A sum of ₹5000 is lent out in two parts, one at 6% simple interest and the other at 9% simple interest. If the annual interest is ₹390, the sum lent at 6% is:

Correct — A. Let the part at 6% be ₹x, so ₹(5000 − x) earns 9%.

Interest: 0.06x + 0.09(5000 − x) = 390
Expand: 0.06x + 450 − 0.09x = 390
Collect: 0.03x = 450 − 390 = 60
Divide by 0.03: x = ₹2000 → option (a)

Check: 6% of 2000 = 120 and 9% of 3000 = 270, total ₹390

Why the others are wrong

• (b)
  ₹2500 — ₹2500 at each rate earns 150 + 225 = ₹375, short of ₹390. Equal halves average 7.5%, and the actual average is 7.8%.
• (c)
  ₹3000 — ₹3000 is the part at 9%, not at 6%. Putting ₹3000 at 6% and ₹2000 at 9% earns 180 + 180 = ₹360.
• (d)
  ₹3500 — ₹3500 at 6% leaves ₹1500 at 9%: 210 + 135 = ₹345, well below ₹390. More money at the lower rate means less interest.

When one sum is split between two rates, the total interest fixes the average rate: 390 ÷ 5000 = 7.8%.

Alligation then gives the split: (9 − 7.8) : (7.8 − 6) = 1.2 : 1.8 = 2 : 3. The part at 6% takes 2 of the 5 parts, 2⁄5 × 5000 = ₹2000.

Key facts

• Simple interest for one year = principal × rate ⁄ 100.
• Average rate = total interest ÷ total principal × 100, per year.
• Alligation: part at the lower rate : part at the higher rate = (higher − average) : (average − lower).

Common traps

• Solving for the 9% part and marking ₹3000, when the question asks for the sum lent at 6%.
• Turning the alligation ratio round. The larger part goes to the rate nearer the average, here 9%.

No directly related past PYQ was found.