# Amanda borrowed some money at 4.36% per annum, compounded quarterly. Two years later, she paid \$7,000 toward the principal and interest. After another 4 years, she paid the remainder, which totaled \$9,000. How much money did she originally borrow?

Amanda initially borrowed \$11,750.27.

Let the amount initially borrowed by Amanda be P. The rate of interest at which this is borrowed is 4.36% per annum, compounded quarterly.

The present value of an amount P after t terms, borrowed at a rate of interest r, is given by `PV = P/(1+r)^t`

Here, as compounding is done quarterly, the rate of interest is 1.09% per term, and the number of terms after t years is 4*t.

Amanda pays \$7,000 after 2 years. The present value of this is

`PV_1 = 7,000/(1+0.019)^(2*4)`

After another 4 years, the remainder is \$9,000. The present value of this is

`PV_1 = 9,000/(1+0.019)^(6*4)`

The sum of these is as follows:

`PV_1 + PV_2 `

`= 7,000/(1+0.019)^(2*4)+ 9,000/(1+0.019)^(6*4)`

`= 7,000/(1+0.019)^8+ 9,000/(1+0.019)^24`

`= 11,750.27`.

The amount borrowed initially was \$11,750.27.

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