Multiple choice Suppose you invested 5000 in three...

Multiple choice Suppose you invested $\$ 5000$ in three different funds for one year. The funds paid simple interest of $8 \%, 10 \%$ , and 7$\%$ , respectively. The total interest at the end of one year was $\$ 405 .$ You invested $\$ 500$ more at 10$\%$ than at 8$\%$ . How much did you invest in the 10$\%$ fund? $\begin{array}{ll}{\text { A } \$ 150} & {\text { B } \$ 1000}\end{array}$ C $\$ 1500 \quad$ D $\$ 2500$

Multiple choice Suppose you invested $\$ 5000$ in three different funds for one
year. The funds paid simple interest of $8 \%, 10 \%$ , and 7$\%$ , respectively.
The total interest at the end of one year was $\$ 405 .$ You invested $\$ 500$ more
at 10$\%$ than at 8$\%$ . How much did you invest in the 10$\%$ fund?
$\begin{array}{ll}{\text { A } \$ 150} & {\text { B } \$ 1000}\end{array}$
C $\$ 1500 \quad$ D $\$ 2500$

Key Concepts

Algebraic Modeling

Algebraic modeling involves translating real-world situations or scenarios into mathematical equations by assigning variables to unknown quantities and forming equations based on the relationships described. This process is crucial in solving problems that involve multiple constraints and relationships, such as different interest rates and investment amounts.

Simple Interest

Simple interest is a method of calculating interest where the interest earned is a fixed percentage of the principal amount. It does not take into account any compounding effect, meaning that interest is always calculated solely on the original principal over the specified time period.

System of Linear Equations

A system of linear equations consists of multiple equations used to represent relationships between several unknowns. In financial problems like this one, where multiple relationships (total investment, interest earned, and differences in investment amounts) are given, setting up and solving a system of linear equations allows you to determine the value of each unknown variable.

Transcript

00:01 So in this problem, you've invested $5 ,000 in three different funds.

00:05 Okay, so you have one fund that has a 10 % interest.

00:09 We had another fund that has an 8 % interest, and we have a third fund that has 7 % interest.

00:16 So we know that if i was to take my interest, so i'm going to change my interest into decimals.

00:23 So that's 0 .1.

00:26 This is 0 .08, and then this would be 0 .07.

00:32 So x, y, and z are how much i invested, okay? so if i take all three of those x plus y plus z, i know i invested $5 ,000.

00:46 Then it tells me that after one year, my total interest was $405.

00:51 So if i take my interest rates multiplied by how much i invested for each account, it should give me my total interest for that year, which is $405.

01:05 And lastly, it tells me that i invested $500 more at 10 % than at 8%.

01:13 So i know that whatever i invested at $100 is going to be $500 more than what i invested at 8%.

01:21 I have my theory equations now.

01:24 So what i first want to do is to give under these decimals.

01:26 I'm going to multiply this whole equation by 100.

01:29 So that equation is now going to be 10x plus 8y plus 7.

01:36 7 z equals 40 ,500.

01:47 Okay, so i'm gonna do a little substitution.

01:51 So i know x equals 500 plus y.

01:56 So i'm gonna replace that in both of my equations.

01:59 So this first equation is going to be 500 plus y plus y plus z equals 5 ,000.

02:09 So that equation is going to be 2y 2y plus z, subtract the 500 for both sides, 4 ,500...