Divide ₹ 9,000 into two parts in such a way that S.I. on one part at 16% p.a. and in 2 years is equal to the S.I. on the other part at 6% p.a. and in 3 years.

Let the first part be ₹ $x$ and the second part be ₹ $(9,000 - x)$.

Hence,

P₁ = ₹ $x$

R₁ = 16%

T₁ = 2 years

$\text{S.I.} = \Big(\dfrac{P \times R \times T}{100}\Big)\\[1em] \Rightarrow \text{S.I.} = \Big(\dfrac{x \times 16 \times 2}{100}\Big)\\[1em] \Rightarrow \text{S.I.} = \dfrac{32x}{100}\\[1em] \Rightarrow \text{S.I.} = \dfrac{8x}{25}\\[1em]$

P₂ = ₹ $9,000 - x$

R₂ = $6\%$

T₂ = 3 years

$\text{S.I.} = \Big(\dfrac{P \times R \times T}{100}\Big)\\[1em] \Rightarrow \text{S.I.} = \Big[\dfrac{(9,000 - x) \times 6 \times 3}{100}\Big]\\[1em] \Rightarrow \text{S.I.} = \dfrac{18(9,000 - x)}{100}\\[1em] \Rightarrow \text{S.I.} = \dfrac{9(9,000 - x)}{50}\\[1em]$

Both interest are equal.

$\dfrac{8x}{25} = \dfrac{9(9,000 - x)}{50}\\[1em] \Rightarrow\dfrac{8x}{25} = \dfrac{81,000 - 9x}{50}\\[1em] \Rightarrow\dfrac{8x}{25} + \dfrac{9x}{50} = \dfrac{81,000}{50}\\[1em] \Rightarrow\dfrac{16x}{50} + \dfrac{9x}{50} = 1,620\\[1em] \Rightarrow\dfrac{(16x + 9x)}{50} = 1,620\\[1em] \Rightarrow\dfrac{25x}{50} = 1,620\\[1em] \Rightarrow x = \dfrac{1,620\times 50}{25}\\[1em] \Rightarrow x = \dfrac{81,000}{25}\\[1em] \Rightarrow x = 3,240$

Other amount will be ₹ (9,000 - 3,240) = ₹ 5,760.

Hence, ₹ 3,240 and ₹ 5,760 are the two parts.