Question Video: Solving a System of Linear Equations Involving Percentages

Video Transcript

At the beginning of a year, you invested 10,000 dollars into two accounts, A and B, which receive eight percent simple interest and five percent simple interest, respectively. At the end of that year, you had 10,710 dollars in your combined accounts. How much was invested in each account?

In order to answer this question, we firstly need to consider the simple interest formula. This states that 𝐼 is equal to 𝑃 multiplied by 𝑅 multiplied by 𝑇. 𝐼 is the amount of interest earned. 𝑃 is the principal amount or the amount invested. 𝑅 is the rate of interest given as a decimal. And 𝑇 is the time period. In this question, we’re given information about account A and account B. We’re trying to calculate the amount that was invested in each account. Let’s let the amount invested in account A be 𝑥 dollars. As 10,000 dollars was invested altogether, the amount invested in account B is 10,000 minus 𝑥.

We’re told that the interest rate for account A was eight percent. For account B, this was five percent. To convert from a percentage to a decimal, we divide by 100. So, 𝑅 is equal to 0.08 and 0.05, respectively. As we’re only dealing with one year, the value for 𝑇 in both cases is one. Multiplying our three values for account A gives us 0.08𝑥. Therefore, the interest earned in account A is 0.08𝑥 dollars.

Repeating this process for account B gives us 𝐼 is equal to 0.05 multiplied by 10,000 minus 𝑥. Distributing the parentheses by multiplying 0.05 by 10,000 and then by negative 𝑥 gives us 500 minus 0.05𝑥. At the end of the year, we were told we have 10,710 dollars in the combined accounts. The total interest earned can, therefore, be calculated by subtracting 10,000 from 10,710. This is equal to 710 dollars. We will now clear some space to solve the equations.

We know that the total interest from account A and account B is equal to 710. Writing this as an equation, we have 0.08𝑥 plus 500 minus 0.05𝑥 is equal to 710. Grouping or collecting like terms on the left-hand side gives us 0.03𝑥 plus 500. Our next step is to subtract 500 from both sides of the equation. This gives us 0.03𝑥 is equal to 210. Finally, we divide both sides by 0.03. This gives us a value of 𝑥 equal to 7,000. This was the amount invested in account A, 7000 dollars. As the total amount invested was 10,000 dollars, then the amount invested in account B was 3,000 dollars.

Investing these amounts in two accounts with simple interest rates of eight percent and five percent for one year would accrue 710 dollars of interest. 560 dollars of these would come from account A as 0.08 multiplied by 7,000 is 560. This means that 150 dollars would come from account B as 710 minus 560 is 150.