Oscar invests 20,000 in three investments earning 6% APY,...

Oscar invests $\$ 20,000$ in three investments earning 6$\%$ APY, 8$\%$ APY, and 10$\%$ APY. He invests $\$ 9000$ more in the 10$\%$ investment than in the 6$\%$ investment. How much does he have invested at each rate if he receives $\$ 1780$ interest the first year?

Key Concepts

Interest Calculation with Percentages

This concept pertains to calculating the amount of interest earned on an investment using percentage rates. Understanding how to convert percentage rates into decimal form and apply them to different principal amounts is crucial for solving interest-related problems.

Substitution Method

The substitution method is a technique used to solve a system of equations by expressing one variable in terms of another and replacing it in the other equations. This method simplifies the process of finding the values of the unknowns in investment problems where one amount is defined in relation to another.

Algebraic Modeling

This concept involves translating real-world problems into mathematical expressions or equations. In investment problems, variables are used to represent unknown amounts, and the relationships between these amounts, such as differences or totals, are expressed as equations.

System of Linear Equations

A system of linear equations is used when there are multiple unknowns that must satisfy several conditions simultaneously. In investment scenarios, the total invested amount and the conditions on the interest earned or the amounts invested at various rates are represented by such a system.

Transcript

00:03 In this problem, we have oscar investing money in different kinds of investments.

00:08 There's one that earns 6%, another that earns 8%, and another that earns a 10 % annual percentage yield.

00:15 And we know his total amount invested and the interest rate he receives in one year.

00:20 So we can use all of that information to set up a system of three equations and three unknowns and solve for the amount invested in each account.

00:28 So i chose to use variables x, y, and z for the amounts invested in the respective accounts.

00:33 And now one of the things we know is the total investment is $20 ,000.

00:39 So i'm going to write a total investment equation.

00:48 So he invests x dollars in one account, y dollars in the second account, and z dollars in the third account for a total of $20 ,000.

00:59 So that's our first equation.

01:02 Now we also have information about the interest earned.

01:05 So i'm going to make an interest equation and we know that to calculate simple interest for one year, we just multiply the interest rate as a decimal by the amount of money invested.

01:15 So in the first account, the interest rate would be 0 .06, multiply that by x, the amount invested.

01:22 In the second account, the interest rate is 0 .08.

01:25 We multiply that by y, the amount invested.

01:27 And in the third account, the interest rate is 0 .1.

01:30 And we multiply that by z, the amount invested.

01:33 And the total amount of interest is $1 ,180.

01:38 Now for our third equation, we look back through the information in the problem.

01:42 And it says that he invests 9 ,000 more in the 10 % investment than in the 6 % investment.

01:58 Okay, so we can write an equation for that.

02:00 The 10 % investment was z, and he invests 9 ,000 more than in the 6%.

02:07 So z equals x plus 9 ,000.

02:11 And we can just call that our other equation.