7. What is the amount of a quarterly salary of $225 in arrears for 4 years, allowing 1 per cent. interest per quarter, and compounding the interest quarterly? Ans. $4034.78+.

8. What is the amount of a semi-annual dividend of $500 in arrears for 4 years, allowing 3 per cent. interest for 6 months time, and compounding the interest semi-annually?

9. What is the amount of a biennial salary of $10000 in arrears for 8 years, allowing 12 per cent. interest for 2 years, compounding the interest biennially?

10. What is the amount of an annual rent of $300, in arrears for 19 years?

373. PROB. 3. To find the present worth of a certain annuity at compound interest,*

RULE 1.

*

- Find the present worth of each instalment, and the sum of these will be the present worth of the annuity; or, RULE 2. Find the amount of the annuity as though it were in arrears, and then discount this amount for the time to elapse before the last instalment becomes due.

NOTE. - These two rules will give the same result, but the 2d is the easier to apply.

*To find the present worth of a certain annuity, discounting at simple interest, some authors have given this rule: - Find the present worth of each instalment separately, and the sum of these will be the present worth of the annuity. Others find the amount of the annuity as though it were in arrears, and then discount this amonnt for the time to elapse before the last instalment is due.

These rules will give different results, but the difference is unimportant; for to purchase an annuity by either of these rules would be in the highest degree absurd, since the present worth of an annuity for about 25 years at 6 per cent. by the 2d rule, or 30 years by the 1st, would be so great that its annual interest would be more than the annual instalment of the annuity; e. g. the present worth of an annuity of $100 for 25 years, found by the 2d rule, is $1720. Now the loan of $1720 will entitle the lender to $103.20 interest annually, forever, and the principal would still be due; whereas the purchase of the annuity of $100 for 25 years by the payment of $1720, its present worth, will only secure the payment of $100 annually for 25 years, d neither instalment nor the refunding of purchase money subsequently.

Ex. 1. What is the present worth of $100 annuity, payable

annually for 4 years?

$100 × 1.063

present worth of 1st instalment. = present worth of 2d instalment. = present worth of 3d instalment. = present worth of 4th instalment. = present worth of annuity, Ans.

OPERATION BY RULE 2.

$119.1016 last term of series (360); ($119.1016-$100) (1.06-1)+ $119.1016=$437.4616 amount of $100 annuity in arrears for 4 years (363); $437.46161.064 $346.51708804, Ans. as before.

=

These operations may be much abridged by using the following TABLE,

Showing the present worth of the annuity of $1, £1, etc., at 4, 5, 6 and 7 per cent., for any number of years not exceeding 20.

Ex. 2. What is the present worth of an annuity of $60 per annum to continue 20 years, at 6 per cent. compound interest? The present worth of $1 by the table is $11.469921,

.. $11.469921 × 60

= $688.19526, Ans. 3. What is the present worth of an annuity of $175 per annum to continue 15 years, at 7 per cent. compound interest? Ans. $1593.884+.

4. What is the present worth of an annual pension of $150 for 12 years at 5 per cent.? Ans. $1329.4878.

5. A young man buys a farm for $2000, which he agrees to pay in 16 equal annual instalments, the first in 1 year from the time of purchase. Allowing 6 per cent., what ready money will the debt? Ans. $1263.236+.

pay

6. What is the present worth of a semi-annual salary of $500, to continue 8 years, allowing 4 per cent. interest for the time between two successive payments? Ans. $5826.145.

374. PROB. 4.—To find the present worth of a perpetuity.

The present worth of a perpetuity is, evidently, a sum whose interest for the interval between two successive payments is equal to one instalment; now, interest is found by multiplying the principal by the rate per cent.; .., conversely, the principal equals the interest divided by the rate per cent. Hence,

RULE.-Divide the instalment by the rate per cent. and the quotient will be the present worth of the perpetuity.

Ex. 1. What is the present worth of a perpetuity of $60 per annum at 6 per cent.? $60.06 $1000, Ans. 2. What is the value of a perpetuity of $1200 per annum at 6 per cent.? Ans. $20000. 3. What is the present worth of a perpetuity of $900 per annum at 3 per cent? Ans. $30000.

375. PROB. 5.-To find the present worth of an annuity certain, in reversion.

RULE 1.-Find the value of the annuity if entered on immediately and then discount that value for the time in reversion.

NOTE.-A like rule will give the value of a perpetuity in reversion.

RULE 2. Find the present worth of a like annuity for the time in reversion, also for the whole time from the present till the last instalment is due, and the difference of these will be the present worth of the annuity in reversion.

Ex. 1. What is the present worth of $500 annuity to commence in 3 years and continue 4 years, at 6 per cent.

Present worth for 3 years = $1336.506.

Difference $1454.684+, Ans.

2. What is the value of a perpetuity of $1000 to commence in 2 years, discounting at 6 per cent.? Ans. $14833.274. 3. What is the present worth of an annual pension of $300 to commence in 8 years and continue 12 years, discounting at 4 per cent.? Ans. $2057.273.

376. PROB. 6.-To find an annuity, its present worth being given,

RULE.-Divide the given present worth by the present worth of $1 annuity for the given rate and time.

Ex. 1. The present worth of an annuity for 3 years is $500; what is the annuity?

$500 2.673012= $187.055 —, Ans.

2. The present worth of an annual rent for 10 years is $6000; what is the rent, discounting at 5 per cent.? Ans. $777.027.

377. PROB. 7.—To find an annuity, its amount being given, RULE.-Divide the given amount by the amount of $1 annuity for the given rate and time.

Ex. 1. The amount of an annuity for 4 years is $600; what is the annuity, discounting at 6 per cent.?

378. When several things are placed in a line in every possible order of succession, so that each shall enter every result, and enter it but once, they are said to be permuted, and each order of succession is called a permutation; thus, the single letter, a, can have but 1 position, i. e., it cannot stand either before or after itself; the 2 letters, a and b, furnish the 2 permutations, ab the number of which is expressed by the product of bas

1 X 2 =

2; and if a 3d letter, c, be introduced, we have (cab, cba)

a cb, bc a ; i. e., the new letter, c, may stand 1st, 2d, or 3d abc, bac)

in each of the 2 permutations of a and b; hence the number of permutations of 3 things is expressed by the product, 1 × 2 × 3

6. If a 4th letter, d, be taken, it may stand as 1st, 2d, 3d, or 4th, in each of the 6 permutations of a, b and c, and, of course, furnish 4 times 6=1 X2 X3 X 424 permutations.

By the above, it is evident that the No. of permutations