Investments A man invests his savings in two accounts, one...

Investments A man invests his savings in two accounts, one paying 6$\%$ and the other paying 10$\%$ simple interest per year. He puts twice as much in the lower-yielding account be- cause it is less risky. His annual interest is $\$ 3520 .$ How much did he invest at each rate?

Key Concepts

Simple Interest

Simple interest is calculated by multiplying the principal amount by the rate and the time period. It provides a straightforward, linear relationship that is useful in determining how much interest an investment will generate annually.

Algebraic Modeling

Algebraic modeling involves representing real-world situations, such as investment distributions, using variables and equations. This approach enables one to translate word problems into solvable mathematical problems.

Systems of Linear Equations

Systems of linear equations arise when there are multiple relationships between unknown quantities, such as different interest rates and a total interest amount. These systems are solved simultaneously to find the values that satisfy all equations.

Ratio Reasoning

Ratio reasoning helps interpret proportional relationships between different quantities. In the context of investments, it is used to determine the specific amounts allocated based on a given ratio, like investing twice as much in one account as in another.

Transcript

00:01 Hello, hope you're doing well.

00:03 So for this problem, we're going to let x be equal to the amount of money invested in the lower interest account, and y equal to the amount of money invested in the higher interest, higher paying account.

00:16 So we know that the lower twice as much as invested in this lower interest account as the higher ones.

00:23 So that means x is going to be equal to 2y.

00:25 We also know that the interest rate for this lower paying account is 6%.

00:32 And this interest rate for the higher paying account is 10%.

00:35 And the total amount of interest made is $3 ,520.

00:43 So this means that 0 .06 times the amount of money invested in the lower paying account, plus 0 .1 times the amount of money invested in the higher paying account is equal to 35, and this right here is our system of equations that we want to solve.

01:03 So we are going to solve this using the substitution method.

01:06 So we've got this first equation already solved for x.

01:09 We're going to take that, i'm sorry, i'm going to take that and plug it into x into our second equation here.

01:14 So in doing that, we're going to write, rewrite the second equation, 0 .06 times instead of x, it's 2y, plus 0 .1y, is equal to 3520.

01:27 0 .06 times 2y is 0 .12x times 2y is 0 .12 y is equal to 3520.

01:36 0 .12 y plus 0 .1 .1 y is 0 .22 y and it's equal to 3520.

01:43 And then we can divide both sides of our side's equation by 0 .22, we end up with y is equal to 16 ,000...