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Chapter 2: Problem 58

You invested \(\$ 20,000\) in two accounts paying \(7 \%\) and \(9 \%\) annual interest. If the total interest earned for the year is \(\$ 1550,\) how much was invested at each rate? (Section P.8, Example 5 )

Short Answer

Expert verified
The amount invested at 7% is $12500 and the amount invested at 9% is $7500.

Step by step solution

01

Define the Variables

Let 'x' be the amount invested at 7%, and 'y' be the amount invested at 9%.
02

Create the System of Equations

From the problem, we can create the following two equations: \n1) \(x + y = 20000\) (total amount invested is $20000) \n2) \(0.07x + 0.09y = 1550\) (total interest earned is $1550)
03

Solve the System of Equations

We can start by solving the first equation for one of the variables, for example: \(y = 20000 - x\). \nThen, substitute this solution into the second equation: \(0.07x + 0.09(20000-x) = 1550\). \nThis equation simplifies to \(0.07x + 1800 - 0.09x = 1550\), and further down to \(0.02x = 250\). \nDividing both sides by 0.02, we get \(x = 12500\). \nSubstitute \(x = 12500\) into the first equation to solve for y: \(12500 + y = 20000\) which gives \(y = 7500\).
04

Verify the Solution

Substitute \(x = 12500\) and \(y = 7500\) into both original equations to make sure they still hold true. Both equations are satisfied, so our solution is correct.

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