$876.37 from April 5 to June 11, 67 da., = $9.79 = 3.77 = 0.00 2.45 313.37 from June 11 to July 28, 47 da., = 1.67 $145,38 from June 11 to Aug. 8, 58 da., = $1.41 Sum of losses, Excess of gain over loss,. $27.48 = .46 $ 3.54 23.94 As Mr. Pond gains this interest on money which he owes, he ought to pay the debt ($361.08, the balance of the account) as many days before June 11, 1852, as it will take for it to gain $23.94 interest. This gives for the equated time 398 days before June 11, 1852, which is May 10, 1851. The sum necessary to settle the account after the equated time will be the amount of the balance, $361.08, from the equated time to the time of settlement. 3. When was the balance of the following account due? 4. When was the balance of the following account due? Dr. George Black in account with John Brown. Cr. NOTE. - 4 mo., 3 mo., &c., means that goods were sold at so many months' credit. 5. What was due on the following account Jan. 1, 1853? 6. What was due on the following account Jan. 1, 1853, interest being 7 per cent, and 4 months' credit being allowed on each entry? Dr. David H. Daniels in account with George W. Dean. Cr. 195. To find the Principal, or Interest, from the Amount, Rate, and Time. (a.) When the amount, time, and rate are given to find the principal or interest, we find what part any principal, or (if the interest be required) its interest for the given time, at the given rate, is of its amount, and then take this part of the given amount. (b.) The first step towards this is to find the fraction expressing what part any interest for the given time, at the given rate, is of its principal. This fraction will always be the same part of the given annual rate that the given time is of 1 year, or 360 days; or, if the rate is 6 per cent, it will equal the fraction expressing the part which the given time is of 200 months, or 6000 days. The amount, of course, will equal the principal, plus the fractional part of it which the interest equals. = 19 219 (c.) Thus, interest for 1 yr. 7 mo., or 19 months, at 6 per cent per year, of the principal, and the amount = = 288 +20%: zig of the amount, of the principal. Hence, zoo of the principal 200 19 and the entire principal: = and the interest 29, of the amount or 19 mo. at 6 per cent. (The same fractions would have been obtained by considering the interest to be of Tʊ of the principal.) 4 100 1 800 (d) Again. Interest for 19 months at 4 per cent per year = of = 13 of 280 = 35% of the principal, and the amount = 888 857 of the principal. Hence, the principal 57, of the amount for 19 mo. at 4 per cent. = interest = 812 6000 203 = 1500 = 800 , and the (e.) Again. Interest for 2 yr. 3 mo. 2 da., or 812 days, at 6 per cent per year,= of the principal, and the amount = 1588 +150%=1783 of the principal. Hence, the principal = 18%, and the interest = 203 , of the amount for 2 yr. 3 mo. 2 da. at 6 per cent. (The same fractions would have been obtained by considering the interest to be of 18 of the principal.) = (f) Again. Interest for 5 mo. 14 da., or 164 days, at 7 per cent per year, of Toσ = # of Too of the principal, and the 9387 of the principal. Hence, the prin 9299, and the interest = 927, of the amount for 5 mo. 14. da. at 7 per cent. 1. What principal on interest at 6 per cent per year will amount to $884.125 in 1 yr. 2 mo. 10 da. ? Since, at 6 per cent per year, interest for 1 day = 5000 of the principal, interest for 1 yr. 2 mo. 10 da., or 430 days, must equal , or, of the principal, and the amount must equal 888 + , or 8, of the principal. Hence, o of the principal must equal , and the principal itself must equal 4 of the amount. 42 of $884.125 = $825 = principal required. The mere numerical work may be indicated thus: principal required. Second Solution. The amount of $1 for 1 yr. 2 mo. 10 da. = $1.07}, and if one dollar is required to gain $1.07, as many dollars will be required to gain $884.125 as there are times $1.07 in $884.125 But $884.125 ÷ $1.07 = $884.125 ÷ 643 = 9 of $884.125 = $825, as before. Proof. The amount of $825 for 1 yr. 2 mo. 10 da. is $884.125 2. What principal on interest at 7 per cent per year will amount to $703.551 in 4 mo. 27 da. ? Solution. = days, = 34 3 - At 7 per cent per year, interest for 4 mo. 27 da., or 147 147 of 100 = 12000 To of the principal, and the amount = 12888 +1268012343 of the principal. Hence the principal must equal 193 of the amount, which in this case is 129 of $703.551 $684. The mere numerical work may be indicated thus:4 mo. 27 da. = 147 da. of Too 12000 12000 13480 + 12888-13343. 2000. 129 of $703.551 = $684 = = 343 principal required. The amount of $1 at 7 per cent for 4 mo. 27 da. = = $1.02123; and if one dollar is required to gain $1.02123, as many dollars would be required to gain $703.551 as there are times $1.0223 in $703.551. But $703.551 ÷ $1.02128 = $703.551 ÷ 12348 = Proof. The amount of $684 for 4 mo. 27 da. at 7 per cent equals $703.551. What principal will amount to 3. $569.296 in 8 mo. 16 da. at 6 per cent? 4. $573.16 in 2 yr. 9 mo. 10 da. at 6 per cent? 5. $922.13 in 1 yr. 4 mo. at 8 per cent? 7. $57.72 in 8 mo. 20 da. at 10 per cent? 8. 196. Discount and Present Worth. (a.) DISCOUNT, as it is technically called, furnishes the most common application of the processes of the preceding article to the problems of business life. (b.) It is obvious that the true present value of a debt due at a future time is that sum of money which, put on interest at the present time, will amount to the given debt at the time it becomes due. Thus, when the rate of interest is 6 per cent per year, a debt of $106 due in one year is worth the same as a debt of $100 due now; for if the money received on the second debt be put on interest, it will amount to $106 in one year; that is, when the first debt becomes due. (c.) A debt due at a future time may be regarded, then, as the amount of a principal on interest from the present time to the time when the debt will become due. This principal is usually called the PRESENT WORTH of the debt, and its interest is called the DISCOUNT, because, if discounted or deducted from the debt, it leaves the present worth. (d.) From this, it follows that to ask what is the present worth of $651, due in 6 mo. 20 da., money being worth 6 per cent per year, is equivalent to asking what principal on interest at 6 per cent will amount to $651 in 6 mo. 20 da.; and that the solution of the first question is the same as that of the second. To test the correctness of any result, see if the amount of the present worth equals the given debt. 1. What is the present worth of $438.18 due in 1 yr. mo. at 6 per cent per year ? 6 Partial Solution. The present worth required is that sum of money which, put on interest at 6 per cent, will amount to $438.18 in 1 yr. 6 mo., or 18 mo. The interest for 18 mo. = pal, and the amount, &c., as in the last article. What is the present worth of T80 of the princi 2. $83.45 due in 4 yr. 2 mo. at 6 per cent? |