One year ago, you invested 10,000 dollars into two accounts that pay simple interest annually: Account A, which pays three percent interest, and account B, which pays 2.5 percent interest. Given that, after one year, you earned 283 dollars 50 cents in interest, how much was invested into each account?
In order to answer this question, we recall that simple interest is equal to 𝑃 multiplied by 𝑅 multiplied by 𝑇, where 𝑃 is the principal amount or amount invested, 𝑅 is the rate of interest as a decimal, and 𝑇 is the time. We know that 10,000 dollars was invested altogether. However, we do not know how much was invested into each account. If we let the amount invested in account A equal 𝑥, then 𝑃 is equal to 𝑥. The rate of interest in account A was three percent; therefore, 𝑅 is equal to 0.03. To convert from a percentage to a decimal, we divide by 100. As we’re dealing with just one year, 𝑇 is equal to one.
The amount of simple interest in account A is therefore equal to 𝑥 multiplied by 0.03 multiplied by one, which is 0.03𝑥. As there was 10,000 dollars invested altogether, and we invested 𝑥 dollars in account A, the amount invested in account B will be 10,000 minus 𝑥. 𝑅 will be equal to 0.025 as 2.5 divided by 100 is 0.025. Once again, 𝑇 is equal to one. The total amount of interest in account B will be equal to 10,000 minus 𝑥 multiplied by 0.025 multiplied by one.
Distributing the parentheses or expanding the brackets gives us 250 minus 0.025𝑥. We now have the amount of interest in terms of 𝑥 for account A and account B. The total amount of interest earned was 283 dollars and 50 cents. This means that 0.03𝑥 plus 250 minus 0.025𝑥 must equal 283.50. Collecting or grouping our like terms gives us 0.005𝑥. And subtracting 250 from both sides gives us 33.5 on the right-hand side. Dividing both sides of this equation by 0.005 gives us 𝑥 is equal to 6,700.
This means that 6,700 dollars was invested in account A. Subtracting this from 10,000 gives us 3,300. So 3,300 dollars was invested in account B. We could calculate the interest earned from each account by multiplying these values by 0.03 and 0.025, respectively. 6,700 multiplied by 0.03 is 201. Therefore, 201 dollars of interest was earned from account A. 3,300 multiplied by 0.025 gives us 82.5; therefore, 82 dollars and 50 cents was earned from account B. These have a total of 283 dollars and 50 cents.