Brennen B. answered • 02/08/24
Experienced High School Tutor, Math and Music
It helps to set this up as a full equation. To do this, we can set up the initial investments with their returns as a function of "y", like so:
y = 8000(1.06) + (1.03)x
Why 1.06 and 1.03, instead of 0.06 and 0.03? The added 1 in both expressions accounts for the initial investments of $8000 and $x. Instead of multiplying 8000 by .06 and getting 480, we multiply by 1.06 and see the initial investment plus the return, getting 8480.
Now this is all well and good, but we still have a problem: we have no idea how to solve for x unless we take literally every value under the sun and guess and check it. I don't know about you, but that seems like it would take way too long.
Fortunately, there's a better way. We have to ask ourselves: what does y represent? Well, y represents the total amount of money after both of Lamar's initial investments gave an average return of 3%. The key here is in the wording. A total means we're adding something. Adding what? The amount of Lamar's initial investments, which we can show mathematically like so: (8000 + x). Now, if this gave an average return of 3%, the total amount of money with investments and returns can be represented as (8000 + x)(1.03).
Now that we have a better way to write y, let's rewrite our equation:
(8000 + x)(1.03) = 8480 + (1.01)x
First, let's multiply out the left side. 8240 + (1.03)x = 8480 + (1.01)x
Now, let's get numbers on one side, variables on the other. (1.03)x - (1.01)x = 8480 - 8240.
Subtract. (0.02)x = 240
Divide both sides by 0.02. x = 240/(0.02) = 12000
And that's Lamar's second initial investment, $12000.
And if we check: $12000 with a 1% yield gives 12120. 20000 (both investments together) with a 3% yield gives 20600, which is 8480 + 12120.
Hope this helped!
