An investor invests 23,000 dollars into two bonds where one pays four percent in simple interest and the other two percent. If the investor earns 710 dollars in annual interest, how much is invested in each account?
We recall that in order to calculate simple interest, we multiply 𝑃 by 𝑅 by 𝑇. 𝑃 is the principal amount or amount invested, 𝑅 is the rate of interest written as a decimal, and 𝑇 is the time. In this question, we are told that there are two bonds which we will call bond 𝐴 and bond 𝐵. We know that the total investment was 23,000 dollars and we need to calculate the amount invested in each account. If we let the amount invested in bond 𝐴 be 𝑥 dollars, then the amount invested in bond 𝐵 will be 23,000 minus 𝑥. These are our values of 𝑃.
The first bond pays four percent in simple interest; therefore, 𝑅 is equal to 0.04. To convert from a percentage to decimal, we divide by 100. The interest rate for bond 𝐵 is two percent. Therefore, 𝑅 is equal to 0.02. In both cases, we’re looking at the annual interest. Therefore, 𝑇 is equal to one. We calculate the amount of interest in each account by multiplying these three values. 𝑥 multiplied by 0.04 multiplied by one is 0.04𝑥. For bond 𝐵, we need to multiply 23,000 minus 𝑥 by 0.02 and then multiply this answer by one. Distributing the parentheses or expanding the brackets gives us 460 minus 0.02𝑥. When we multiply this by one, it remains the same. Therefore, the amount of interest in bond 𝐵 is 460 minus 0.02𝑥.
We are told that the total interest is 710 dollars. This means that 0.04𝑥 plus 460 minus 0.02𝑥 is equal to 710. Collecting or grouping our like terms on the left-hand side gives us 0.02𝑥. Subtracting 460 from both sides means that this is equal to 250. Dividing both sides of this equation by 0.02 gives us 𝑥 is equal to 12,500. The amount invested in bond 𝐴 was 12,500 dollars. Subtracting this from 23,000 gives us 10,500. Therefore, the amount invested in bond 𝐵 was 10,500 dollars.
We could calculate the amount of interest from each bond by multiplying these values by 0.04 and 0.02, respectively. 12,500 multiplied by 0.04 is 500 and 10,500 multiplied by 0.02 is 210. The investor earned 500 dollars from bond 𝐴 and 210 dollars from bond 𝐵, giving him a total interest of 710 dollars.