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**Hint:**Here we will simply use the formula of the simple interest. We will put the values in the formula and solve it to get the time which will be required to get the given interest. Simple interest is the interest incurred on a certain principal amount at a certain rate over a fixed period of time.

**Formula used:**

Simple interest \[ = \dfrac{{PRT}}{{100}}\] where, \[P\] is the principal amount, \[R\] is the rate of interest and \[T\] is the time period.

**Complete step by step solution:**

Given principal amount is \[\ $9000\], annual rate of interest is \[8\%\] and interest is \[\$180\].

Now we will simply use the formula of the simple interest and put all the given values in the equation and solve it to get the value of the time period. Therefore, we get

Simple interest \[ = \dfrac{{PRT}}{{100}}\]

Now we will put all the given values i.e. interest, principal amount and annual rate of interest in the above equation. Therefore, we get

\[ \Rightarrow 180 = \dfrac{{9000 \times 8 \times T}}{{100}}\]

Now we will solve this equation to get the value of the time period. Therefore, we get

\[ \Rightarrow 180 = 90 \times 8 \times T\]

\[ \Rightarrow T = \dfrac{{180}}{{90 \times 8}}\]

Now by simply dividing we will get the value of the time period. Therefore, we get

\[ \Rightarrow T = \dfrac{1}{4} = 0.25\]

**So the time period is equal to 0.25 which is almost equal to 3 months.**

Hence, 3 months long it will take a $ \$9000$ investment to earn $ \$180$ interest at an annual interest rate of 8%.

Hence, 3 months long it will take a $ \$9000$ investment to earn $ \$180$ interest at an annual interest rate of 8%.

**Note:**

Here we should note that while calculating the interest we should take the rate of interest in percentage in the formula of the interest. In simple interest, the interest amount per year remains constant over the period of time and in case of the compound interest, interest per year varies and it goes on increasing over the period of time.

Formula of the compound interest, compound interest \[ = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}\] where, P is the principal amount, R is the rate of interest and T is the time period.

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