516. From these illustrations we derive the following

RULE FOR MULTIPLICATION OF DUODECIMALS.

I. Place the several terms of the multiplier under the corresponding terms of the multiplicand.

II. Multiply each term of the multiplicand by each term of the multiplier separately, beginning with the lowest denomination in the multiplicand, and the highest in the multiplier, and write the first figure of each partial product one place to the right of that of the preceding product, under its corresponding denomination. (Art. 515.)

III. Finally, add the several partial products together, carrying 1 for every 12 both in multiplying and adding, and the sum will be the answer required.

OBS. 1. It is sometimes asked whether the inches in duodecimals, are linear, square, or cubic. The answer is, they are neither. An inch is 1 twelfth of a foot. Hence, in measuring surfaces an inch is 1⁄2 of a square foot; that is, a surface 1 foot wide and 1 inch long. In measuring solids, an inch denotes of a cubic foot. In measuring lumber, these inches are commonly called carpenter's inches.

2. Mechanics, also surveyors of wood and lumber, in taking dimensions of their work, lumber, &c., often call the inches a fractional part of a foot, and then find the contents in feet and a fraction of a foot. Sometimes inches are

regarded as decimals of a foot.

3. We have seen that one of the factors in multiplication, is always to be considered an abstract number. (Art. 82. Obs. 2.) How then, can feet be multiplied by feet, inches by inches, &c.

It should be observed, that when one geometrical quantity is multiplied by another, some particular extent is to be considered the unit. It is immaterial what this extent is, provided it remains the same in the different parts of the same calculation. Thus, if one of the factors is one foot and the other half a foot, the former being 12 in., and the latter 6 in., the product is 72 in. Though it would be nonsense to say that a given length is repeated as often as another is long, yet there is no impropriety in saying that one is repeated as many times as there are feet or inches in another.

4. On the principles of duodecimals, it has been supposed that pounds shillings, pence, and farthings can be multiplied by pounds, shillings, pence, and farthings. But it may be asked, what denomination shillings multiplied by pence, or pence by farthings, will produce? It is absurd to say that 2s. and 6d. is repeated 2s. and 6d. times.

QUEST.-516. What is the rule for multiplication of duodecimals? Obs. What kind of inches are those spoken of in measuring surfaces by duodecimals? In measuring solids ?

Ex. 2. How many square feet are there in a piece of marble 9 ft. 7 in. 2" long, and 3 ft. 4 in. 7" wide?

Note. It is not absolutely necessary to begin to multiply by the highest denomination of the multiplier, or to place the lower denomination to the right of the multiplicand. The result will be the same if we begin with the lowest denomination of the multiplier, and place the first figure of each partial product under the figure by which we multiply.

3. How many square feet are there in a board 15 ft. 7 in. long, and 1 ft. 10 in. wide?

4. How many cubic feet in a stick of timber 15 ft. 3 in. long, 2 ft. 4 in. wide, and 1 ft. 8 in. thick?

5. How many cubic feet in a block of granite 18 ft. 5 in. long, 4 ft. 2. in. wide, and 3 ft. 6 in. thick?

6. How many square feet in a stock of 10 boards, 15 ft. 8 in. long, and 1 ft. 6 in. wide?

7. How many square feet in a stock of 15 boards, 20 ft. 3 in. long, and 2 ft. 5 in. wide ?

8. Multiply 16 ft. 3' 4" by 6 ft. 5' 8" 10".

9. Multiply 20 ft. 4' 8" 5" by 7 ft. 6' 9" 4"". 10. Multiply 18 ft. 0' 5' 10" by 4 ft. 8' 7" 9!!!.

11. Multiply 50 ft. 6'0' 2" 6" by 3 ft. 10' 5''.

12. How many cords in a pile of wood 50 ft. 6 in. long, 8 ft. 3 in. wide, and 7 ft. 4 in. high?

13. If a cistern is 30 ft. 10 in. long, 12 ft. 3 in. wide, and 10 ft. 2 in. deep, how many cubic feet will it contain ?

14. What will it cost to plaster a room 20 ft. 6 in. long, 18 ft. wide, and 10 ft. high, at 124 cts. per square yard ?

15. How many bricks 8 in. long, 4 in. wide, and 2 in. thick, will make a wall 50 ft. long, 10 ft. high, and 2 ft. 6 in. thick?

SECTION XVI.

EQUATION OF PAYMENTS.

ART 517 EQUATION OF PAYMENTS is the process of finding the equalized or average time when two or more payments due at different times, may be made at once, without loss to either party.

OBS. The equalized or average time for the payment of several debts, due at different times, is often called the mean time.

518. From principles already explained, it is manifest, when the rate is fixed, the interest depends both upon the principal and the time. (Art. 404.) Thus, if a given principal produces a certain interest in a given time,

Double that principal will produce twice that interest;

Half that principal will produce half that interest; &c.

In double that time the same principal will produce twice that interest;

In half that time, half that interest; &c.

519. Hence, it is evident that any given principal will preduce the same interest in any given time, as

One half that principal will produce in double

One third that principal will

Twice that principal will
Thrice that principal will

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that time

"thrice

that time;

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half

that time;

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a third of that time; &c.

For example, at any given per cent.

The int. of $2 for 1 year, is the same as the int. of $1 for 2 yrs. ;

The int. of $3 for 1 year,

The int. of $4 for 1 mo.

The int. of $5 for 1 mo.

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66

QUEST.-517. What is equation of payments? Obs. What is the average time for the payment of several debts sometimes called? 518. When the rate is fixed, upon wha does the interest depend?

520. The interest, therefore, of any given principal for 1 year, or 1 month, &c., is the same, as the interest of 1 dollar for as many years, or months, as there are dollars in the given principal.

Ex. 1. Suppose you owe a man $15, and are to pay him $5 in 10 months, and $10 in 4 months, at what time may both payments be made without loss to either party?

Analysis.-Since the interest of $5 for 1 month is the same as the interest of $1 for 5 months, (Art. 519,) the interest of $5 for 10 months must be equal to the interest of $1 for 10 times 5 months. And 5 mo. X 10-50 mo. In like manner the interest of $10 for 4 months is equal to the interest of $1 for 4 times 10 months; and 10 mo. X440 months. Now 50 months added to 40 months make 90 months; that is, you are entitled to the use of $1 for 90 months. But $1 is of $15, consequently you are entitled to the use of $15, of 90 months, and 90÷156.

Proof.

Ans. 6 months.

The interest of $5 at 6 per cent. for 10 mo. is $5 X.05=$.25 The interest of $10

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4 mo. is $10X.02.20

Sum of both $.45

The interest of $15 at 6 per cent. for 6 mo. is 15X.03=$.45. 521. From these principles we derive the following general

RULE FOR EQUATION OF PAYMENTS.

First multiply each debt by the time before it becomes due; ther divide the sum of the products thus obtained by the sum of the debts, and the quotient will be the average time required.

OBS. 1. If one of the debts is paid down, its product will be nothing; but in finding the sum of the debts, this payment must be added with the others. 2. When there are months and days, the months must be reduced to days, or the days to the fractional part of a month.

3. This rule is based upon the supposition that discount and interest paid in advance are equal. But this is not exactly true; consequently, the rule, though in general use, is not strictly accurate. (Art. 432. Obs. 1.)

QUEST.-521. What is the rule for equation of payments?
15*

2. If you owe a man $60, payable in 4 months, $120 payable in 6 months, and $180 payable in 3 months, at what time may you justly pay the whole at once?

$ 60X4

$120X6

$180X3

Operation.

$240, the same as $1 for 240 months. (Art. 520.)

$360 debts. $1500, sum of products.

Now 1500-360-4 months. Ans.

3. A merchant bought one lot of goods for $1000 on 5 months; another for $1000 on 4 months; another for $1500 on 8 months: what is the average time of all the payments?

4. If a man has one debt of $150, due in 3 months; another of $200, due in 4 months; another of $500, due in 7 months: what is the average time of the whole ?

5. A man bought a house for $3500, and agreed to pay $500 down, and the balance in 6 equal annual instalments: at what time may he pay the whole?

6. If you owe one bill of $175, due in 30 days; another of $81, due in 60 days; another of $120, due in 65 days, and another of $200, due in 90 days: when may you pay the whole at once?

PARTNERSHIP.

522. PARTNERSHIP is the associating of two or more individuals together for the transaction of business. (Art. 464.) The persons thus associated are called partners; and the association itself, a company or firm.

The money employed is called the capital or stock; and the profit or loss to be shared among the partners, the dividend.

CASE I.—When stock is employed an equal length of time.

Ex. 1. A and B formed a partnership; A furnished $600 capital, and B $900; they gained $300: what was each partner's share of the gain?

QUEST.-522. What is partnership? What are the persons thus associated called? What is the association itself called? What is the money employed called? What the profit or loss?