Elena has $18,500 to invest. She invests some of it at 8.4 % 8.4\% annual simple interest for 1 1 year, and the remainder at 11.6 % 11.6\% annual simple interest for 8 8 months. At the end of the year, her total interest earned is $1,500. How much did she invest at each rate?

In this problem, Elena has $$18,500 to invest. She invests some of it at 8.4% annual simple interest for one year, and the remainder at 11.6% annual simple interest for eight months. At the end of the year, her total interest earned is 1$$,500. So how much did she invest at each rate? So since this is a simple interest problem, we need our simple interest formula, which says that the interest earned is equal to the principal amount invested times the interest rate times the amount of time that has passed. So now, to better understand this problem, let's draw a picture. Let's say this rectangle represents the total amount of interest earned, which we know is $$1,500. And to create this total, we have two parts. Part of this money was invested at 8.4% interest, and the other part was invested at 11.6% interest. Another thing that is different about these two parts of our mixture is the length of time. Our 8.4% was invested for one year, whereas our 11.6 was invested for eight months, which we could also say is eight twelfths of a year. Now that we better understand what's going on, let's assign some variables. For our amount invested at 8.4%, we'll call that x, and for the amount invested at 11.6%, we'll call that y. Next, let's build our equation. In order to get our total of 1$$,500, we need to use our formula and multiply the principal times the rate times the time for each of our two quantities, and then sum them up. So for our first quantity, we're going to have our principal amount of x times its rate of 8.4% as a decimal of 0.084, times its amount of time, which is one year. We're gonna add to that our second amount, y, times its rate of 11.6% as a decimal is 0.116, times its amount of time, which is eight twelfths of the year. Now that we have our equation, we need to put everything in terms of one variable in order to solve it. So here, we have our two variables x and y, which each represent the amount of money invested at that rate. So it's another equation that we could write that relates these two quantities. Well, we know that the total amount of money invested was $$18,500. So we could say that 18,500 is equal to x plus y. And now, we can solve for one of these variables. Let's go ahead and solve for x by subtracting y from both sides. So we have 18,500 minus y is equal to x. And now, I can take this part of my equation and plug it in four x into my main equation. And doing that and simplifying, we get 1,500 is equal to 18,500 minus y, times 0.084, plus, y times point one one six times eight twelfths is 0.07733 repeating. Now that everything's in terms of one variable, we're ready to solve for y. First, let's simplify by distributing this in. We get 1,500 is equal to 18,500 times 0.084 is 1,554. That's gonna be minus y times 0.084, plus y times 0.07733 repeating. Now, we need to collect all variable terms to one side, and all other terms to the other side. So we're gonna subtract 1,554 from both sides, which leaves us with negative 54 on the left hand side, and combining these two quantities, we get y times negative 0.0066 repeating. Isolating y, we'll divide both sides by negative zero zero six six, negative 0.0066, and we get that y is equal to 8,000 1 100.08 after rounding. So remember, our y quantity represented the amount invested at 11.6%. So we can go ahead and state that part of our answer, that we invested 8$$,100.08 at 11.6%, And to find the amount invested at 8.4%, we simply plug our y value into our second equation to get x. So we'll have 18,500 minus 8108¢ is equal to x, which means x equals 10,399.92. So this is our amount invested at 8.4%. So we can state the second part of our answer as $10,399.92 invested at 8.4%. And that is our final answer.

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Multiple Choice

Elena has $18,500 to invest. She invests some of it at $8.4 percent sign$ annual simple interest for $1$ year, and the remainder at $11.6 percent sign$ annual simple interest for $8$ months. At the end of the year, her total interest earned is $1,500. How much did she invest at each rate?

\$11,000 at 8.4% and \$7,500 at 11.6%

\$7,500 at 8.4% and \$11,000 at 11.6%

\$10,399.92 at 8.4% and \$8100.08 at 11.6%

\$8100.08 at 8.4% and \$10,399.9 at 11.6%

Verified step by step guidance

Define variables to represent the amounts invested at each interest rate. Let $x$ be the amount invested at 8.4% and $y$ be the amount invested at 11.6%.

Write an equation for the total amount invested: $x + y = 18500$.

Convert the 8 months investment period into years for the 11.6% investment, since interest rates are annual. So, 8 months is $\frac{8}{12} = \frac{2}{3}$ of a year.

Write an equation for the total interest earned using the simple interest formula $I = P \times r \times t$, where $P$ is principal, $r$ is rate (as a decimal), and $t$ is time in years. The total interest is $1500$, so: $0.084x \times 1 + 0.116y \times \frac{2}{3} = 1500$.

Use the system of equations from steps 2 and 4 to solve for $x$ and $y$. Substitute $y = 18500 - x$ into the interest equation and solve for $x$, then find $y$.

4:10m

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Elena has \(18,500 to invest. She invests some of it at $8.4 percent sign$ annual simple interest for $1$ year, and the remainder at $11.6 percent sign$ annual simple interest for $8$ months. At the end of the year, her total interest earned is \)1,500. How much did she invest at each rate?