Monica receives an 80,000 inheritance. She invests part of...

Monica receives an $\$ 80,000$ inheritance. She invests part of it in CDs (certificates of deposit) earning 6.7$\%$ APY (annual percentage yield), part in bonds earning 9.3$\%$ APY, and the remainder in a growth fund earning 15.6$\%$ APY. She invests three times as much in the growth fund as in the other two combined. How much does she have in each investment if she receives $\$ 10,843$ interest the first year?

Key Concepts

System of Linear Equations

This concept involves creating and solving multiple equations simultaneously to determine unknown values. In finance and many application problems, relationships between various quantities are expressed as equations, which are then solved using techniques such as substitution or elimination.

Mixture Problems in Finance

Mixture problems require combining different components—such as investments or ingredients—with varying properties to meet a set of requirements. In financial contexts, this involves allocating funds among various investment vehicles with distinct interest rates, while also satisfying constraints on the total amount or other conditions.

Percentage and Interest Calculations

This concept centers on converting percentage rates, like APYs, into decimal form to calculate interest earnings. Understanding how to manipulate percentages is essential for determining the actual dollar amount of interest earned from various investments over a given time period.

Problem Modeling with Variables

Defining variables to represent unknown quantities is a key step in translating word problems into algebraic expressions. In investment and allocation problems, assigning variables to the amounts invested in different options allows the construction of equations that model the relationships and constraints presented.

Transcript

00:03 Okay, in this problem, monica has received an inheritance, and she has decided to invest it in different ways.

00:10 She's going to invest part of it in cds, part of it in bonds, and part of it in growth funds, and we have their respective percentages of annual yield, annual percentage yield.

00:21 And i've chosen to use variables c, b, and d for c, d, b, b, and g for growth funds.

00:28 And so what we want to do to find the values of c, b, and g is to write three equations, with those three unknowns.

00:36 So one of the things we notice in the problem is that the total amount of money invested is $80 ,000.

00:42 So we could call that the total investment equation.

00:53 So she invests c dollars in the cds plus b dollars in the bonds plus g dollars in the growth funds for a total of $80 ,000.

01:04 Now we need a second equation.

01:07 So we know that her interest earned is $10 ,843.

01:12 And we know the way to calculate interest in one year's time is just to take the interest rate times the amount invested.

01:20 And so we'll call that the interest earned equation.

01:29 So for the first type of investment, the cds, she earns 6 .7 % interest.

01:35 So we convert that to a decimal .067 and multiply it by the amount invested c.

01:42 And then we add to that the interest from the second investment, which earned 9 .3%.

01:47 So we make that 0 .093 multiplied by the amount invested b.

01:52 And do the same for the growth fund, 0 .156, multiplied by the amount invested g.

01:59 And that will give us the total amount of interest earned $10 ,843.

02:07 So at this point, we have two equations, and we need one more...

Please give Ace some feedback