CAT Difference between SI and CI for two years and three years MCQ

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5 Questions around this concept.

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If the difference between compound interest at 8% p.a. and simple interest at 13/2% p.a. on a certain sum of money for 2 years is Rs. 1820, then find the sum.

Rs. 50000

Rs. 40000

Rs. 32000

Rs. 25000

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Concepts Covered - 1

Difference between SI and CI for two years and three years

One of the notable differences between SI and CI arises when we look at multiple years, especially for two and three years. This is because while SI remains constant every year, CI grows as it compounds on the previous year's principal + interest.

Formula:

1. For two years:

- Simple Interest for 2 years:

$\left(2 \times \frac{P \times R}{100}\right)$
- Compound Interest for 2 years:

$\left(P \times \frac{R}{100} \times\left(2+\frac{R}{100}\right)\right)$

The difference between CI and SI for two years:

$\left(P \times\left(\frac{R}{100}\right)^2\right)$

2. For three years

- Simple Interest for 3 years:

$\left(3 \times \frac{P \times R}{100}\right)$
- Compound Interest for 3 years:
$\mathrm{ P \times \frac{R}{100} \times\left(3+3 \times \frac{R}{100}+\left(\frac{R}{100}\right)^2\right) }$
The difference between Cl and SI for three years:

$\mathrm{ P \times R^2 \times(300+R) \div 100^3 }$

Tip 1: The difference between SI and CI for two years is the interest on the first year's interest. For three years, it includes interest on interest for both the first and second years.

Foundation Building Questions:

Question: On a principal of ₹5,000, find the difference between compound interest and simple interest for 2 years at a rate of 10% per annum.

Solution:

Using the formula for the difference between CI and SI for two years:

Difference $\mathrm{ =P \times\left(\frac{R}{100}\right)^2 \\ }$

Difference $\mathrm{ =5,000 \times\left(\frac{10}{100}\right)^2 }$

Difference = ₹500

Tip 2: For problems that specifically ask for the difference between CI and SI, jump directly to the formula for the difference instead of calculating both CI and SI separately.

Application of Previous Concepts (Compound Interest):

The difference between CI and SI particularly highlights the power of compound interest. While the initial differences are small, over long periods, the effect becomes pronounced due to the compounding effect.

Solved Example:

Question: On a sum of ₹8,000, the difference in compound and simple interest for 3 years at a rate of 5% per annum is?

Solution:

Using the formula for the difference between CI and SI for three years:

Difference = $\mathrm{ P \times R^2 \times(300+R) \div 100^3 }$

Difference = $\mathrm{ 8,000 \times 5^2 \times(300+5) \div 100^3 }$

Difference = ₹820

Tip 3: When working with rates that aren't whole numbers, it's helpful to convert everything to decimals to ensure accurate calculations.

Tip 4: Always make sure you're using the right formula for the specified time frame. Mixing up two-year and three-year formulas can lead to significant errors.

Understanding the difference between simple and compound interest over multiple years reinforces the power of compounding. By recognizing the growing disparity between the two over time, students can better appreciate the potential impact of compound growth, whether it relates to finance, populations, or any other application. Regular practice with varied problems will solidify this understanding.