1. How long will it take for your money to double at $$ 6.5 \% $$ simple interest? Solution:

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To determine how long it will take for your money to double at a simple interest rate of $$ 6.5\% $$, we can use the formula for simple interest:

$$
A = P(1 + rt)
$$

Where:
- $$ A $$ is the amount of money accumulated after time $$ t $$,
- $$ P $$ is the principal amount (the initial amount of money),
- $$ r $$ is the rate of interest (as a decimal),
- $$ t $$ is the time in years.

In this case, we want to find $$ t $$ when the amount $$ A $$ is double the principal $$ P $$. Therefore, we can set $$ A = 2P $$.

Substituting into the formula, we have:

$$
2P = P(1 + rt)
$$

We can divide both sides by $$ P $$ (assuming $$ P \neq 0 $$):

$$
2 = 1 + rt
$$

Now, we can isolate $$ rt $$:

$$
2 - 1 = rt
$$
$$
1 = rt
$$

Next, we substitute $$ r = 0.065 $$ (since $$ 6.5\% = 0.065 $$):

$$
1 = 0.065t
$$

Now, we can solve for $$ t $$:

$$
t = \frac{1}{0.065}
$$

Calculating this gives:

$$
t \approx 15.38
$$

Thus, it will take approximately **15.38 years** for your money to double at a simple interest rate of $$ 6.5\% $$.
It will take approximately 15.38 years for your money to double at a simple interest rate of 6.5%.

1. How long will it take for your money to double at \( 6.5 \% \) simple interest? Solution:

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