Simple Interest; Compound Interest; Time and Work

Simple Interest

10.1 Formula

SI = (P × R × T) / 100

Amount = P + SI = P(1 + RT/100)

Where:

P = Principal (initial amount)
R = Rate of interest (% per annum)
T = Time (in years)

10.2 Derivations

To Find	Formula
Principal (P)	P = (SI × 100) / (R × T)
Rate (R)	R = (SI × 100) / (P × T)
Time (T)	T = (SI × 100) / (P × R)

10.3 Converting Time Units

Days to years: T = days/365
Months to years: T = months/12

Example: P = 15000, R = 18%, from Jan 11 to Jun 3 (146 days):

T = 146/365 = 2/5 years
SI = 15000 × 18 × 2/5 × 1/100 = 30 × 18 × 2 = Rs. 1080
Amount = 15000 + 1080 = Rs. 16,080

10.4 Finding Principal When Amount at Different Times is Given

Key Insight: In SI, interest is CONSTANT every year.

Amount after n₁ years = A₁

Amount after n₂ years = A₂

SI per year = (A₂ - A₁) / (n₂ - n₁)
Principal = A₁ - n₁ × SI per year

Example: Amount after 3 years = Rs. 815, after 4 years = Rs. 854:

SI per year = 854 - 815 = Rs. 39
SI for 3 years = 39 × 3 = Rs. 117
Principal = 815 - 117 = Rs. 698

Compound Interest

11.1 Formula

A = P × (1 + R/100)ⁿ

CI = A - P = P[(1 + R/100)ⁿ - 1]

Where n = number of compounding periods (years if compounded annually)

11.2 Different Compounding Frequencies

Compounded	Formula
Annually	A = P(1 + R/100)ⁿ
Half-yearly	A = P(1 + R/200)^(2n)
Quarterly	A = P(1 + R/400)^(4n)
Monthly	A = P(1 + R/1200)^(12n)

11.3 CI vs SI Comparison

For same P, R, T:

CI > SI always (for T > 1 year)
For T = 1 year: CI = SI

Difference = CI - SI = P(R/100)² for 2 years

Difference = P(R/100)²(3 + R/100) for 3 years

11.4 Depreciation Formula

When value decreases at r% per year:

V = V₀ × (1 - r/100)ⁿ

Example: Machine worth Rs. 10,000 depreciates at 10% per year. Value after 2 years:

V = 10000 × (0.9)² = 10000 × 0.81 = Rs. 8100

11.5 Quick Formulas

CI for 2 years:

CI = P[2R/100 + (R/100)²]

CI for 3 years:

CI = P[3R/100 + 3(R/100)² + (R/100)³]

Example: CI on Rs. 2000 at 10% for 2 years:

CI = 2000 × [(1.1)² - 1] = 2000 × 0.21 = Rs. 420
Amount = 2000 + 420 = Rs. 2420

Time and Work

12.1 Fundamental Concept

Work = Efficiency × Time

If a person completes work in N days → his work per day = 1/N

12.2 Combined Work Formula

If A completes in 'a' days and B completes in 'b' days, together they complete in:

T = ab / (a + b) days

For three persons A, B, C with completion times a, b, c:

T = abc / (ab + bc + ca) days

12.3 Work Efficiency Table

Person	Days to Complete	1-day Work	Efficiency (LCM Method)
A	10 days	1/10	3 units/day (if LCM=30)
B	15 days	1/15	2 units/day
A+B	6 days	1/6	5 units/day

12.4 LCM Method (Most Efficient Approach)

Step 1: Find LCM of all given days (this = total work) Step 2: Calculate each person's efficiency = LCM / their days Step 3: Combine efficiencies as needed

Example: A completes in 10 days, B in 15 days. How long together?

LCM(10, 15) = 30 (Total work = 30 units)
A's efficiency = 30/10 = 3 units/day
B's efficiency = 30/15 = 2 units/day
Combined = 5 units/day
Time = 30/5 = 6 days

12.5 Work Done Simultaneously — Multiple Phases

When A works for x days, then B joins:

Work done by A in x days = x/a
Remaining work = 1
x/a
Time for B to complete remaining = (1 - x/a) × b

12.6 Important Formulas

Scenario	Formula
A and B work together	T = ab/(a+b)
A can do 1/3 work in n days	A completes total in 3n days
If M men do W work in D days	M×D = W (unit of work)
M₁D₁H₁/W₁ = M₂D₂H₂/W₂	General work formula

12.7 Man-Day Concept (Work Effort)

Work = Manpower × Time

If 10 men can do a work in 8 days → 20 men can do in 4 days
M₁ × D₁ = M₂ × D₂ (if same work amount)
M₁D₁H₁/W₁ = M₂D₂H₂/W₂ (general formula including hours and work quantity)