Type 5: When Principal is Divided into Parts
What This Type is About
In this type of simple interest question, the total principal is divided into two or more parts and invested at different rates of interest. The total interest earned from all parts is usually given, and you need to find how the money was distributed.
These questions are commonly solved using the alligation method, which is faster than forming lengthy equations.
Main Concept
When money is divided into different parts, each part earns interest according to its own rate.
Therefore:
Total Interest = Interest from Part 1 + Interest from Part 2 + …
To find how the money is distributed, we first determine the average rate of interest and then apply the alligation method.
Average Rate Concept
The average rate represents the combined effect of all investments.
Formula
Average Rate = (Total Interest ÷ Total Principal) × 100
Once the average rate is known, it can be compared with the given rates to find the ratio of investment.
Alligation Method
Follow these steps:
Calculate the average rate.
Compare it with the given rates.
Find the cross differences.
Form the ratio.
Divide the principal according to that ratio.
This method is quick, accurate, and widely used in competitive exams.
Solved Example
₹10,000 is invested at 5% and 10%. The total interest earned in one year is ₹700. Find the amount invested at each rate.
Step 1: Find the Average Rate
Average Rate
= (700 ÷ 10000) × 100
= 7%
Step 2: Apply Alligation
Rates: 5% and 10%
Average Rate: 7%
Cross differences:
10 − 7 = 3
7 − 5 = 2
Therefore,
Amount at 5% : Amount at 10% = 3 : 2
Step 3: Find the Actual Amounts
Total ratio parts = 3 + 2 = 5
Value of one part
= 10000 ÷ 5
= ₹2000
Therefore:
Amount at 5% = 3 × 2000 = ₹6000
Amount at 10% = 2 × 2000 = ₹4000
Answer: ₹6000 at 5% and ₹4000 at 10%
Another Example
₹15,600 is invested at 7% and 9%. The total annual interest is ₹1200. Find the distribution of money.
Step 1: Find the Average Rate
Average Rate
= (1200 ÷ 15600) × 100
≈ 7.69%
Step 2: Apply Alligation
Cross differences:
9 − 7.69 = 1.31
7.69 − 7 = 0.69
Ratio
= 1.31 : 0.69
After simplifying the ratio, divide ₹15,600 accordingly to obtain the required amounts.
Why Alligation Works
The average rate must always lie between the given rates. The cross differences show how much each rate contributes to achieving that average.
This allows us to find the ratio directly without creating algebraic equations, making the solution faster and simpler.
Common Mistakes to Avoid
Calculating the average rate incorrectly.
Taking the cross differences in the wrong order.
Mixing up the ratio obtained from alligation.
Using lengthy equations when a simple ratio method can solve the problem.
Forgetting to check whether the average rate lies between the given rates.
Exam Strategy
When you see a question involving different rates and a total interest value:
Calculate the average rate immediately.
Apply the alligation method.
Find the ratio of investment.
Divide the principal according to that ratio.
This approach saves time and reduces calculation errors.
Key Takeaways
Total interest is the sum of interest earned from all parts.
The average rate is the starting point for solving these questions.
Alligation is usually the fastest method.
Cross differences provide the ratio of investment.
Once the ratio is known, the principal can be divided easily.
Type 6: Increase or Decrease in Rate/Time
What This Type is About
In this type of simple interest question, the rate of interest or the time period changes, causing the simple interest to increase or decrease. Instead of calculating the complete interest in both situations, we focus only on the change in interest.
These questions are common in competitive exams and can often be solved quickly using the difference between the two cases.
Main Concept
When the principal remains the same, any change in simple interest is caused by a change in the rate, the time, or both.
Therefore, it is usually unnecessary to calculate the full simple interest separately. Working directly with the difference saves time and simplifies the problem.
Golden Rule
The most important formula for this type is:
Difference in SI = (Difference in Rate × Principal × Time) ÷ 100
This formula helps find the change in simple interest when the rate changes.
Change in Rate
When the rate increases or decreases, first find the difference between the two rates and then apply the Golden Rule.
Example
The rate decreases from 18% to 15%, and the loss in simple interest for one year is ₹750. Find the principal.
Step 1: Find the rate difference
18% − 15% = 3%
Step 2: Apply percentage logic
3% of Principal = ₹750
Principal = (750 × 100) ÷ 3
Principal = ₹25,000
Answer: ₹25,000
Change in Rate for Multiple Years
When the change continues for more than one year, multiply the rate difference by the number of years.
Example
The rate decreases from 12% to 10%, and the loss in simple interest for 2 years is ₹400. Find the principal.
Step 1: Find the rate difference
12% − 10% = 2%
Step 2: Include time
2% × 2 years = 4%
Step 3: Find the principal
4% of Principal = ₹400
Principal = (400 × 100) ÷ 4
Principal = ₹10,000
Answer: ₹10,000
Increase in Rate
The same method applies when the rate increases.
Example
The rate increases from 5% to 8% for 3 years, and the gain in simple interest is ₹450. Find the principal.
Step 1: Find the rate difference
8% − 5% = 3%
Step 2: Include time
3% × 3 years = 9%
Step 3: Find the principal
9% of Principal = ₹450
Principal = (450 × 100) ÷ 9
Principal = ₹5000
Answer: ₹5000
Change in Time
Sometimes the rate remains constant, but the time period changes. In such cases, the difference in simple interest comes from the additional years.
Example
Simple interest for 2 years is ₹200, and simple interest for 3 years is ₹300. Find the yearly interest.
Step 1: Find the difference in interest
₹300 − ₹200 = ₹100
Step 2: Find the difference in time
3 − 2 = 1 year
Step 3: Calculate yearly interest
Yearly Interest = ₹100 ÷ 1 = ₹100
Answer: ₹100 per year
Shortcut Method
Most questions can be solved using these steps:
Find the difference in rate or time.
Multiply by time if required.
Treat the result as a percentage of the principal.
Find the required value using basic percentage calculations.
This method is faster than calculating complete simple interest in both cases.
Common Mistakes to Avoid
Calculating the full simple interest for both situations unnecessarily.
Forgetting to multiply the rate difference by time.
Using the original rate instead of the rate difference.
Making errors while converting percentages into values.
Avoiding these mistakes can save time and improve accuracy.
Exam Strategy
When you see words such as increase, decrease, gain, or loss, immediately focus on the difference rather than the complete values.
Find the rate difference or time difference first and then apply the Golden Rule. This approach is usually the fastest way to solve these questions.
Key Takeaways
Focus on the change in simple interest rather than the complete interest values.
The Golden Rule is the foundation of this type.
Always include time when the change continues for multiple years.
Gain and loss follow the same method.
Working with differences is faster and more efficient than calculating full simple interest.
Type 7: Successive Change in Rate
What This Type is About
In this type of simple interest question, the rate of interest changes during different time periods. Instead of earning interest at a single rate throughout the investment period, the money earns interest at different rates for different durations.
Since simple interest is always calculated on the original principal, the principal remains the same even when the rate changes.
Main Concept
To solve these questions:
Divide the total time into separate periods.
Calculate the interest for each period using the same principal.
Add all the interests to get the total simple interest.
The key point is that only the rate changes; the principal remains constant.
Why the Principal Remains Constant
In simple interest, interest is always calculated on the original principal.
Unlike compound interest, the interest earned in one period is not added to the principal. Therefore, even if the rate changes several times, every calculation uses the same principal amount.
Concept Method
Follow these steps:
Step 1
Break the total time into different periods.
Step 2
Calculate the interest for each period separately.
Step 3
Add all the interests to obtain the total simple interest.
This method is straightforward and works for all questions of this type.
Solved Example
₹10,000 is invested at:
5% for 2 years
10% for 3 years
Find the total simple interest.
First Period
5% × 2 years = 10%
Interest = 10% of ₹10,000 = ₹1,000
Second Period
10% × 3 years = 30%
Interest = 30% of ₹10,000 = ₹3,000
Total Interest
₹1,000 + ₹3,000 = ₹4,000
Answer: ₹4,000
Shortcut Method
Instead of calculating each interest separately, calculate the total percentage first.
Example
5 × 2 = 10%
10 × 3 = 30%
Total Percentage = 10% + 30% = 40%
Interest = 40% of ₹10,000
= ₹4,000
This method is faster and very useful in competitive exams.
Another Example
₹5,000 is invested at:
8% for 1 year
12% for 2 years
Find the total simple interest.
Calculate Total Percentage
8 × 1 = 8%
12 × 2 = 24%
Total Percentage = 32%
Find Interest
32% of ₹5,000
= ₹1,600
Answer: ₹1,600
Finding Principal
Sometimes the total interest is given, and the principal is unknown.
Example
The total simple interest is ₹3,000.
The money earns:
10% for 2 years
20% for 3 years
Find the principal.
Step 1: Find Total Percentage
10 × 2 = 20%
20 × 3 = 60%
Total Percentage = 80%
Step 2: Find Principal
80% of Principal = ₹3,000
Principal = (3000 × 100) ÷ 80
Principal = ₹3,750
Answer: ₹3,750
Common Mistakes to Avoid
Adding rates directly without considering time.
Forgetting to multiply the rate by the number of years.
Assuming the principal changes after each period.
Trying to solve the entire question in one step without separating the periods.
Avoiding these mistakes makes the solution much easier.
Exam Strategy
When you see different rates for different periods:
Separate the time periods immediately.
Multiply each rate by its corresponding time.
Add the percentages.
Apply the total percentage to the principal.
This approach is usually the fastest way to solve these questions.
Key Takeaways
The rate changes, but the principal remains constant.
Total simple interest is the sum of interest from all periods.
The shortcut method uses total percentage directly.
Always multiply rate by time before combining percentages.
Breaking the question into parts makes it easier to solve.
Type 8: When Simple Interests Are Equal
What This Type is About
In this type of simple interest question, two different investments earn the same simple interest. The principal, rate, or time may be different, but the final interest remains equal.
The objective is usually to find an unknown principal, rate, or time by comparing the two investments.
Main Concept
When the simple interests of two investments are equal, we can directly equate their principal-rate-time products.
Formula
P₁ × R₁ × T₁ = P₂ × R₂ × T₂
Since the denominator 100 appears on both sides of the simple interest formula, it cancels automatically.
This makes calculations faster and simpler.
Why This Formula Works
We know that:
Simple Interest = (P × R × T) ÷ 100
If two simple interests are equal, then:
(P₁ × R₁ × T₁) ÷ 100 = (P₂ × R₂ × T₂) ÷ 100
Cancelling 100 from both sides gives:
P₁ × R₁ × T₁ = P₂ × R₂ × T₂
This is the foundation of all questions in this type.
Concept Method
Follow these steps:
Step 1
Write the equation:
P₁ × R₁ × T₁ = P₂ × R₂ × T₂
Step 2
Cancel common values whenever possible.
Step 3
Solve for the unknown value.
This method avoids unnecessary calculations and makes the solution easier.
Finding Time
Example
Simple interest on ₹5,000 at 10% for 2 years equals simple interest on ₹10,000 at 5% for T years.
5000 × 10 × 2 = 10000 × 5 × T
Cancelling common factors:
20 = 10T
T = 2 years
Answer: 2 years
Finding Principal
Example
Simple interest on ₹X at 10% for 3 years equals simple interest on ₹6,000 at 5% for 6 years.
X × 10 × 3 = 6000 × 5 × 6
X × 30 = 6000 × 30
X = ₹6000
Answer: ₹6000
Finding Rate
Example
Simple interest on ₹4,000 at R% for 2 years equals simple interest on ₹2,000 at 10% for 4 years.
4000 × R × 2 = 2000 × 10 × 4
8000R = 80000
R = 10%
Answer: 10%
Shortcut Method
For most questions:
Write only P × R × T on both sides.
Ignore the denominator 100 because it cancels.
Cancel common factors before multiplying.
Solve the remaining equation.
This approach saves time and reduces calculation errors.
Common Mistakes to Avoid
Writing the complete simple interest formula on both sides unnecessarily.
Forgetting that the denominator 100 cancels.
Not cancelling common values before calculation.
Mixing up principal, rate, and time while forming the equation.
Making multiplication errors due to large numbers.
Avoiding these mistakes helps solve questions faster and more accurately.
Exam Strategy
When you see two investments earning the same simple interest:
Immediately write the P × R × T equation.
Cancel common values wherever possible.
Simplify before multiplying.
Solve the remaining equation directly.
This method is usually the quickest way to solve such questions.
Key Takeaways
Equal simple interests imply equal P × R × T products.
The denominator 100 cancels automatically.
Most questions can be solved using a simple comparison equation.
Early cancellation reduces calculations significantly.
This type is based on logical comparison rather than lengthy formulas.
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