0
0

Different Time Periods Comparison

Introduction

Many SI questions compare interest earned over different time periods - for example, interest earned in months vs years, or interest for uneven time spans on the same principal. Mastering conversion of time units and setting up the SI formula correctly makes these problems straightforward.

Pattern: Different Time Periods Comparison

Pattern

Key concept: Always convert all time periods to the same unit (years) and use SI = (P × R × T)/100 - where T is in years.

Useful conversions and reminders:
• Months → years: months ÷ 12 (e.g., 9 months = 9/12 = 0.75 years).
• Days → years: days ÷ 365 (use 365 unless the problem says otherwise).
• When comparing two interests, equate SI expressions or compute each separately and compare the numeric values.

Step-by-Step Example

Question

A sum of money is lent at 6% per annum. The interest for 9 months is ₹90. Find the principal.

Options:

  1. ₹2,000
  2. ₹1,800
  3. ₹2,200
  4. ₹2,500

Solution

  1. Step 1: Convert and identify given values

    R = 6%, SI = ₹90, T = 9 months = 9/12 = 0.75 years.

  2. Step 2: Use the SI formula with T in years

    Apply SI = (P × R × T) / 100.

  3. Step 3: Substitute values

    90 = (P × 6 × 0.75) / 100

  4. Step 4: Simplify

    6 × 0.75 = 4.5 → 90 = (P × 4.5) / 100 → P × 4.5 = 9,000

  5. Step 5: Solve

    P = 9000 / 4.5 = 2000

  6. Final Answer:

    ₹2,000 → Option A

  7. Quick Check:

    Yearly SI = (2000×6)/100 = 120 → For 0.75 yr → 120×0.75 = 90 ✅

Quick Variations

1. Compare SI for different time spans on the same principal: compute yearly SI and scale by each time.

2. When rates differ, compute each SI separately with correct time units before comparing.

3. If one interest is given in months and the other in years, convert both to years first.

4. For multiple segments, add times as decimals (e.g., 1 + 0.5 = 1.5 years) or compute segment-wise.

Trick to Always Use

  • Step 1 → Convert ALL time values into years.
  • Step 2 → Use SI = (P × R × T)/100.
  • Step 3 → Compare interests numerically or algebraically as required.

Summary

Summary

  • Convert all time periods to years.
  • Use SI = (P × R × T)/100 with T in years.
  • Compare SI values only after converting time uniformly.
  • Quick check: compute yearly SI = (P×R)/100 and scale by time.

Example to remember:
At 6% for 9 months, SI = ₹90 → Principal = ₹2,000

Practice

(1/5)
1. A sum of ₹2000 is lent at 6% per annum for 9 months. Find the simple interest.
easy
A. ₹90
B. ₹100
C. ₹110
D. ₹120

Solution

  1. Step 1: Convert months to years

    Given P = 2000, R = 6% per annum, T = 9 months = 9/12 = 0.75 years.

  2. Step 2: Apply simple interest formula

    SI = (P × R × T) / 100 = (2000 × 6 × 0.75) / 100 = (2000 × 4.5) / 100 = 9000 / 100 = 90.

  3. Final Answer:

    SI = ₹90 → Option A.

  4. Quick Check:

    Yearly SI = (2000 × 6)/100 = 120; for 0.75 year → 120 × 0.75 = 90 ✅
Hint: Convert months to years (9/12 = 0.75) before using SI formula.
Common Mistakes: Using months as if they were years (e.g., treating 9 as 9 years).
2. If the SI on a sum of money for 15 months at 8% per annum is ₹1200, find the principal.
easy
A. ₹9600
B. ₹10,000
C. ₹11,000
D. ₹12,000

Solution

  1. Step 1: Convert months to years

    Given SI = 1200, R = 8% per annum, T = 15 months = 15/12 = 1.25 years.

  2. Step 2: Substitute in SI formula

    Use SI = (P × R × T)/100 → 1200 = (P × 8 × 1.25)/100.

  3. Step 3: Simplify the factor

    (8 × 1.25)/100 = 10/100 = 0.10, so 1200 = 0.10 × P.

  4. Step 4: Solve for principal

    P = 1200 / 0.10 = 12000.

  5. Final Answer:

    Principal = ₹12,000 → Option D.

  6. Quick Check:

    (12000 × 8 × 1.25)/100 = 1200 ✅
Hint: Convert months to years (15/12 = 1.25) and simplify the factor (R×T)/100 before dividing.
Common Mistakes: Forgetting to convert months to years or miscomputing (R×T)/100.
3. A sum of ₹5000 is lent at 10% per annum for 2 years. The same sum at the same rate is lent for 18 months. Find the difference in the interests.
easy
A. ₹250
B. ₹300
C. ₹350
D. ₹400

Solution

  1. Step 1: Write given values

    Given P = 5000, R = 10% per annum.

  2. Step 2: Compute SI for 2 years

    SI₁ = (5000 × 10 × 2)/100 = 1000.

  3. Step 3: Convert 18 months to years and compute SI

    18 months = 1.5 years → SI₂ = (5000 × 10 × 1.5)/100 = 750.

  4. Step 4: Compare the two interests

    Difference = SI₁ - SI₂ = 1000 - 750 = 250.

  5. Final Answer:

    Difference = ₹250 → Option A.

  6. Quick Check:

    Extra 0.5 year interest = (5000 × 10 × 0.5)/100 = 250 ✅
Hint: Compute yearly SI and scale by time (2.0 vs 1.5 years) to find the difference quickly.
Common Mistakes: Not converting 18 months to 1.5 years.
4. A sum of ₹8000 earns simple interest of ₹600 in 9 months at a certain rate. What will be the interest for 2 years at the same rate?
medium
A. ₹1200
B. ₹1600
C. ₹1400
D. ₹1800

Solution

  1. Step 1: Convert given time to years

    Given P = 8000, SI₁ = 600 for T₁ = 9 months = 0.75 years.

  2. Step 2: Find rate using SI formula

    R = (SI × 100)/(P × T) → R = (600 × 100)/(8000 × 0.75) = 60000/6000 = 10% per annum.

  3. Step 3: Use rate to compute new interest

    SI₂ = (8000 × 10 × 2)/100 = 1600.

  4. Final Answer:

    Interest for 2 years = ₹1600 → Option B.

  5. Quick Check:

    Yearly interest = (8000 × 10)/100 = 800; ×2 = 1600 ✅
Hint: First find the annual rate from the short period, then reuse it for the longer period.
Common Mistakes: Forgetting to convert 9 months to 0.75 years when finding R.
5. If ₹10,000 earns simple interest of ₹800 in 16 months, find the rate of interest per annum.
medium
A. 9%
B. 8%
C. 6%
D. 12%

Solution

  1. Step 1: Convert months to years

    Given P = 10000, SI = 800, T = 16 months = 16/12 = 4/3 years.

  2. Step 2: Apply rate formula

    R = (SI × 100)/(P × T) = (800 × 100)/(10000 × 4/3).

  3. Step 3: Simplify the fraction

    Denominator = 10000 × 4/3 = 40000/3 → R = 80000 / (40000/3) = 80000 × (3/40000) = 6.

  4. Final Answer:

    Rate = 6% → Option C.

  5. Quick Check:

    SI = (10000 × 6 × 4/3)/100 = 800 ✅
Hint: Use fraction simplification: 16/12 = 4/3 to keep arithmetic exact.
Common Mistakes: Rounding intermediate steps or not simplifying the fraction 16/12 to 4/3.

Mock Test

Ready for a challenge?

Take a 10-minute AI-powered test with 10 questions (Easy-Medium-Hard mix) and get instant SWOT analysis of your performance!

10 Questions
5 Minutes