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7. A man agreed to work 70 days on this condition, that for every day he worked he should receive $ ,80, and for every day he was idle he should pay $ ,30; and at the expiration of the time he received $31,80. How many days did he work and how many was he idle ?
Ans. he worked 48 days, and was idle 22. 8. A farmer having driven his cattle to market received for them all $546, being paid for every ox $30, for every cow $20, and for every calf 3. There were twice as many cows as oxen, and three times as many calves as cows; how many had he of each?
Ans. 6 oxen 12 cows 36 calves. 9. A man gave his estate to his three sons in the following manner, viz. : to B he gave half, lacking $130, to C one-third, and to D the rest, which was $75 less than the share of C; what was the amount of the whole estate, and how much was each one's portion ? Ans. The whole amount was $1230, and
B had $485; C $410, and D $335. 10. Three men are to share a certain sum of money, as follows, viz.: the first is to have twice as much as the third; and the second two-thirds as much as the first, and the shares of the second and third, added together, are $1435; what is the share of each ?
Ans. The first $1230, the second $820, the third $615.
Questions. 1. What is Position ?
6. In Single Position, how is the num: 2. How is it divided ?
ber sought always multiplied or divi3. What is Single Position ?
ded? 4. What is the Rule for Single Posi 7. How in Double Position? tion ?
8. What is the Rule for Double Posi5. What is Double Position ?
DUODECIMALS. Duodecimals are fractions of a foot. They are so called from the Latin word duodecim, which signifies twelve. The foot being divided into 12 equal parts, called inches, or primes, and each of these parts again divided into 12 other equal parts; called seconds, and each second again
divided into twelve other equal parts called thirds, and each third into twelve equal parts called fourths; and so on to any extent. Thus, 1 inch or prime is
of a foot. 1 second is in
of a foot. 1 third is is of 1 of 1 of 1
of a foot.
1728 1 fourth is
of a foot, &c. It is usual to distinguish inches by one mark, thus () seconds by two ("') thirds by 3 marks, ("'') fourths by 4 marks, ('''') &c. These marks are called indices.
Twelve of each of the less denominations are equal to one of the next greater, as in the following
TABLE. 12""' fourths make
1" third. 12'" thirds make
1" second. 12" seconds make
l' inch or prime. 12' inches or primes make
1 foot. Hence they are added and subtracted in the same manner as compound numbers.
MULTIPLICATION OF DUODECIMALS Is used in finding the contents of surfaces and solids.
(Note. F stands for feet, and I or' for inches.) Fx F give
X" give That is, the product of any two denominations will al. ways be of that denomination denoted by the sum of their indices:
1. Write the multiplier under the corresponding denominations of the multiplicand.
2. Multiply each denomination in the multiplicand by the highest denomination in the multiplier, carrying 1 to the next higher for every 12 in the lower denomination.
3. Multiply each denomination in the multiplicand by the second denomination in the multiplier, and set the result of each denomination one place removed to the right of the
former products; and so on for each of the other denominations of the multiplier, always placing the product by a smaller denomination one place farther to the right than that of its superior denomination.
1. How many square feet are contained in a board 6 feet 7 inches long, and 2 feet 5 inches wide ?
Illustration. We first multiply 6 7
the multiplicand, beginning with 2 5
the inches, by the two feet in the 13 2'
multiplier ; thus, 7x2F=14=1F
2in. 2 8
We place the 2' under the inches, and
the 1 to the feet; 15 10' 11'
thus, 6F X2F=12, and 1 to carry =13F. We then multiply by the 5' in the multiplier ; thus, 7' X 5'=35"=2' 11"; we set the 11" one place to the right, and carry the 2' to the next; thus, 6F x 5'=30'; and 2' to carry make 32'=2F 8', which we set down in their proper places, and the products added together give the answer, 15F 10' 11", or 15F 1011 inches.
2. How many square feet are contained in a board 15 feet 5 inches long, and 1 foot 8 inches wide ?
5' x1ft. =5', and 15ft. x 1ft.=15ft.
Then, 5' X 8'=40"=3'4", and 15ft. X 15 5'
8:=120', and 3' to carry make 123'= 10 3' 4"
10 feet 3'. Ans. 25
8 4" 3. How many solid feet in a block 2ft. 6 inches long; ift. 9 inches wide, and ift. 5 inches thick ?
2ft. 6' length
Note. The length multiplied 4 4' 6"
by the breadth, and that product 1 5 thickness by the thickness, gives the cú
bic, or solid contents. 4 4' 6''
1 9' 10" 6" Ans: 6 2' 4" 6"
4. How many square feet are contained in a floor 16ft. 9 inches long, and 12ft. 4 inches wide ? Ans. 206ft. 7in.
5. How much wood in a pile 6ft. long, 3ft. 9in. high, and 4ft. 6in. wide ?
Ans. 101ft. 3in. 6. How many square feet are contained in 15 boards, each 18ft. 5in. long, and ift. 7in. wide ? Ans. 437f. 4' 9".
First find the square contents in 1 board which multiply by 15, using the factors.
7. Multiply 16ft. 9in. by 7ft. llin. Ans. 132ft. 71 in. 8. Multiply 10ft. 4' 3'by 2ft. 8'. Ans. 27ft. 7 4. 9. How many square feet in a board 36ft. 8in. long, and 2ft. 4' 9" wide ?
Ans. 87ft. 10' 2'. 10. How much wood in a load 8ft. long, 4ft. 5' high, and 3ft. 9. wide ?
Ans. 132ft. 6'=l cord, 4 ft
MISCELLANEOUS QUESTIONS FOR EXERCISE IN THE
FOREGOING RULES. 1. What is the sum of 19484 added to itself?
Ans. 38961 2. There are three numbers, their sum is 2302, the first is 311, and the second 695, what is the third ?
Ans. 1296. 3. What is the difference between 31 eagles, and 3099 dimes ?
Ans. 10cts. 4. If the minuend be 3441, and the remainder 365, what is the subtrahend ?
Ans. 3806. 5. What number being multiplied by 5, will make just as much as 25 multiplied by 49 ?
Ans. 245. 6. What number is that which being divided by 96 the quotient will be 128 ?
Ans. 12288. 7. There are 6 chests of drawers, each containing 18 drawers—in each drawer is 6 divisions, and each division contains $36. How many dollars in all ? Ans. $23328.
A line or vinculum drawn over several numbers, denotes that the numbers under it are to be taken jointly, or as one whole number.
Ans. 84. 10. 40+7–15 X 12–3+7=How many ? Ans. 512.
11. A man being asked his age replied, I have 7 sons, the difference between whose ages is just 2 years—I was 34 years old when my oldest son was born, and that is now the age of my youngest. What was his
Ans. 80 years. 12. If from a staff 8 feet in length,
A shadow, 5 is made,
Ans. 48 yards. 13. What number, being multiplied by 15, the product will be
Ans. j = 1=zo. 14. What number is that which being divided by 15, the product will be ?
Ans. 2X15=111. 15. What number is that, to which if you add }, will be
Ans. -= 16. What number is that, from which if you subtract the remainder will be i?
Ans. 17. B and C traded together and gained $100; B put in $640, C put in so much that he must receive $60 of the gain. How much did C put in ?
Ans. $960. 18. Divide $356 between B and C, so that the shares shall be in proportion to each other as 3 to 5. 3+5=8 : 356; 53: $133. B's part.
5 : 222] C's part. 19. Bought cloth at $1,25 per yard, and lost 25 per cent by the sale of it. How was it sold per yard ?
Ans. $93,7 söm. 20. Thomas sold 150 pine apples at 33 cts, apiece, and received as much money as Harry received for a certain number of watermelons which he sold for 25 cents apiece. How mụch money did each receive, and how many melons had Harry ?
Ans, each received $50, and Harry had 200 melons. 21. B and C depart from the same place and travel in opposite directions, B goes east 23 miles a day, and C travels west 35 miles a day. How far will they be apart at the end of 6 days, and how many miles will each have traveled ?
Ans. 348 miles. B will have traveled 138m. and C 210. • 22. A .certain pasture will last 936 sheep 7 weeks. –