Finding Rate & Time — Doubling Sums

🎯 Learning Objective

Rearrange the SI formula for R or T and crack doubling-tripling questions instantly.

💡 Concept

• R = (100 × SI)/(P × T) and T = (100 × SI)/(P × R) — same formula, rearranged
• Sum DOUBLES → SI = P → R × T = 100
• Doubles in n years → R = 100/n; at R% it doubles in 100/R years
• Sum becomes k times in n years → R = (k − 1) × 100/n
• SI grows linearly: doubles in 8 years → triples in 16 years (one extra P every 8 years)

🧮 Key Formulas

R = 100 × SI/(P × T)

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Doubles in n years → R = 100/n

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k times in n years → R = (k − 1) × 100/n

✏️ Easy Example

Q. At what rate per annum will ₹4,000 earn ₹1,000 as simple interest in 5 years?

1. R = (100 × 1000)/(4000 × 5)
2. = 100000/20000

Answer: 5%

🇮🇳 Real-Life Example

The classic post-office promise 'paisa double in 8 years' is quietly telling you the rate: 100/8 = 12.5% per annum simple.

📝 Exam-Level Example

Q. A sum of money doubles itself in 8 years at simple interest. Find the rate per annum.

1. Double → SI = P
2. R = 100/n = 100/8

Answer: 12.5%

📝 Exam-Level Example

Q. A sum becomes 3 times in 10 years at simple interest. In how many years will it become 5 times?

1. 3 times → SI = 2P in 10 years
2. So SI = P every 5 years
3. 5 times → SI = 4P → 4 × 5

Answer: 20 years

🪄 Memory Trick

Doubling table: 100/n gives the rate. k-times questions — count how many extra P you need, each extra P takes the same time.

⚠️ Common Mistakes

• ❌ Using k × 100/n instead of (k − 1) × 100/n for k-times questions
• ❌ Applying compound-interest doubling logic to SI questions
• ❌ Forgetting that the doubled amount includes the principal

🏆 Exam Tips

• ✅ Memorise: double in 10 yrs → 10%, in 8 yrs → 12.5%, in 5 yrs → 20%
• ✅ k times means SI = (k−1)P — write this before anything else

📌 Summary

• R and T come from the same SI formula, rearranged
• Double → SI = P → R × T = 100
• k times in n years → R = (k−1) × 100/n
• SI is linear — extra multiples take equal extra time