Divide Rs.6000 into two parts so that the simple interest on the first part for 9 months at 12% annum is equal to the simple interest on the second part for $1\dfrac{1}{2}$ years at 10% per annum.

Hint: To solve this question, we have to divide the given amount, that is Rs.6000 into two parts such that one is ‘x’ and the other part is (6000 – x). We have to use the equation to find the simple interest, that is,
Simple interest = $\dfrac{PRT}{100}$
where ‘P’ is initial value, ‘R’ is rate of interest and ‘T’ is time in years.
It is given that the simple interest of both the parts are equal so we have to find it using the equation and solve it to find the value ‘x’ and ’6000 - x’.

Complete step-by-step answer:
Let us assume that one part is x and the other part is (6000 - x).
Now, we have to find the simple interest of both the parts. So,
Simple interest of part one = PRT
where ‘P’ is initial value, ‘R’ is rate of interest and ‘T’ is time in years.
So, we know that for part one the initial value is x, rate of interest = $12$ and the time is $\dfrac{9}{12}$ years.
So, simple interest = $\dfrac{PRT}{100}$
$\Rightarrow x\times \dfrac{12}{100}\times \dfrac{9}{12}$
$\Rightarrow \dfrac{9x}{100}$
Now, we have to find the simple interest of the second part, that is,
Simple interest of second part = $\dfrac{PRT}{100}$
So, here the initial value is (6000 - x), rate of interest = $10$ and the time is $=1\dfrac{1}{2}=\dfrac{3}{2}$ years.
So, simple interest = PRT
$\Rightarrow (6000-x)\times \dfrac{10}{100}\times \dfrac{3}{2}$
$\Rightarrow 900-\dfrac{3x}{20}$
Now, it is given that the simple interest of one part is equal to the other part. So, we can write it as,
Simple interest of part one = Simple interest of second part
$\dfrac{9x}{100}=900-\dfrac{3x}{20}$
Now, take the x terms to one side,
$\dfrac{9x}{100}+\dfrac{3x}{20}=900$
Now, we have to take the L.C.M,
$\dfrac{9x}{100}+\dfrac{15x}{100}=900$
$\Rightarrow \dfrac{24x}{100}=900$
$\begin{align}
  & \Rightarrow 24x=900\times 100 \\
 & \Rightarrow x=\dfrac{90000}{24} \\
 & \Rightarrow x=3750 \\
\end{align}$
So, (6000 - x) = 6000 – 3750 = 2250
Therefore, one part is Rs.3570 and the other part is Rs.2250.

Note: The equation for simple interest and compound interest are different so do not get confused between that. The time is always taken in terms of years, so we should always convert it into years if it is given in months else the answer will be wrong.