Proportional Reasoning II — Class 8 Mathematics (Ganita Prakash)
"Compound interest is the eighth wonder of the world. Those who understand it, earn it. Those who don't, pay it." — Albert Einstein (attributed)
1. About the Chapter
This chapter extends 'Proportional Reasoning' (Ch 7) with more advanced applications:
- Compound Interest (the most important new concept)
- Time and Work problems
- Time, Speed, and Distance
- Mixtures and Alligation
- Partnership (sharing profits)
These are the bread and butter of competitive exams and real-world finance.
2. Compound Interest (CI)
Definition
Compound Interest is interest calculated on the principal PLUS accumulated interest from previous periods. Unlike simple interest (which is fixed), CI grows exponentially.
Formula
A = P(1 + R/100)ⁿ
- A = Amount after n periods
- P = Principal
- R = Rate of interest per period (%)
- n = Number of compounding periods
Compound Interest
CI = A − P
How It Works
Example: ₹10,000 at 10% per annum, compounded annually for 3 years.
- After 1 year: 10000 × 1.10 = ₹11,000
- After 2 years: 11000 × 1.10 = ₹12,100
- After 3 years: 12100 × 1.10 = ₹13,310
Using formula: A = 10000 × (1.10)³ = 10000 × 1.331 = ₹13,310 ✓ CI = 13310 − 10000 = ₹3,310
Comparison with Simple Interest
For the same P, R, n:
- SI grows linearly: a fixed amount per period
- CI grows exponentially: each period's interest is bigger than the last
Compounding Periods
Interest may be compounded:
- Annually (yearly) — formula as above
- Half-yearly (twice a year) — use R/2 as rate, 2n as periods
- Quarterly (4 times) — use R/4, 4n periods
- Monthly (12 times) — use R/12, 12n periods
Example (half-yearly): ₹8,000 at 10% per annum, compounded half-yearly, 2 years.
- Half-year rate = 10/2 = 5%
- Periods = 2 × 2 = 4
- A = 8000 × (1.05)⁴ = 8000 × 1.21551 ≈ ₹9,724
3. Time and Work
Basic Principle
If a person can complete a job in T days, they do 1/T of the job per day.
Combined Rate
If A does job in a days and B in b days:
- A's daily rate = 1/a
- B's daily rate = 1/b
- Combined rate = 1/a + 1/b = (a+b)/(ab)
- Time together = ab/(a+b)
Example
A does a job in 12 days, B does it in 18 days. How long together?
- Combined rate = 1/12 + 1/18 = 3/36 + 2/36 = 5/36
- Time = 36/5 = 7.2 days
Workers Adding/Leaving
Example: A and B together do a job in 12 days. A alone takes 18 days. How long does B alone take?
- A's rate = 1/18, Together = 1/12
- B's rate = 1/12 − 1/18 = 3/36 − 2/36 = 1/36
- B alone takes 36 days
4. Time, Speed, Distance
Fundamental Formula
Distance = Speed × Time
- d = s × t
- t = d/s
- s = d/t
Unit Conversions
- 1 km = 1000 m
- 1 hr = 3600 s
- km/hr to m/s: multiply by 5/18
- m/s to km/hr: multiply by 18/5
Example
A train travels at 72 km/hr. What is its speed in m/s?
- 72 × 5/18 = 20 m/s
Average Speed
For one trip at two different speeds: NOT the arithmetic mean.
If you travel distance d at speed s₁ going, and same d back at s₂:
- Average speed = 2s₁s₂ / (s₁ + s₂) (harmonic mean)
Train Problems
- Length of train crossing a pole: distance = length of train
- Crossing a bridge: distance = length of train + length of bridge
- Two trains in same direction: relative speed = difference
- Two trains in opposite direction: relative speed = sum
5. Mixtures and Alligation
Mixing Two Items at Different Prices
If we mix:
- a₁ units at price p₁ per unit
- a₂ units at price p₂ per unit
- Average price = (a₁p₁ + a₂p₂) / (a₁ + a₂)
Alligation (Inverse Approach)
To find ratio of mixing two items to get a desired mean price:
- (Cheaper) : (Dearer) = (Dearer − Mean) : (Mean − Cheaper)
Example: Mix milk (₹30/L) and water (₹0/L) to get a mixture costing ₹20/L.
- Ratio = (30 − 20) : (20 − 0) = 10 : 20 = 1 : 2
- So 1 part milk : 2 parts water? Wait, that's wrong. Let me redo:
- Actually milk:water = (Dearer − Mean) : (Mean − Cheaper)
- Milk is dearer (30); water is cheaper (0); mean is 20
- Wait, the formula gives (mean − cheaper) : (dearer − mean)
- = (20 − 0) : (30 − 20) = 20:10 = 2:1
- So milk:water = 2:1
- Check: 2L milk at ₹30 + 1L water at ₹0 = ₹60 for 3L = ₹20/L ✓
6. Partnership
Basic Idea
When two or more people invest different amounts for different times, the profit is shared in the ratio of (investment × time).
Formula
Profit ratio = I₁ × T₁ : I₂ × T₂ : I₃ × T₃ : ...
Example
A invests ₹5,000 for 6 months. B invests ₹4,000 for 12 months. Profit ₹1,800 — how is it shared?
- A's share = 5000 × 6 = 30,000
- B's share = 4000 × 12 = 48,000
- Ratio = 30000 : 48000 = 5 : 8
- A's share = (5/13) × 1800 = ₹692.31
- B's share = (8/13) × 1800 = ₹1107.69
7. Worked Examples
Example 1: Compound Interest
Find CI on ₹5,000 at 8% per annum for 2 years.
- A = 5000(1 + 8/100)² = 5000(1.08)² = 5000 × 1.1664 = ₹5,832
- CI = 5832 − 5000 = ₹832
Example 2: Half-yearly CI
₹20,000 at 10% per annum, compounded half-yearly, 1.5 years.
- Half-year rate = 5%, periods = 3
- A = 20000(1.05)³ = 20000 × 1.157625 = ₹23,152.50
- CI = ₹3,152.50
Example 3: Time and Work
A can do a job in 15 days, B in 20 days. They work together for 4 days, then B leaves. How long does A take to finish?
- Combined daily rate = 1/15 + 1/20 = 7/60
- Work done in 4 days = 4 × 7/60 = 28/60 = 7/15
- Remaining work = 1 − 7/15 = 8/15
- A's rate = 1/15. Days needed = (8/15) / (1/15) = 8 days
- Answer: A takes 8 more days.
Example 4: Speed-Distance-Time
A train 120 m long passes a pole in 8 seconds. Find its speed in km/h.
- Speed = 120/8 = 15 m/s
- Convert: 15 × 18/5 = 54 km/h
Example 5: Trains Meeting
Two trains, 150 m and 100 m long, travel in opposite directions on parallel tracks. Speeds: 60 and 30 km/h. How long do they take to cross each other?
- Total distance = 150 + 100 = 250 m
- Relative speed = 60 + 30 = 90 km/h = 90 × 5/18 = 25 m/s
- Time = 250/25 = 10 seconds
Example 6: Mixture
A 60-L mixture has milk and water in ratio 5:1. How much water to add to make ratio 5:2?
- Initial: 50 L milk, 10 L water
- After adding x L water: 50 milk : (10+x) water = 5:2
- 50/(10+x) = 5/2
- 100 = 50 + 5x
- 5x = 50
- x = 10 L of water to add
Example 7: Partnership
A invests ₹15,000 for 1 year, B invests ₹20,000 for 9 months. If profit is ₹6,300, find each share.
- A: 15000 × 12 = 180,000
- B: 20000 × 9 = 180,000
- Ratio = 1:1
- Each gets ₹3,150
8. Common Mistakes
-
Using SI formula instead of CI
- SI: PRT/100 (linear)
- CI: P(1 + R/100)ⁿ − P (exponential)
-
Forgetting to adjust for non-annual compounding
- Half-yearly: R/2, 2n
- Quarterly: R/4, 4n
-
Wrong unit conversion
- km/h → m/s: multiply by 5/18
- m/s → km/h: multiply by 18/5
-
Average speed = arithmetic mean
- WRONG when distances are equal
- Use 2s₁s₂/(s₁+s₂) instead
-
Confusing 'with stream' and 'against stream' in boat problems
- With: boat speed + stream speed
- Against: boat speed − stream speed
9. Real-World Applications
Banking
- Fixed deposits use CI (usually compounded quarterly)
- Loans use compound interest reverse
- Credit cards charge CI on outstanding amounts
Real Estate
- Property values typically grow with CI-like patterns
- Mortgage payments use compound interest
Business
- Inflation compounds yearly (~6% in India typically)
- Salary growth often modelled with CI
Daily Life
- Travel time calculations
- Cooking (mixtures, recipes)
- Comparing prices (alligation)
10. Tips for Mastery
For CI
- Memorise A = P(1 + R/100)ⁿ
- Know how to adjust for half-yearly/quarterly
- Practise 10 problems mixing P, R, n
For Time and Work
- Always think in 'per-day rates'
- Add rates for working together
- Subtract for one working
For Speed-Distance
- Memorise conversion factor 5/18
- Draw diagrams for train and boat problems
Practice
- Solve 5 CI problems
- Solve 5 time-work problems
- Solve 5 speed-distance problems
- Practise alligation for mixtures
11. Conclusion
'Proportional Reasoning II' is the chapter that prepares you for life. CI is the foundation of all banking and investment. Time-and-work and speed-distance problems are core to logical reasoning. Mixtures and partnership are essential for business arithmetic.
Master these topics, and you'll be:
- Financially literate (CI, EMI calculations)
- Logically sharp (time-work problems)
- Practically smart (mixtures, alligation)
Every competitive exam (NTSE, IMO, JEE Foundation, CAT) tests these heavily. Build a strong foundation now.
