509. When the first terms have factors common to the second or third terms.

Cancel the factors which are common, then divide the product of those remaining in the second and third terms by the product of those remaining in the first, and the quotient will be the answer.

PROOF.-Place the answer in the denominator, or on the left of the perpendicular line, and if the factors of the divisor and dividend exactly cancel each other, the work is right.

OBS. 1. Instead of placing points between the antecedents and consequents of the left hand couplets of the proportion, it is sometimes more convenient to put a perpendicular line between them, as in division of fractions. (Art. 232.) This will bring all the terms whose product is to be divided on the right of the line, and those whose product is to form the divisor, on the left. In this case the third term should be placed below the second terms, with the sign of proportion (::) before it, to show its origin, and its relation to the answer.

2. It will be observed that Cancelation can be applied in Compound Proportion to all those examples whose first terms have factors common to the second terms, or to the third term.

8. If 24 men can saw 90 cords of wood in 6 days, when the days are 9 hours long, how many cords can 8 men saw in 36 days, when they are 12 hours long?

9. If 6 men can make 120 pair of boots in 20 days, working 8 hours a day, how long will it take 12 men to make 360 pair, working 10 hours a day?

10. If 12 men can build a wall 30 ft. long, 6 ft. high, and ft. thick, in 18 days, how long will it take 36 men to build on 360 ft. long, 8 ft. high, and 6 ft. thick.

11. If a horse can travel 120 miles in 4 days when the day are 8 hours long, how far can he travel in 30 days when the days are 10 hours long?

QUEST.-509. When the first terms have factors cominon to the second or third terms, how proceed?

12. If $250 gain $30 in 2 years, what will be the interest of $750 for 5 years?

13. What will be the interest of $500 for 4 years, if $600 will gain $42 in 1 year?

14. If $360 gain $14.40 in 8 months, what will $4800 gain in 32 months?

15. If a family of 8 persons spend $200 in 9 months, how much will 18 persons spend in 12 months?

16. If 15 men, working 12 hours a day, can hoe 60 acres in 20 days, how long will it take 30 boys, working 10 hours a day, to hoe 96 acres, 6 men being equal to 10 boys?

CONJOINED PROPORTION.

510. When each antecedent of a compound ratio is equal in value to its consequent, the proportion is called Conjoined Proportion.

OBS. Conjoined Proportion is often called the chain rule. It is chiefly used in comparing the coins, weights and measures of two countries, through the medium of those of other countries, and in the higher operations of exchange. The odd term is sometimes called the demand.

17. If 20 lbs. United States make 12 lbs. in Spain; and 15 lbs. Spain 20 lbs. in Denmark; and 40 lbs. Denmark 60 lbs. in Russia: how many pounds in Russia are equal to 100 lbs. U. S.?

Operation.

20 lbs. U. S.=12 lbs. Spain
15 lbs. Spain= 20 lbs. Den.

40 lbs. Den. 60 lbs. Rus. How many lbs. R.=100 lbs. U. S.

Arrange the given terms in pairs, making the first term the antecedent, and its equal the consequent; then since it is required to find how many of the last kind are equal to a

given number (100 lbs.) of the first, place the odd term or demand under the consequents.

Then, 20X15X40: 12X20X60:100: Ans.

511. From these illustrations we derive the following

RULE FOR CONJOINED. PROPORTION.

I. Taking the terms in pairs, place the first term on the left of the sign of equality or a perpendicular line for the antecedent, and its equal on the right for the consequent, and so on. Then, if the answer is to be of the same kind as the first term, place the odd term under the antecedents; but if not, place it under the consequents.

II. Cancel the factors common to both sides, and if the odd term falls under the consequents, divide the product of the factors remaining on the right by the product of those on the left, and the quotient will be the answer; but if the odd term falls under the antecedents, divide the product of the factors remaining on the left by the product of those on the right, and the quotient will be the

answer.

PROOF.-Reverse the operation, taking the consequents for the antecedents, and the answer for the odd term, and if the result thus obtained is the same as the odd term in the qiven question, the work is right.

OBS. In arranging the terms, it should be observed that the first antecedent and the last consequent will always be of the same kind.

18. If 100 lbs. United States, make 95 lbs. Italian; and 19 lbs, Italian, 25 lbs. in Persia; how many pounds in the U. S. are equal to 50 lbs. in Persia? Ans. 40 lbs.

19. If 10 yds. at New York make 9 yds. at Athens; and 90 yds. at Athens, 112 yds. at Canton; how many yds. at Canton are equal to 50 yds. at New York?

20. If 50 yds. of cloth in Boston are worth 45 bbls. of flour in Philadelphia; and 90 bbls. of flour in Philadelphia 127 bales of cotton in New Orleans; how many bales of cotton at New Orleans are worth 100 yds. of cloth in Boston?

21. If $18 U. S. are worth 8 ducats at Frankfort; 12 ducats at Frankfort 9 pistoles at Geneva; and 50 pistoles at Geneva, 12 rupees at Bombay: how many rupees at Bombay are equal to $100 United States?

SECTION XV

DUODECIMALS.

ART. 512. DUODECIMALS are a species of compound numbers, the denominations of which increase and decrease uniformly in a twelvefold ratio. The denominations are feet, inches or primes, seconds, thirds, fourths, fifths, &c.

Note. The term duodecimal is derived from the Latin numeral duodecim, which signifies twelve.

OBS. The accents used to distinguish the different denominations below feet, are called Indices.

513. Duodecimals may be added and subtracted in the same manner as the other compound numbers. (Arts. 300, 302.)

MULTIPLICATION OF DUODECIMALS.

514. Duodecimals are principally applied to the measurement of surfaces and solids. (Arts. 285, 286.)

Ex. 1. How many square feet are there in a board 12 ft. 7 in. long, and 4 ft. 3 in. wide?

QUEST.-512. What are duodecimals? What are the denominations? Note. What is the meaning of the term duodecimal? Repeat the Table. Obs. What are the accents called, which are used to distinguish the different denominations? 513. How are duodecimals added and subtracted? 514. To what are duodecimals chiefly applied?

Operation. 12 ft. 7/

9"

We first multiply each denomination of the multiplicand by the feet in the multiplier, beginning at the right hand. Thus, 4 times 7' are 28', equal to 2 ft. and 4'. Set the 4' under inches, and carry the 2 feet to the next product. 4 times 12 ft. are 48 ft. and 2 to carry make 50 ft. Again, since 3' of a of a ft. 21", or 1' and 9'

of a ft., 3' into 7' is 2

ft. and 7

3

12

44

3 12

Write the 9" one place to the right of inches, and carry the 1' to the next product. Then 3' or of a ft. multiplied into 12 ft.19 of a ft., or 36', and 1' to carry make 37'; but 37'=3 ft. and 1'. Now adding the partial products, the sum is 53 ft. 5′ 9′′.

12

OBS. It will be seen from this operation, that feet multiplied into feet, produce feet; feet into inches, produce inches; inches into inches, produce seconds, &c. That is, the product of any two factors has as many accents as the factors themselves have. Hence,

515. To find the denomination of the product of any two factors in duodecimals.

Add the indices of the two factors together, and the sum will be the index of their product.

Thus, feet into feet, produce feet; feet into inches, produce inches; feet into seconds, produce seconds; feet into thirds, produce thirds; &c.

Inches into inches, produce seconds; inches into seconds, produce thirds; inches into fourths, produce fifths, &c.

Seconds into seconds, produce fourths; seconds into thirds, produce fifths; seconds into sixs, produce eighths, &c.

Thirds into thirds, produce sixths; thirds into fifths, produce eighths; thirds into sevenths, produce tenths, &c.

Fourths into fourths, produce eighths; fourths into eighths, produce twelfths, &c.

Note.The foo'. is considered the unit and has no index.

QUEST.-515. How find the denomination of the product in duodecimals? What do feet into feet produce? Feet into inches? Feet into seconds? What do inches into inches produce? Inches into thirds? Inches into fourths? Seconds into seconds? Seconds into thirds? Seconds into eighths? Thirds into thirds? Thirds into sixths?