Type 9: When Amounts Are Equal
What This Type is About
In this type of simple interest question, two different investments produce the same final amount. The principal, rate, or time may be different, but the total amount at the end is equal.
Remember:
Amount = Principal + Simple Interest
So, in these questions, we compare total amounts, not just the interest earned.
Main Concept
When amounts are equal:
Amount₁ = Amount₂
Using the amount formula:
Amount = P × (100 + RT) / 100
we get:
P₁ × (100 + R₁T₁) = P₂ × (100 + R₂T₂)
This relation is enough to solve most questions of this type.
Percentage Method
Instead of calculating interest separately, convert each investment directly into amount percentage.
Amount Percentage = 100 + (Rate × Time)
For example:
10% for 2 years → Amount = 120% of Principal
5% for 4 years → Amount = 120% of Principal
20% for 3 years → Amount = 160% of Principal
This method makes calculations much faster.
Finding Time
Example
Amount on ₹4,000 at 10% for 2 years equals the amount on ₹3,000 at 20% for T years.
First investment:
Amount = 120% of ₹4,000 = ₹4,800
Second investment:
Amount = (100 + 20T)% of ₹3,000
Form the equation:
4800 = (100 + 20T)% of 3000
4800 / 3000 = (100 + 20T) / 100
160 = 100 + 20T
T = 3 years
Finding Principal
Example
Amount on ₹X at 10% for 2 years equals the amount on ₹2,000 at 20% for 2 years.
Second investment:
Amount = 140% of ₹2,000 = ₹2,800
First investment:
Amount = 120% of X
Equation:
1.2X = 2800
X = ₹2,333.33
Shortcut Method
For exam questions:
Convert each investment into amount percentage.
Compare the total amounts.
Form a simple equation.
Solve for the unknown value.
This approach is much faster than calculating simple interest separately.
Common Mistakes
Comparing only simple interest instead of total amount.
Forgetting to include the principal.
Using the equal-interest method from Type 8.
Making mistakes while converting into amount percentage.
Using lengthy formulas when a percentage method is sufficient.
Exam Strategy
When you see “amounts are equal”:
Convert both sides into amount percentages.
Write a simple equation.
Cancel common values whenever possible.
Solve directly without unnecessary calculations.
Key Takeaways
Amount = Principal + Simple Interest.
Equal amounts mean total values are equal.
Use Amount Percentage = 100 + RT.
Compare total amounts, not just interest.
Percentage comparison is usually the fastest method for exams.
20 Practice MCQs
1. A sum of money at 10% simple interest becomes ₹1500 in 5 years. Find the principal.
A) ₹900
B) ₹1000
C) ₹1200
D) ₹1250
Answer: B) ₹1000
Solution:
SI=\frac{P\times R\times T}{100}
SI = 50% of principal in 5 years.
Amount = Principal + SI
1500 = 150% of Principal
Principal = 1500 × 100 / 150 = ₹1000
2. In how many years will a sum double itself at 12.5% simple interest?
A) 6 years
B) 8 years
C) 10 years
D) 12 years
Answer: B) 8 years
Solution:
To double, interest = 100%
Time = 100 / 12.5 = 8 years
3. A sum becomes 5 times in 20 years at simple interest. Find the rate.
A) 15%
B) 18%
C) 20%
D) 25%
Answer: C) 20%
Solution:
5 times means interest = 4 parts
Rate = (4 / 20) × 100 = 20%
4. If ₹8000 earns ₹2400 simple interest in 4 years, find the rate.
A) 6%
B) 7.5%
C) 8%
D) 10%
Answer: B) 7.5%
Solution:
SI=\frac{P\times R\times T}{100}
2400 = (8000 × R × 4)/100
R = 7.5%
5. A sum becomes 3 times in 10 years. In how many years will it become 7 times?
A) 25 years
B) 30 years
C) 35 years
D) 40 years
Answer: B) 30 years
Solution:
3 times → interest = 2 parts in 10 years
7 times → interest = 6 parts
Time = (6/2) × 10 = 30 years
6. Difference between SI for 6 years and 2 years is ₹1200. Find yearly interest.
A) ₹200
B) ₹250
C) ₹300
D) ₹400
Answer: C) ₹300
Solution:
Difference in years = 4
Yearly interest = 1200 / 4 = ₹300
7. The difference between SI for 5 years and 3 years is ₹500 at 5%. Find the principal.
A) ₹4000
B) ₹5000
C) ₹6000
D) ₹7000
Answer: B) ₹5000
Solution:
Difference = interest for 2 years
Yearly interest = 500 / 2 = 250
5% of principal = 250
Principal = 250 × 100 / 5 = ₹5000
8. ₹10,000 is divided into two parts at 5% and 10%. Total interest for 1 year is ₹700. Amount invested at 10% is:
A) ₹3000
B) ₹4000
C) ₹5000
D) ₹6000
Answer: B) ₹4000
Solution:
Average rate = 700/10000 × 100 = 7%
Alligation:
10 − 7 = 3
7 − 5 = 2
Ratio = 3 : 2
At 10% = 2/5 × 10000 = ₹4000
9. A rate decreases from 12% to 10%, and SI decreases by ₹400 in 2 years. Find principal.
A) ₹8000
B) ₹10000
C) ₹12000
D) ₹15000
Answer: B) ₹10000
Solution:
Rate difference = 2%
For 2 years = 4%
4% = ₹400
100% = ₹10000
10. ₹5000 is invested at 8% for 2 years and then at 12% for 3 years. Find total SI.
A) ₹2200
B) ₹2400
C) ₹2600
D) ₹2800
Answer: C) ₹2600
Solution:
8 × 2 = 16%
12 × 3 = 36%
Total = 52%
52% of 5000 = ₹2600
11. SI on ₹4000 at R% for 2 years equals SI on ₹2000 at 10% for 4 years. Find R.
A) 8%
B) 10%
C) 12%
D) 15%
Answer: B) 10%
Solution:
4000 × R × 2 = 2000 × 10 × 4
8000R = 80000
R = 10%
12. Amount on ₹5000 at 10% for 2 years is equal to amount on ₹4000 at 20% for T years. Find T.
A) 2.5 years
B) 3 years
C) 4 years
D) 5 years
Answer: A) 2.5 years
Solution:
First amount = 120% of 5000 = 6000
6000 = (100 + 20T)% of 4000
150 = 100 + 20T
T = 2.5 years
13. A sum trebles itself in 16 years. In how many years will it become 9 times?
A) 48 years
B) 56 years
C) 64 years
D) 72 years
Answer: C) 64 years
Solution:
Treble → interest = 2 parts in 16 years
9 times → interest = 8 parts
Time = (8/2) × 16 = 64 years
14. Find SI on ₹12,000 at 15% for 2 years.
A) ₹3000
B) ₹3200
C) ₹3600
D) ₹4000
Answer: C) ₹3600
Solution:
15 × 2 = 30%
30% of 12000 = ₹3600
15. A sum becomes ₹8400 in 4 years and ₹9600 in 6 years at SI. Find the principal.
A) ₹6000
B) ₹7200
C) ₹8000
D) ₹9000
Answer: A) ₹6000
Solution:
Difference = 9600 − 8400 = 1200
2 years’ interest = 1200
1 year interest = 600
4 years’ interest = 2400
Principal = 8400 − 2400 = ₹6000
16. What rate percent per annum will make ₹800 become ₹920 in 3 years at SI?
A) 4%
B) 5%
C) 6%
D) 7%
Answer: B) 5%
Solution:
Interest = 920 − 800 = 120
SI=\frac{P\times R\times T}{100}
120 = (800 × R × 3)/100
R = 5%
17. A sum becomes 6 times in 25 years. Find rate percent.
A) 16%
B) 18%
C) 20%
D) 25%
Answer: C) 20%
Solution:
6 times → interest = 5 parts
Rate = (5/25) × 100 = 20%
18. SI on a certain sum for 5 years at 8% is ₹2400. Find principal.
A) ₹5000
B) ₹6000
C) ₹7000
D) ₹8000
Answer: B) ₹6000
Solution:
2400 = (P × 8 × 5)/100
2400 = 40% of P
P = ₹6000
19. The simple interest on a sum at 9% for 4 years is ₹720. Find the sum.
A) ₹1800
B) ₹2000
C) ₹2200
D) ₹2500
Answer: B) ₹2000
Solution:
36% of principal = 720
Principal = 720 × 100 / 36 = ₹2000
20. A person invested ₹6000 at 5% and another ₹4000 at 8%. Find total SI for 2 years.
A) ₹1240
B) ₹1280
C) ₹1320
D) ₹1360
Answer: A) ₹1240
Solution:
Interest from ₹6000 = 6000 × 5 × 2 /100 = 600
Interest from ₹4000 = 4000 × 8 × 2 /100 = 640
Total = ₹1240
Conclusion
Simple interest may look like a small topic, but it builds a very strong base for understanding how money grows over time. Once you clearly understand the concept that interest is constant every year, all 9 types become easy and logical instead of confusing. By using simple thinking, percentage methods, and smart simple interest tricks, you can solve questions faster and with more confidence in exams.
The key to mastering this topic is not memorizing formulas, but understanding the logic behind each type and practicing regularly. When you focus on concepts like yearly interest, ratio, and comparison, even difficult questions become simple. This approach not only helps in exams but also improves your real-life financial understanding.
In the future, we will move to the next important topic, compound interest, where interest is added to the principal and grows faster over time. It may look a little tricky at first, but with the same simple approach and clear concepts, you will be able to understand it easily. Stay tuned 👍
Frequently Asked Questions (FAQs)
Q1. What will I learn from this simple interest article?
A: You will learn all 9 types of simple interest with clear concepts, easy explanations, and smart tricks to solve questions faster.
Q2. Is this article useful for beginners?
A: Yes, it is written in very simple English and explained step-by-step, so even beginners can understand easily.
Q3. Do I need to memorize formulas to understand this topic?
A: No, this article focuses more on concepts and logical thinking rather than memorizing formulas.
Q4. How can these simple interest tricks help in exams?
A: These tricks reduce calculation time and help you solve questions quickly and accurately.
Q5. What should I learn after completing this article?
A: After mastering simple interest, you should learn compound interest, which is the next important topic.
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