Question Video: Solving a System of Linear Equations Involving Percentages

You invested $2300 in account 1, and $2700 in account 2, which both pay simple interest annually. If the total amount of interest after one year is $254, and account 2 has 1.5 times the interest of account 1, what are the interest rates?

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Video Transcript

You invested 2300 dollars in account one and 2700 dollars in account two, which both pay simple interest annually. If the total amount of interest after one year is 254 dollars and account two has 1.5 times the interest of account one, what are the interest rates?

In order to answer this question, we recall that in order to calculate the simple interest, we multiply 𝑃 by 𝑅 by 𝑇. 𝑃 is the principal amount or amount invested, 𝑅 is the interest rate written as a decimal, and 𝑇 is the time. In account one, the principal amount is 2300 dollars. We will let the interest rate for this account be 𝑥. As we are dealing with one year, the time is equal to one.

For account two, 𝑃 is equal to 2700 dollars. The time is also equal to one. As the interest rate is 1.5 times the interest of account one, 𝑅 is equal to 1.5𝑥. The amount of interest for account one is therefore equal to 2300 multiplied by 𝑥 multiplied by one. This is equal to 2300𝑥. For account two, we need to multiply 2700 by 1.5𝑥 by one. This is equal to 4050𝑥.

We are also told in the question that the total amount of interest is 245 dollars. This means that 2300𝑥 plus 4050𝑥 is equal to 254. The left-hand side simplifies to 6350𝑥. Dividing both sides of this equation by 6350 gives us 𝑥 is equal to 254 over 6350. This fraction simplifies to one over 25. Therefore, 𝑥 is equal to 0.04 as a decimal.

To convert from a decimal to a percentage, we multiply by 100. This means that the interest rate for account one is four percent. The interest rate for account two was 1.5 times this, and four multiplied by 1.5 is six. This means that the interest rate for account two is six percent. Four percent simple interest on 2300 dollars and six percent simple interest on 2700 dollars gives a total of 254 dollars of interest after one year.

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