1+r\); \(n\) periods multiply it by \((1+r)^n\).
Year-wise interest
\[I_n=P(1+r)^{n-1}r\]
Each year’s interest is \((1+r)\) times the preceding year’s interest.
Compounding is a repeated multiplier
Do not add the same interest every year. Apply the growth factor to the updated balance each period.
\[P\to P(1+r)\to P(1+r)^2\to\cdots\to P(1+r)^n\]
EThe compounding clock: annual, half-yearly and quarterly
| Compounding | Rate per period | Periods in \(T\) years | Amount |
|---|---|---|---|
| Annual | \(R\%\) | \(T\) | \(P(1+R/100)^T\) |
| Half-yearly | \(R/2\%\) | \(2T\) | \(P(1+R/200)^{2T}\) |
| Quarterly | \(R/4\%\) | \(4T\) | \(P(1+R/400)^{4T}\) |
| \(m\) times yearly | \(R/m\%\) | \(mT\) | \(P(1+R/(100m))^{mT}\) |
Two things change together
If compounding becomes more frequent, divide the annual nominal rate and multiply the number of periods by the same frequency.
\[R\to\frac{R}{m},\qquad T\to mT\]
Odd time with annual compounding
For \(n\) complete years plus a fractional year \(f\), many elementary SSC questions apply CI for \(n\) years and SI on the resulting amount for the fraction, unless a different compounding rule is stated.
\[A=P(1+r)^n(1+fr)\]
Exact periods only
For half-yearly compounding, \(18\) months gives \(3\) complete half-years:
\[A=P\left(1+\frac{R}{200}\right)^3\]
Always count the actual compounding periods.
FVariable rates and successive change
Different annual rates
\[A=P\prod_{i=1}^{n}\left(1+\frac{R_i}{100}\right)\]
Multiply the year-wise factors; never average rates blindly.
Growth followed by fall
\[A=P\left(1+\frac{x}{100}\right)\left(1-\frac{y}{100}\right)\]
Net successive percentage
\[x+y+\frac{xy}{100}\]
Use signed values: a decrease of \(d\%\) means \(y=-d\).
Rates form a chain of multipliers
Order does not change the final product when each rate acts on the entire current amount, but intermediate balances differ.
\[P(1+r_1)(1+r_2)=P(1+r_2)(1+r_1)\]
GGrowth and depreciation models
Repeated growth
\[V_n=V_0\left(1+\frac{R}{100}\right)^n\]
Used for population, production, prices and value appreciation.
Repeated depreciation
\[V_n=V_0\left(1-\frac{R}{100}\right)^n\]
Depreciation is calculated on the reduced value each period.
Original value from present value
\[V_0=\frac{V_n}{(1\pm R/100)^n}\]
Reverse the multiplier by division.
Up escalator vs down escalator
A percentage rise and the same percentage fall do not cancel because the bases differ.
\[(1+r)(1-r)=1-r^2\]
\[\text{net loss}=\frac{R^2}{100}\%\]
HEffective annual rate
General effective rate
If nominal annual rate \(R\%\) is compounded \(m\) times yearly:
\[R_{\mathrm{eff}}=100\left[\left(1+\frac{R}{100m}\right)^m-1\right]\%\]
Half-yearly shortcut
\[R_{\mathrm{eff}}=R+\frac{R^2}{400}\]
Here \(R\) is the nominal annual percentage rate.
Two successive rates
\[R_{\mathrm{eff}}=R_1+R_2+\frac{R_1R_2}{100}\]
This returns the net percentage for one combined cycle.
Effective rate includes interest on interest
More frequent compounding produces a larger effective yearly yield for the same positive nominal annual rate.
\[\left(1+\frac{R}{200}\right)^2\gt1+\frac{R}{100}\quad(R\gt0)\]
IDifference between CI and SI
For one year
\[\mathrm{CI}-\mathrm{SI}=0\]
Both use only the original principal during the first year.
For two years
\[\boxed{\mathrm{CI}-\mathrm{SI}=P\left(\frac{R}{100}\right)^2}\]
For three years
\[\boxed{\mathrm{CI}-\mathrm{SI}=P\left(\frac{R}{100}\right)^2\left(3+\frac{R}{100}\right)}\]
Binomial origin
\[\mathrm{CI}=P\left[(1+r)^n-1\right]\]
\[\mathrm{SI}=Pnr\]
\[\mathrm{CI}-\mathrm{SI}=P\left[\binom n2r^2+\binom n3r^3+\cdots+r^n\right]\]
Two-year reverse formula
If the two-year difference is \(D\):
\[P=\frac{10000D}{R^2}\qquad R=100\sqrt{\frac DP}\]
Valid for the same annual rate under annual compounding.
The gap begins after year one
The extra compound interest is the interest earned on earlier interest. For two years, it is simply one year’s interest on the first year’s interest.
\[\text{gap}=\left(\frac{PR}{100}\right)\frac{R}{100}\]
JPresent value and reverse compounding
Present value
\[P=\frac{A}{(1+R/100)^T}\]
Move backward through time by dividing by each growth factor.
Variable-rate reverse
\[P=\frac{A}{\prod_{i=1}^{n}(1+R_i/100)}\]
Depreciation reverse
\[V_0=\frac{V_n}{(1-R/100)^n}\]
Do not subtract depreciation percentages directly from the final value.
Forward means multiply; backward means divide
This single picture handles present value, original population, original machine price and missing principal.
KComparison bank and edge conditions
| Situation | Correct rule | Watch condition |
|---|---|---|
| SI vs CI for positive \(P,R,T\) | For more than one full annual period, CI exceeds SI. | For one period they are equal. |
| Same nominal rate | More frequent compounding gives higher amount. | Assumes positive rate and same duration. |
| Equal rise and fall | Net loss \(=R^2/100\%\). | The second percentage uses a changed base. |
| Successive rates | Multiply factors. | Use a negative signed rate for decrease. |
| Zero rate | \(A=P\). | Both SI and CI are zero. |
| Rate \(100\%\) depreciation | Value becomes zero after the first period. | A depreciation rate above \(100\%\) is not meaningful in the ordinary value model. |
| Fractional period | Follow the stated compounding convention. | Do not invent quarterly or monthly compounding. |
03SAVE TIME
Short Tricks & Magic Formulas
Translate every percentage into a multiplier before doing long arithmetic.
1Factor method
Multiplier bank
\[10\%\to\frac{11}{10},\quad20\%\to\frac65,\quad25\%\to\frac54\]
\[12.5\%\to\frac98,\quad16\frac23\%\to\frac76\]
Depreciation factors
\[10\%\downarrow\to\frac9{10},\quad20\%\downarrow\to\frac45,\quad25\%\downarrow\to\frac34\]
Cancel before multiplying
For \(25\%\) growth over \(3\) years:
\[A=P\left(\frac54\right)^3\]
Cancel factors with \(P\) before evaluating powers.
2Instant SI recovery
Given two amounts
\[j=\frac{A_2-A_1}{T_2-T_1},\qquad P=A_1-jT_1\]
Given amount ratio
If SI amount after \(T\) years is \(kP\):
\[R=\frac{100(k-1)}{T}\]
Rate–time swap
Under SI, the product \(RT\) controls the percentage interest.
\[R_1T_1=R_2T_2\Rightarrow I_1=I_2\quad\text{for equal }P\]
3CI–SI difference shortcuts
Two years
\[D_2=Pr^2\]
If first-year interest is \(j=Pr\), then \(D_2=jr\).
Three years from two-year gap
\[D_3=D_2(3+r)\]
Here \(r=R/100\), not \(R\).
Second-year interest
\[I_2=I_1(1+r)\]
Thus \(r=I_2/I_1-1\).
4Successive-rate speed rules
Two increases
\[x\%\uparrow,y\%\uparrow\Rightarrow x+y+\frac{xy}{100}\%\uparrow\]
Increase then decrease
\[x\%\uparrow,y\%\downarrow\Rightarrow x-y-\frac{xy}{100}\%\]
Equal opposite rates
\[x\%\uparrow,x\%\downarrow\Rightarrow\frac{x^2}{100}\%\downarrow\]
5Option elimination and estimation
Monotonicity test
For positive \(P,R,T\), the amount rises when any one rises. Reject options violating this direction.
CI lower bound
\[(1+r)^n\gt1+nr\quad(n\gt1,r\gt0)\]
So CI must exceed SI for more than one period.
Approximation for small rate
\[(1+r)^n\approx1+nr+\binom n2r^2\]
Use only for estimation or option elimination, not when exact values are required.
SSC interest speed dashboard
Read in this order: identify SI or CI, align time units, convert to a factor, cancel, and only then calculate.
\[\text{model}\to\text{clock}\to\text{factor}\to\text{cancel}\to\text{answer}\]
04DON'T RUSH
The SSC / TCS Traps — Red Flags 🚩
Most distractors come from a wrong clock, a wrong base or confusing interest with amount.
Trap 1: rate changed, periods forgotten
- For half-yearly compounding, halve the rate and double the periods.
- For quarterly compounding, divide rate by \(4\) and multiply periods by \(4\).
- Changing only one creates the classic distractor.
Trap 2: interest is not amount
- The compound formula gives \(A\), not CI.
- Always subtract principal: \(\mathrm{CI}=A-P\).
- If the question gives interest, add \(P\) before reversing the factor.
Trap 3: percentages act on changing bases
- An equal percentage rise and fall produces a loss, not zero change.
- Do not average variable rates unless the question’s weights justify it.
- Depreciation is on reducing value, not original value.
SI doubling rule used on CI
\(T=100/R\) is an SI rule. Compound doubling requires solving \((1+r)^T=2\) or using stated factors.
Fractional year guessed
Use the convention explicitly stated. Annual compounding plus extra months is not automatically monthly compounding.
Three-year gap misremembered
The multiplier is \(3+r\), where \(r=R/100\). Writing \(3+R\) is dimensionally wrong.
05LOCK IT IN
Memory Hooks & Mnemonics
Turn every interest rule into a picture that survives exam pressure.
“Simple means same step”
SI adds equal interest each year because the principal base stays fixed.
“Compound rolls a snowball”
Old interest joins the base, so the next period earns interest on a larger amount.
“Slice rate, multiply time”
For \(m\) compoundings per year, divide \(R\) by \(m\) and multiply \(T\) by \(m\).
“Future multiply, past divide”
Move forward with growth factors; recover the original value by dividing them away.
“The gap is interest on interest”
For two years, apply the rate once more to the first year’s interest: \(D_2=Pr^2\).
“Same up, same down — still down”
Equal opposite percentages leave a loss of \(R^2/100\%\) because the bases differ.
SI engine\(PRT/100\)
CI engine\(P(1+r)^n\)
Half-year clockrate half, periods double
Successive ratesmultiply factors
Two-year gap\(Pr^2\)
Reverse journeydivide by factors