Determine how much is borrowed at each given rate

Question

A city zoo borrows $2,000,000 at simple annual interest to construct a breeding facility. Some of the money is borrowed at 9%, some at 10%, and some at 13%. Use a system of linear equations to determine how much is borrowed at each rate given that the total annual interest is $204,500 and the amount borrowed at 9% is twice the amount borrowed at 13%. Solve the system of linear equations using matrices

Answer 1

Good question!

So we will denote P for principal and use 9, 10, and 13 as subscripts on P denoting amounts borrowed at that interest rate.

So

P = $2,000,000 = P₉ + P₁₀ + P₁₃

Given from problem statement: P₉ = 2 * P₁₃

Simple interest: I = Prt

Since we are just given an annual interest, t = 1 year

I₉ = 0.09*P₉

I₁₀ = 0.1*P₁₀

I₁₃ = 0.13*P₁₃

Given from problem statement: I₉ + I₁₀ + I₁₃ = $204,500

I will solve here using substitution and will post matrices in a comment below if the formatting allows.

First, substitute interest formulas into sum of interests.

0.09*P₉ + 0.1*P₁₀ + 0.13*P₁₃ = $204,500

Then substitute given, P₉ = 2 * P₁₃

0.09*2*P₁₃ + 0.1*P₁₀ + 0.13*P₁₃ = $204,500

0.18*P₁₃ + 0.1*P₁₀ + 0.13*P₁₃ = $204,500

0.1*P₁₀ + 0.31*P₁₃ = $204,500

From earlier, P = $2,000,000 = P₉ + P₁₀ + P₁₃

Given from problem statement: P₉ = 2 * P₁₃

So, P₁₀ + 3*P₁₃ = $2,000,000

P₁₀ = $2,000,000 - 3*P₁₃

Substitute this into our last equation, 0.1*P₁₀ + 0.31*P₁₃ = $204,500

0.1*($2,000,000 - 3*P₁₃) + 0.31*P₁₃ = $204,500

$200,000 - 0.3*P₁₃ + 0.31*P₁₃ = $204,500

0.01*P₁₃ = $4,500

P₁₃ = $450,000

Now, substitute into earlier equations.

P₁₀ = $2,000,000 - 3*P₁₃ = $2,000,000 - 3*$450,000 = $2,000,000 - $1,350,000 = $650,000 = P₁₀

P₉ = 2 * P₁₃ = 2*$450,000 = $900,000 = P₉

Now if you add all the principals, it adds up to the full principle of $2 million. Yay, we solved it!

Again, I'll look into how easy it is to do matrix notation in this format in a comment below. But I believe the substitution will still be useful for most.

Update: I don't see any way here to show matrix notation, please follow an online guide or book a session with myself or someone else who is good with matrices.