Christopher takes out a $10,000 bank loan, from which the bank deducts the interest in advance. if the interest rate is 8% and the loan is for one year, how much is the principal (or face value) of this loan?

Suppose that the principal value is P and that the interest is I. We have then that P+I=10000 holds. Also, I=8%*P since the interest is 8% on the principal value of the loan; if it were on the total, it would have been as if we paid interest for money we never got. Hence, we have that P+8%P=10000. Thus, 108%*P=10000. Solving for P, P=10000*100/108=9.259$ (approximating to the nearest dollar). (If we just took 8%*10000=800, we would only get 9200$; these 59$ are the error of "paying interest on interest").