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Phase 1: Owning the Dataset

Understanding the 15/12 Discount Loan

Let's use a concrete example to understand this loan structure. If we borrow \(\mathrm{n = 1000}\) rupees:

Visual Representation:

Loan Amount (n) = 1000 rupees
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Interest (15%) = 150 rupees (deducted upfront)
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Borrower Receives = 1000 - 150 = 850 rupees
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After 12 months → Borrower Repays = 1000 rupees

Key insight: In a discount loan, interest is deducted upfront from what the borrower receives, but they must repay the full loan amount.

Phase 2: Understanding the Question

We need to find expressions for:

1. Repayment amount: The total amount paid back to fully repay the loan
2. Loan proceeds: The actual amount the borrower receives

From the problem:

• Loan amount = n rupees
• Interest rate = \(\mathrm{15\%}\) (discounted, meaning paid upfront)
• Interest amount = \(\mathrm{15\% \text{ of } n = 0.15n}\)
• Loan proceeds = loan amount - interest = \(\mathrm{n - 0.15n}\)

Phase 3: Finding the Answer

Calculating Loan Proceeds

Loan proceeds = \(\mathrm{n - 0.15n = n(1 - 0.15) = 0.85n}\)

Determining Repayment Amount

In a discount loan, the borrower must repay the full loan amount:
Repayment amount = \(\mathrm{n = 1.00n}\)

Let's verify with our example:

• Loan amount: 1000 rupees
• Interest (\(\mathrm{15\%}\)): 150 rupees
• Borrower receives: 850 rupees (which is \(\mathrm{0.85 \times 1000}\))
• Borrower repays: 1000 rupees (which is \(\mathrm{1.00 \times 1000}\)) ✓

Phase 4: Solution

For a 15/12 discount loan with amount n rupees:

• Repayment amount = 1.00n
• Loan proceeds = 0.85n

These selections satisfy the loan terms where the borrower receives the loan amount minus \(\mathrm{15\%}\) interest upfront, but must repay the full loan amount after 12 months.