Phyllis invested $10,500, a portion earning a simple interest rate of 8 1/4% per year and the rest earning a rate of 8% per year.

Question

Phyllis invested $10,500, a portion earning a simple interest rate of 8 1/4% per year and the rest earning a rate of 8% per year. After one year the total interest earned on these investments was $856.25. How much money did she invest at each rate?

at 8 1/4% per year = in dollars

at 8% per year = in dollars

Please provide an in-depth answer

Answer 1

The simple interest formula is as follows:

A = P + Prt = P(1 + rt), where

A = the total Accrued Amount (principal + interest)

P = Principal Amount, or the amount invested

r = Rate of Interest per year in decimal

t = Time Period involved in months or years

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The interest portion of the formula is represented by Prt.

Let's convert the word problem into a formula with both rates:

I = P₁r₁t + P₂r₂t, where

P₁ = the portion Phyllis invested in the simple interest rate of 8 1/4% per year

r₁ = the simple interest rate of 8 1/4% per year

P₂ = the portion Phyllis invested in the simple interest rate of 8% per year

r₂ = the simple interest rate of 8% per year

I = the total interest Phyllis made on the investments, $856.25

We also know that P₁ + P₂ = P = $10,500 (the total amount she invested initially) and t = 1 year.

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We have 2 equations now. Let's plug in the values

Eqn 1. P₁ + P₂ = $10,500

Eqn 2. P₁(0.0825)(1) + P₂(0.08)(1) = $856.25

One way to solve this is to solve for P₁ in terms of P₂.

Using Eqn 1: P₁ = 10500 - P₂

Plug this into Eqn 2 and solve for P₂.

(10500 - P₂)(0.0825) + P₂(0.08) = 856.25

866.25 - 0.0825P₂ + 0.08P₂ = 856.25

P₂(-0.0825 + 0.08) = 856.25 - 866.25

-0.0025P₂ = -10

P₂ = $4000

We can plug this value back into Eqn 1 to get P₁.

P₁ + 4000 = 10,500

P₁ = $6500

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So, Phyllis invested $6500 at 8 1/4% per year and $4000 at 8% per year.