The interest accrued on a sum of ₹16,000 lent at a simple interest rate of 14% p.a. for the time in those many years, which gives a simple interest of three-fifths of a certain sum at 10% p.a., is:

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Let the time period be T years.

Simple Interest (SI) on ₹16,000 at 14% p.a. for T years = (16000 × 14 × T) / 100 = 2240T

This interest is equal to three-fifths of the simple interest on a certain sum at 10% p.a. for T years.

Let the certain sum be P.

Simple Interest on P at 10% for T years = (P × 10 × T) / 100 = (P × T) / 10

According to the problem:
2240T = (3/5) × (P × T / 10)

Cancel T from both sides (assuming T ≠ 0):
2240 = (3/5) × (P / 10)

Multiply both sides by 10:
22400 = (3/5) × P

Multiply both sides by 5/3:
P = (22400 × 5) / 3 = 112000 / 3 ≈ 37333.33

Now, we want to find the interest accrued on ₹16,000 at 14% for T years.

We need to find T first.

From the equation:
2240T = (3/5) × (P × T / 10)

We already canceled T, so T can be any positive number. But the problem states "for the time in those many years," implying T is the number of years.

Since T cancels out, the interest accrued is 2240T.

To find T, use the fact that the interest accrued on ₹16,000 at 14% for T years equals three-fifths of the interest on P at 10% for T years.

We can pick T such that the interest accrued is one of the options.

Alternatively, since the problem is about the interest accrued on ₹16,000 at 14% for T years, and the interest is 2240T, the interest must be one of the options.

Check each option:

Option 1: ₹13,440

2240T = 13440

T = 13440 / 2240 = 6 years

Check if this T satisfies the other condition:

Interest on P at 10% for 6 years = (37333.33 × 10 × 6) / 100 = 37333.33 × 0.6 = 22400

Three-fifths of this = (3/5) × 22400 = 13440

This matches the interest on ₹16,000 at 14% for 6 years.

Hence, the interest accrued is ₹13,440.

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