252. Suppose that A was engaged to pay B £350 at the end of four years from this time, and that it is agreed between them that the debt shall be paid immediately; suppose, also, that money can be employed at 5 per cent, simple interest; it is plain that A ought not to pay the whole sum, £350, because, if he did, he would lose 4 years’ interest of the money, and B would gain it. It is fair, therefore, that he should only pay to B as much as will, _with interest_, amount in four years to £350, that is (251), £291. 13. 4. Therefore, £58. 6. 8 must be struck off the debt in consideration of its being paid before the time. This is called DISCOUNT;[55] and £291. 13. 4 is called the _present value_ of £350 due four years hence, discount being at 5 per cent. The rule for finding the present value of a sum of money (251) is: Multiply the sum by 100, and divide the product by 100 increased by the product of the rate per cent and number of years. If the time that the debt has yet to run be expressed in years and months, or months only, the months must be reduced to the equivalent fraction of a year. [55] This rule is obsolete in business. When a bill, for instance, of £100 having a year to run, is _discounted_ (as people now say) at 5 per cent, this means that 5 per cent of £100, or £5, is struck off. EXERCISES. What is the discount on a bill of £138. 14. 4, due 2 years hence, discount being at 4½ per cent? _Answer_, £11. 9. 1. What is the present value of £1031. 17, due 6 months hence, interest being at 3 per cent? _Answer_, £1016. 12. 253. If we multiply by _a_ + _b_, or by _a_-_b_, when we should multiply by _a_, the result is wrong by the fraction _b_ _b_ --- + _b_, or ---------, _a_ _a_ - _b_ of itself: being too great in the first case, and too small in the second. Again, if we divide by _a_ + _b_, where we should have divided by _a_, the result is too small by the fraction _b_/_a_ of itself; while, if we divide by _a_-_b_ instead of _a_, the result is too great by the same fraction of itself. Thus, if we divide by 20 instead of 17, the result is ³/₁₇ of itself too small; and if we divide by 360 instead of 365, the result is too great by ⁵/₃₆₅, or ¹/₇₃ of itself. If, then, we wish to find the interest of a sum of money for a portion of a year, and have not the assistance of tables, it will be found convenient to suppose the year to contain only 360 days, in which case its 73d part (the 72d part will generally do) must be subtracted from the result, to make the alteration of 360 into 365. The number 360 has so large a number of divisors, that the rule of Practice (230) may always be readily applied. Thus, it is required to find the portion which belongs to 274 days, the yearly interest being £18. 9. 10, or 18·491.