Jason invests $2000 at a certain annual interest rate and $1000
at an annual interest rate 1/2 of a percent lower. If he receive a
total of $190 interest, at what rate did he invest the $1000?


Let the annual interest rate for the $2000 investment be denoted as $ r $ (in decimal form). Then, the annual interest rate for the $1000 investment is $ r - 0.005 $, as it is half a percent lower than $ r $.

The total interest earned is the sum of the interest from both investments, and we know that the total interest is $190. The interest from an investment can be calculated using the formula:

\[
\text{Interest} = \text{Principal} \times \text{Rate}
\]

For the $2000 investment:
\[
\text{Interest from $2000} = 2000 \times r
\]

For the $1000 investment:
\[
\text{Interest from $1000} = 1000 \times (r - 0.005)
\]

The total interest is given as:
\[
2000 \times r + 1000 \times (r - 0.005) = 190
\]

Now, solve the equation:

\[
2000r + 1000(r - 0.005) = 190
\]

Expanding the terms:
\[
2000r + 1000r - 5 = 190
\]

Combine like terms:
\[
3000r - 5 = 190
\]

Add 5 to both sides:
\[
3000r = 195
\]

Solve for $ r $:
\[
r = \frac{195}{3000} = 0.065
\]

So, the annual interest rate for the $2000 investment is 6.5%. The rate for the $1000 investment, which is 0.5% lower, is:
\[
r - 0.005 = 0.065 - 0.005 = 0.06
\]

Thus, Jason invested the $1000 at a rate of **6%**.

Would you like further clarification or more details?

Here are 5 related questions for practice:
1. What would the total interest be if Jason invested $3000 at 6%?
2. How much interest would Jason earn on a $500 investment at 6% annually?
3. How would the total interest change if the second rate were 1% lower instead of 0.5% lower?
4. If Jason invested $1500 at 7%, how much interest would he earn in one year?
5. What is the relationship between the principal, rate, and time in calculating interest?

**Tip:** When solving word problems involving percentages, converting percentages to decimals (e.g., 6% to 0.06) helps simplify calculations.

Jason invests $2000 at a certain annual interest rate and $1000 at an annual interest rate 1/2 of a percent lower. If he receives a total of $190 interest, at what rate did he invest the $1000?

Learn how to calculate the interest rates for Jason's investments of $2000 and $1000. This problem involves solving linear equations and applying the simple interest formula. Suitable for students in Grades 9-11.

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