A bank loaned out ​$20,000​, part of it at the rate of 9 % per year and the rest at 19 % per year. If the interest received in one year totaled ​$2500​, how much was loaned at 9 % ?                                      How much of the ​$20,000 did the bank loan out at 9 % question mark

Let's define two variables to represent the amounts loaned at 9% and 19% interest.

Let:
- $ x $ be the amount loaned at 9%.
- $ y $ be the amount loaned at 19%.

According to the problem, we know:
1. The total loaned amount is $20,000. Hence, the equation is:
   \[
   x + y = 20,000
   \]
   
2. The total interest received after one year is $2,500. Hence, the equation for the interest is:
   \[
   0.09x + 0.19y = 2,500
   \]

Now, we have a system of two equations:
1. $ x + y = 20,000 $
2. $ 0.09x + 0.19y = 2,500 $

We will solve this system to find the values of $ x $ and $ y $.

### Solving the system:
From the first equation, express $ y $ in terms of $ x $:
\[
y = 20,000 - x
\]

Substitute this expression for $ y $ into the second equation:
\[
0.09x + 0.19(20,000 - x) = 2,500
\]

Now, we can solve for $ x $.The bank loaned out $13,000 at 9% interest and $7,000 at 19% interest.

Would you like more details on how this solution was derived or have any further questions?

Here are 5 related questions to explore further:
1. What is the formula for simple interest?
2. How do you calculate compound interest if it were applied instead?
3. If the bank increased the loan amount by $5,000, how would the interest be affected?
4. How does changing the interest rate impact the total interest earned?
5. What happens if the loan periods differ between the two parts of the loan?

**Tip:** Always double-check that the total interest matches expectations when splitting investments across different interest rates.

Learn how to solve a bank loan problem where $20,000 is loaned out at 9% and 19% interest rates, yielding $2,500 in interest. Discover the solution step-by-step with clear explanations and formulas. Suitable for high school students learning about linear equations and interest calculations.

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