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330. From the foregoing illustrations we derive the following general
RULE FOR COMPOUND PROPORTION.
I. Place that number which is of the same kind as the answer required for the third term.
II. Then take the other numbers in pairs, or two of a kind, and arrange them as in simple proportion. (Art. 327.)
Ill. Finally, multiply together all the second and third terms, divide the result by the product of the first terms, and the quotient will be the fourth term or answer required.
Proof.—Multiply the answer into all of the first terms or antecedents of the first couplets, and if the product is equal to the continued product of all the second and third terms multiplied together, the work is right. (Art. 324.)
Il may also be proved by analysis. (Art. 295.)
Obs. 1. Among the given numbers there is but one which is of the same kind as the answer. This is sometimes called the odd term, and is always to be placed for the third term.
2. Questions in Compound Proportion may be solved by Analysis ; also by Simple Proportion, by making two or more separate statements. (Arts. 302. N. 327.)
3. If 8 men can clear 30 acres of land in 63 days, working 10 hours a day, how many acres can 10 men clear in 72 days, working 12 hours a day?
10hr. : 12hr.
= the Ans. (Art. 330.) Divided by
8 x 63 x 10
QUEST.—230. What is the rule for compound proportion? How are questions in compound proportion proved? Obs. Among the given numbers, how many are of the same kind as the answer? Can questions in compound proportion be solved by simple proportion? How?
Now by canceling equal factors, (Art. 140,) we have
9 10x78x12X30 360 8 x63x10
51% acres, Ans. Hence, 7 331. After stating the question according to the rule above, if the antecedents or first terms have factors common to the consequents or second terms, or to the third term, they should be canceled before performing the multiplication and division.
Note.—Instead of placing points between the first and second terms, that is, between the antecedents and consequents of the left hand couplets of the proportion as above, it is sometimes more convenient to put a perpendicular line between them, as in division of fractions. (Art. 140.) This will bring all the terms whose product is to be the dividend 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.
4. If a man can walk 192 miles in 4 days, traveling 12 hours a day, how far can he go in 24 days, traveling 8 hours a day?
Operation. That is, the product of the antecedents, 4 d. 24 d. 2 4x12, has the same ratio to the product 12 hr. & hr. 2
of the consequents, 24 x 8, as 192 has to :: 192m.
the answer required. Ans. 192x2x2=768 miles.
5. If 8 men can make 9 rods of wall in 12 days, how many men will it require to make 36 rods in 4 days ?
6. If 5 men make 240 pair of shoes in 24 days, how many men will it require to make 300 pair in 15 days ?
7. If 60 lbs. of meat will supply 8 men 15 days, how long will 72 lbs. last 24 men?
8. If 12 men can reap 80 acres of wheat in 6 days, how long will it take 25 men to reap 200 acres ?
9. If 18 horses eat 128 bushels of oats in 32 days, how many bushels will 12 horses eat in 64 days ?
10. If 8 men can build a wall 20 ft. long, 6 ft. high, QUEST.-33). When the antecedents have factors common to the consequents, what should be done with them ?
and 4 st. thick, in 12 days, how long will it take 24 men to build one 200 ft. long, 8 ft. high, and 6 st. thick ?
11. If 8 men reap 36 acres in 9 days, working 9 hours per day, how many men will it take to reap 48 acres in 12 days, working 12 hours per day?
12. If $100 gain $6 in 12 months, how long will it take $400 to gain $18?
13. If $200 gain $12 in 12 months, what will $400 gain in 9 months ?
14. If 8 men spend £32 in 13 weeks, how much will 24 men spend in 52 weeks ?
15. If 6 men can dig a drain 20 rods long, 6 feet deep, and 4 feet wide, in 16 days, working 9 hours each day, how many days will it take 24 men to dig a drain 200 rods long, 8 ft. deep, and 6 ft. wide, working 8 hours per day?
332. Duodecimals are a species of compound numbers, the denominations of which increase and decrease uniformly in a twelvefold ratio. Its 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.
TABLE. 12 fourths ("'') make 1 third, 12 thirds
1 second, 12 seconds
1 inch or prime,
in. or' 12 inches or primes 1 foot,
ft. Hence l'=' of 1 foot. 1"=ia of 1 in. or it of
t's of 1 ft=111 of 1 ft. 1"=12 of 1", or in of i'm of I'm of 1 ft.=1728 of 1 ft. QUEST.332. What are duodecimals? What are its denominations ? 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 ?
Obs. The accents used to distinguish the different denominations below feet, are called Indices.
333. Duodecimals may be added and subtracted in the same manner as other compound numbers. (Arts. 168, 169.)
MULTIPLICATION OF DUODECIMALS.
334. Duodecimals are principally applied to the measurement of surfaces and solids. (Arts. 153, 154.)
Ex. 1. How many square feet are there in a board 8 ft. 9 in. long and 2 ft. 6 in. wide ?
Operation. We first multiply each denomina8 ft. 9' length,
tion of the multiplicand by the feet 2 st. 6' width,
in the multiplier, beginning at the 17 st. 6'
right hand. Thus, 2 times 9' are 18', 4 ft. 4' 6''
equal to 1 ft. and 6'. Set the 6' un
der inches, and carry the 1 ft. to the 21 ft. 10' 6" Ans.
next product. 2 times 8 ft. are 16 ft. and 1 to carry makes 17 ft. Again, since 6'=fe of a ft. and 9'=i'a of a ft., 6' into 9' is 4 of a ft.=54" or 4' and 6". Write the 6" one place to the right of inches, and carry the 4' to the next product. Then 6' or 192 of a foot multiplied into 8 ft.=41 of a ft. or 4 ft. Now adding the partial products, the sum is 21 ft. 10'6", which is the answer required.
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. Hence,
335. 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.
QUEST.-333. How are duodecimals added and subtracted ? 334. To what are duodecimals chiefly applied ? . 335. How find the denomination of the product in duodecimals? What do feet into feet produce ? Feet into inches ? Feet into seconds ?
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 sixths, 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 foot is considered as the unit, and has no index. 336. From these illustrations we have 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 or more places to the right, under its corresponding denomination. (Art. 335.)
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. 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 12 of a square foot; that is, a surface 1 foot long and I inch wide. In measuring solids, an inch denotes in of a cubic foot. In common language, inches in duodecimals are called carpenter's inches.
2. How many square feet are there in a board 18 feet 9 inches long, and 2 feet 6 inches wide ?
3. How many square feet are there in a board 14 feet 10 inches long, and 11 inches wide ?
Quest.-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 ? 336. What is the rule for multiplication of duodecimals? Obs. In duodecimals, what kind of inches are those spoken of in measuring surfaces ? In measuring solids? What are they called ?