On simple interest, a certain sum becomes ₹59,200 in 6 years and ₹72,000 in 10 years. If the rate of interest had been 2% more, then in how many years would the sum have become ₹76,000?

The correct answer is 9 years.

We are given that on simple interest, a certain sum amounts to ₹59,200 in 6 years and ₹72,000 in 10 years. We need to find the original principal and rate of interest first.

Step 1: Find the Simple Interest for 4 years

The difference in the two amounts gives us the simple interest earned over the difference in time (10 - 6 = 4 years):

SI for 4 years = ₹72,000 - ₹59,200 = ₹12,800

Step 2: Find the Simple Interest per year

SI per year = 12,800 ÷ 4 = ₹3,200

Step 3: Find the Principal

SI for 6 years = 3,200 × 6 = ₹19,200

Principal (P) = Amount in 6 years - SI for 6 years

P = 59,200 - 19,200 = ₹40,000

Step 4: Find the original Rate of Interest

Using the SI formula:

R=\frac{SI\times 100}{P\times T}=\frac{\mathrm{19,200}\times 100}{\mathrm{40,000}\times 6}=\frac{\mathrm{19,20,000}}{\mathrm{2,40,000}}=8\%

So the original rate of interest = 8% per annum

Step 5: Apply the new rate of interest

New rate = 8% + 2% = 10% per annum

Step 6: Find the time for the sum to become ₹76,000 at 10%

SI required = ₹76,000 - ₹40,000 = ₹36,000

Using the SI formula to find time (T):

T=\frac{SI\times 100}{P\times R}=\frac{\mathrm{36,000}\times 100}{\mathrm{40,000}\times 10}=\frac{\mathrm{36,00,000}}{\mathrm{4,00,000}}=9\text{years}

Therefore, at the new rate of 10%, the sum of ₹40,000 would become ₹76,000 in 9 years.