A trader owes a merchant ₹9,810 due in 1 year, but the trader wants to settle the account after 6 months. If the rate of simple interest is 9% per annum, how much cash (in ₹) should he pay?

Correct — B. The ₹9,810 is the amount due at the end of one year, not the sum borrowed, so discount it back before doing anything else.

Present worth: PW × (1 + 9 × 1⁄100) = 9,810
PW = 9,810 × 100⁄109 = ₹9,000

He settles at 6 months, so ₹9,000 has run for half a year:
interest = 9,000 × 9⁄100 × 1⁄2 = ₹405

Cash to pay = 9,000 + 405 = ₹9,405 → option (b)

Why the others are wrong

• (a)
  9,550 — Nothing in the data yields ₹9,550. Half a year of simple interest on the present worth of ₹9,000 is fixed at ₹405, so the settlement is ₹9,405.
• (c)
  9,450 — ₹9,450 is 9,000 + 450, and 450 is 5% of ₹9,000. The rate is 9% per annum, so six months costs 4.5% — that is ₹405.
• (d)
  9,540 — ₹9,540 adds ₹540 to the present worth, which is a full year at 6%, a rate this question never gives. Six months at 9% p.a. is ₹405.

This is present worth under simple interest, dressed as a settlement story.

A debt of ₹9,810 payable in a year, at 9% p.a., is worth less today. The sum that grows into it is PW = Amount × 100 ⁄ (100 + R×T) = 9,810 × 100⁄109 = ₹9,000, and the ₹810 gap is the true discount for the year.

Settling at 6 months means the trader has had the ₹9,000 for half a year, so he owes it back with half a year's interest — ₹405.

Discount first, then regrow for the time actually used.

Key facts

• Present worth = Amount × 100 ÷ (100 + R × T) under simple interest.
• Here PW = 9,810 × 100 ÷ 109 = ₹9,000, so the true discount for the full year is ₹810.
• Interest for six months at 9% per annum is 4.5% of the principal, which is ₹405 on ₹9,000.

Common traps

• Deducting six months' interest from ₹9,810 instead of discounting it to present worth first.
• Applying 9% for the half year rather than 4.5%.
• Treating ₹9,810 as the principal when it is the amount due at the end of the year.

No directly related past PYQ was found.