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4. What is the duty on 12 boxes of raisins, each weighing 50 pounds, valued at 10 cents a pound, 10% tare, duty 8% ad valorem?

5. What is the duty, at 25% ad valorem, on 50 tons of steel, tare 20 pounds to the cwt., invoiced at 18 cents a pound?

6. What is the duty, at 45 cents a square yard and 40% ad valorem, on 800 yards of carpet, of a yard wide, invoiced at 10 francs a yard?

ANNUITIES.

291. An Annuity is a definite sum of money payable at regular intervals of time.

The Amount, or Final Value, of an annuity is the sum of the amounts of all the payments on interest from the time each is due until the annuity ceases.

The Present Value of an annuity is a sum which, if put on interest for the given time and rate, will amount to the final value.

292. Annuities at Simple Interest.

1. What is the final value annually, for 4 yr. at 5% ?

ILLUSTRATION.

600

=

The last payment will draw no interest; the last but one will draw interest for 1 year; and so on.

It is thus seen that an annuity at simple interest forms an arithmetical progression in which the annuity is the first term, the interest for 1 year is the common difference, the number of years is the number of terms, and the final value is the sum of the series. Therefore, by ART. 269,

600

The sum of the terms

WRITTEN EXERCISES.

600 600
600

600

=

=

600

630
660

690

of an annuity of $600, payable

=

($600+ $690) = $2580, Ans.

2. Find the present value of an annuity of $800 for 10 yr., at 5%, payable annually.

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ILLUSTRATION.

=

=

600 600 600 600

=

=

$1160

600 (1.05)
600 (1.05)2
600 (1.05)3

=

RULE.

Take the annuity for the first term, the interest for 1 year for the common difference, and the time for the number of terms; find the last term, and then the sum of the terms; this sum will be the final value.

Then find the present value of the final value.

Find the present value

3. Of an annuity of $400 for 5 yr. at 6%.
4. Of an annuity of $600 for 6 yr. at 6%.
5. Of an annuity of $1200 for 10 yr. at 7%.
6. Of an annuity of $1600 for 8 yr. at 4%.
7. Of an annuity of $2000 for 8 yr. at 5%.
8. Of an annuity of $3000 for 10 yr. at 8%.

293. Annuities at Compound Interest.

$65331, Ans.

WRITTEN EXERCISES.

1. Find the final value of an annuity of $600 for 4 yr. at 5% compound interest.

$9800

The last payment will draw no interest; the last but one will draw interest for 1 yr.; the last but two will draw compound interest for 2 yr.; and so on.

It is thus seen that an annuity at compound interest forms a geometrical progression in which the annuity is the first term, 1 plus the rate of interest is the ratio, the number of years is the number of terms, and the final value is the sum of the series. Therefore, by ART. 271,

The last term =

$600 (1.05)3.

By ART. 272,

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The number in parenthesis is the compound interest of $1 for 4 yr. at 5%, which by the table on page 208 is .215506. Hence

The sum = $12000 × .215506

$2586.072, Ans.

Final value

2. Find the present value of an annuity of $500 for 6 yr. at 5%.

=

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=

$500
.05

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$600 (1.05 — 1).

.05

=

$10000 .340096

(1.056 — 1).

1)= .340096. Hence the

$3400.96
1.340096

$3400.96.

dividing by 1.340096, the compound Hence the

=

=

$2537.85.

RULE.

To find the final value, divide the annuity by the rate, and multiply the quotient by the compound interest for the given time and rate.

To find the present value, divide the final value by the compound amount of 1 dollar for the given time and rate. Find the present value

3. Of an annuity of $800 for 7 yr. at 4%.
4. Of an annuity of $1200 for 8 yr. at 6%.
5. Of an annuity of $1500 for 5 yr. at 5%.
6. Of an annuity of $1600 for 9 yr. at 8%.
7. Of an annuity of $2500 for 10 yr. at 5%..

BUILDING ASSOCIATIONS.

294. Building and Loan Associations, or Co-operative Banks, as they are sometimes called, are corporations which receive small deposits at regular periods of time, and invest them in mortgages among their own members or stockholders.

295. The Members of an association are the persons who either subscribe for shares as an investment or borrow money of the association. The former are called "non-borrowers," and the latter "borrowers."

296. The Shares are usually $200, and are issued periodically in series. When the sum of the regular payments and profits equals the par value of a share, the series is closed.

297. The Dues, or Installments, consist of one dollar a month on each share held by the members.

298. The Premium is the amount bid by a member in excess of simple interest for the purpose of securing the loan.

MODES OF LOANING MONEY.

299. There are three modes of loaning money and fixing the interest used by different associations, known as the Installment Plan, the Gross Plan, and the Net Plan.

By the Installment Plan the borrower receives the par value of a share, and pays the premium in monthly installments, together with the dues and interest.

ILLUSTRATION.-In borrowing $200 the borrower would receive the full amount and pay 6% interest, or $1 a month, together with the dues and monthly premium.

By the Gross Plan the borrower receives only the difference between the face value of the loan and the premium bid, but pays interest on the full amount borrowed.

ILL.-In borrowing $200 at a premium of 25%, the borrower receives $200 less $50, or $150, and pays 6% interest on $200, or $1 a month, together with the monthly dues.

By the Net Plan the premium is deducted as in the gross plan, but the borrower is charged interest only on the amount he receives.

ILL.-In borrowing $200 at 25% premium the borrower receives $200 less $50, or $150, and pays 6% interest on this amount-that is, $9 a year, or 75 cts. a month-together with the monthly dues.

300. Fines are penalties imposed for non-payment of dues or interest. By the Building Association Laws of Pennsylvania fines cannot exceed 2% a month on all arrears, and no fine can be imposed on unpaid fines.

301. The Withdrawal Value of a share is found by adding to the amount paid in on the stock, 6% interest on this sum, and such a share of the profits as is determined by the rules of the association.

CASE I.

302. To find the cost of any number of shares.

1. What is the cost of 8 shares of stock in a building association which has been running 6 yr. 4 mo., payments $1 per month?

PROCESS.

6 yr. 4 mo. = 76 mo.

76 $8 $608, Ans.

The dues on 8 shares for 1 mo. will be $8, and for 6 yr. 4 mo., or 76 mo., they will be 76 times $8, or $608.

2. What is the actual cost of 20 shares of stock in a building association which has been running 5 yr. 8 mo., payments $1 per month?

3. I own 10 shares in the 1st series, 16 in the 2d, 20 in the 3d of the West End Building and Loan Association; if the

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