source: ncert
topic: arithmetic
class_level: class_8
chapter: ch8
difficulty: intermediate
Class 8 | Ch8: Comparing Quantities (Compound Interest) RECAP - PROFIT/LOSS AND SIMPLE INTEREST: Profit%=(Profit/CP)×100. Loss%=(Loss/CP)×100. SI=(P×R×T)/100. Amount=P+SI=P(1+RT/100). COMPOUND INTEREST: Interest calculated on principal AND accumulated interest. Formula: A=P(1+R/100)^n where P=principal, R=rate%, n=time in years. CI=A-P=P[(1+R/100)^n - 1]. Compounded half-yearly: A=P(1+R/200)^(2n) (rate halved, time doubled). Compounded quarterly: A=P(1+R/400)^(4n). CI>SI for same P,R,T (except for 1 year when CI=SI). DIFFERENCE FORMULA (for 2 years): CI-SI = P×(R/100)^2. APPLICATIONS: Population growth: P_n=P_0(1+R/100)^n. Depreciation: V_n=V_0(1-R/100)^n. SOLVED EXAMPLES: Example 1: CI on 12000 at 10% for 2 years compounded annually. A=12000(1+10/100)^2=12000×(1.1)^2=12000×1.21=14520. CI=14520-12000=2520. SI=12000×10×2/100=2400. CI-SI=2520-2400=120=12000×(10/100)^2=12000×0.01=120. Verified. Example 2: CI on 8000 at 10% for 1.5 years compounded half-yearly. Rate per half year=5%, number of periods=3. A=8000(1+5/100)^3=8000×(1.05)^3=8000×1.157625=9261. Example 3: Population 20000, grows 5% per year. After 2 years? P=20000(1.05)^2=20000×1.1025=22050. COMMON MISTAKES: For half-yearly compounding: halve the rate AND double the time period. CI and SI are EQUAL only for first year. After that CI>SI.