In the given questions, two quantities are provided, Quantity I and Quantity II. Compare the two quantities and select the correct option.

**Quantity I:** Rs. 1800 is invested such that Rs. *P* is invested at 10% per annum simple interest, and the remaining Rs. (1800 – *P*) is invested at 20% per annum compound interest. The difference between the total simple interest and total compound interest after 2 years is Rs. 80, with compound interest being greater than simple interest. Find the value of *P*.

**Quantity II:** The marked price of a product is 25% more than its selling price, and the profit earned on selling the product is also 25%. The difference between the profit and the difference between the cost price and the marked price is Rs. 200. Find the cost price of the product.

Choose the correct relationship:

Question

In the given questions, two quantities are provided, Quantity I and Quantity II. Compare the two quantities and select the correct option.

**Quantity I:** Rs. 1800 is invested such that Rs. *P* is invested at 10% per annum simple interest, and the remaining Rs. (1800 – *P*) is invested at 20% per annum compound interest. The difference between the total simple interest and total compound interest after 2 years is Rs. 80, with compound interest being greater than simple interest. Find the value of *P*.

**Quantity II:** The marked price of a product is 25% more than its selling price, and the profit earned on selling the product is also 25%. The difference between the profit and the difference between the cost price and the marked price is Rs. 200. Find the cost price of the product.

Choose the correct relationship:

Accepted Answer

**Solution:**

**Step 1: Solve Quantity I**

- Let Rs. *P* be invested at 10% p.a. simple interest.
- Then, Rs. (1800 - *P*) is invested at 20% p.a. compound interest.

Calculate simple interest (SI) for Rs. *P* over 2 years:

$$SI = \frac{P 	imes 10 	imes 2}{100} = \frac{20P}{100} = 0.2P$$

Calculate compound interest (CI) for Rs. (1800 - *P*) over 2 years at 20% p.a.:

Amount after 2 years:

$$A = (1800 - P) 	imes \left(1 + \frac{20}{100}ight)^2 = (1800 - P) 	imes (1.2)^2 = (1800 - P) 	imes 1.44$$

CI = Amount - Principal =

$$1.44(1800 - P) - (1800 - P) = (1800 - P)(1.44 - 1) = 0.44(1800 - P)$$

Given difference between CI and SI is Rs. 80, with CI > SI:

$$0.44(1800 - P) - 0.2P = 80$$

Simplify:

$$0.44 	imes 1800 - 0.44P - 0.2P = 80$$

$$792 - 0.64P = 80$$

$$0.64P = 792 - 80 = 712$$

$$P = \frac{712}{0.64} = 1112.5$$

So, **Quantity I = Rs. 1112.5**

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**Step 2: Solve Quantity II**

- Let the selling price (SP) be Rs. *S*.
- Marked price (MP) is 25% more than SP:

$$MP = S + 0.25S = 1.25S$$

- Profit is 25%, so cost price (CP) is:

$$CP = \frac{SP}{1 + 0.25} = \frac{S}{1.25} = 0.8S$$

- Profit = SP - CP =

$$S - 0.8S = 0.2S$$

- Difference between profit and difference between CP and MP is Rs. 200:

$$	ext{Profit} - (CP - MP) = 200$$

Substitute values:

$$0.2S - (0.8S - 1.25S) = 200$$

Simplify inside parentheses:

$$0.2S - (-0.45S) = 200$$

$$0.2S + 0.45S = 200$$

$$0.65S = 200$$

$$S = \frac{200}{0.65} = 307.69$$

Calculate CP:

$$CP = 0.8S = 0.8 	imes 307.69 = 246.15$$

So, **Quantity II = Rs. 246.15**

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**Step 3: Compare Quantities**

- Quantity I = Rs. 1112.5
- Quantity II = Rs. 246.15

Clearly, Quantity I > Quantity II.

**Answer:** (a) Quantity I > Quantity II

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