To calculate the remaining balance on loan, you can use the formula 𝑅𝐵 equals 𝑂𝐵 times one plus 𝑟 to the 𝑛 power minus 𝑃 times one plus 𝑟 to the 𝑛 power minus one over 𝑟, where 𝑅𝐵 is the remaining balance, 𝑂𝐵 is the original balance, 𝑃 is the payment, and 𝑟 is the rate of payment and 𝑛 is the number of payments. Which of the following gives a formula for 𝑃 in terms of 𝑅𝐵, 𝑂𝐵, 𝑟, and 𝑛?
To get 𝑃 in terms of 𝑅𝐵, 𝑂𝐵, 𝑟, and 𝑛, we’ll need to isolate the 𝑃 variable. Before we do that, we could notice that a lot is happening in this formula, particularly with these rates. To make our work easier and to help insure we don’t make small sign mistakes or drop numbers, we could let 𝑥 be equal to one plus 𝑟 to the 𝑛 power and let 𝑦 be equal to one plus 𝑟 to the 𝑛 power minus one over 𝑟. This allows us to write this equation as 𝑅𝐵 equals 𝑂𝐵 times 𝑥 minus 𝑃 times 𝑦. After we isolate 𝑃, we can plug back in the values for 𝑥 and 𝑦. Because 𝑃 is negative, we can add 𝑃𝑦 to both sides. 𝑃 becomes positive. Now, we have 𝑃𝑦 plus 𝑅𝐵 equals 𝑂𝐵𝑥. And so, we subtract 𝑅𝐵 from both sides. And we’ll have 𝑃𝑦 equals 𝑂𝐵𝑥 minus 𝑅𝐵. To get 𝑃 by itself, we need to divide both sides by 𝑦. And so, we can say that 𝑃 equals 𝑂𝐵𝑥 minus 𝑅𝐵 divided by 𝑦.
At this point, we’ll have to plug back in what we know 𝑥 and 𝑦 are equal to. Instead of 𝑂𝐵𝑥, we’ll have 𝑂𝐵 times one plus 𝑟 to the 𝑛 power minus 𝑅𝐵 over one plus 𝑟 to the 𝑛 power minus one over 𝑟. This is pretty messy because we have a fraction in our denominator. If we write it out horizontally, we notice that we’re dividing by a fraction. And to simplify this, we need to multiply by that fraction’s reciprocal. In this case, we’ll now be multiplying by 𝑟 over one plus 𝑟 to the 𝑛 power minus one. We multiply 𝑟 times what’s already in the numerator. 𝑟 times 𝑂𝐵 one plus 𝑟 to the 𝑛 power minus 𝑅𝐵. And the denominator stays the same: one plus 𝑟 to the 𝑛 power minus one. This is 𝑃 in terms of 𝑅𝐵, 𝑂𝐵, 𝑟, and 𝑛.
If we look at our options, we see that A and D have 𝑟 as the denominator. So we know that they cannot be correct. Both 𝐵 and C have the correct denominators. But if we look closely, only one of them is subtracting 𝑅𝐵 from 𝑂𝐵 times one plus 𝑟 to the 𝑛 power. And that’s option C. Option C has correctly stated 𝑃 in terms of 𝑅𝐵, 𝑂𝐵, 𝑟, and 𝑛.