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When one quantity is contained in another exactly 2, 3, 4, 6, &c. times, it is called an aliquot or even part of that quantity; thus, 25 cents is an aliquot part of a dollar, because 4 times 25 cents is just equal to 1 dollar; and 6 pence is an aliquot part of a shilling, because 2 times six pence just make 1 shilling. The following table exhibits some of the aliquot parts of a dollar :
of a dollar. of a dollar. of a dollar. of a dollar. 12 of a dollar. of a dollar.
From the illustration of the last example, it appears, that, when the price per yard, pound, &c. is one of these aliquot parts of a dollar, the cost may be found, by dividing the given number of yards, pounds, &c. by that number which it takes of the price to make 1 dollar. If the price be 50 cents, we divide by 2; if 25 cts. by 4; if 124 cts. by 8, &c. This manner of calculating the cost of articles, by taking aliquot parts, is usually called Pructice.
5='5 of a dollar.
11. What is the value of 14756 yards of cotton cloth, at 12 cents, or of a dollar, per yard?
12. What is the cost of 18745 pounds of tea, at $50,
dollar, per pound?
13. What is the value of 9366 bushels of potatoes, at 33 cents, or of a dollar, per bushel? 9366 $3122 Ans.
14. What is the value of 48240 pounds of cheese, at '06, of a dollar, per pound?
15. What cost 4870 oranges, at 5 cents, apiece?
16. What is the value of 151020 bushels cents, of a dollar, per bushel ?
of a dollar,
of apples, aɩ 20
17. What will 264 pounds of butter cost, at 12 cents per pound?
18. What cost 3740 yards of cloth, at $125 per yard?
19. What is the cost of 8460 hats, at $1'124 apiece? at $150 apiece?
at $3'20 apiece?
Ans. $9517'50. $12690. $27072. $34368'75.
30. To find the value of articles sold by the 100, or 1000. 1. What is the value of 865 feet of timber, at $5 per
4325= value at $5 per foot.
Were the price $5 per foot, it is plain, the value would be 865 X $5 $ 4325; but the price is $5 for 100 feet; consequently, $ 4325 is 100 times the true value of the timber; and therefore, if we divide this number ($ 4325) by 100, we shall obtain the true value; but to divide by 100 is but to cut off the two right hand figures, or, in federal money, to remove the separa trix two figures to the left. Ans. $43'25.
It is evident, that, were the price so much per thousand, the same remarks would apply, with the exception of cutting off three figures instead of two. Hence we derive the general RULE for finding the value of articles sold by the 100, or 1000-Multiply the number by the price, and, if it be reckoned by the 100, cut off the two right hand figures, and the product will be the answer, in the same kind or denomination as the price. If the article be reckoned by the 1000, cut off the three right hand figures.
EXAMPLES FOR PRACTICE.
2. What is the value of 4250 feet of boards, at $14 per Ans. 59 dollars and 50 cents.
In this example, because the price is at so much per 1000 feet, we divide by 1000, or cut off three figures.
3. What will 3460 feet of timber come to, at $4 per hundred ?
4. What will 24650 bricks come to, at 5 dollars per 1000? 5. What will 4750 feet of boards come to, at $12′25 per 1000 ? Ans. 58'187. 6. What will 38600 bricks cost, at $4'75 per 1000? 7. What will 46590 feet of boards cost, at $10'625 per 1000?
8. What will 75 feet of timber cost, at $4 per 100? 9. What is the value of 4000 bricks, at 3 dollars per 1000;
DIVISION OF FEDERAL MONEY.
31. 1. If3 yards of cloth cost $5'25, what is that a yard?
Answer, 175 cents, = $175.
$5'25 is 525 certs, which divided by 3, the quotient is 175 cents, which, reduced to dollars, is $175, the answer.
2. Bought 4 bushels of corn for $3; what was that a bushel?
4 is not contained in 3; we may, however, reduce the $3 to cents, by annexing two ciphers, thus:
Ans. 75 cents.
300 cents divided by 4, the quotient is 75 cents, the price of each bush. of
3. Bought 18 gallons of brandy, for $42'75; what did it cost a gallon?
18)42 75 (2375 mills, = $2'375, the answer.
$42'75 is 4275 cents. After bringing down the last figure in the dividend, and dividing, there is a remainder of 9 cents, which, by annexing a cipher, is reduced to mills, (90,) in which the divisor is contained 5 times, which is 5 mills, and there is no remainder. Or, we might have reduced $42'75 to mills, before dividing, by annexing a cipher, 42750 mills, which, divided by 18, would have given the same result, 2375 mills which, reduced to dollars, is $2'375, the answer.
4. Divide $59'387 by &.
Quotient, 7'4238, that is, 7 dollars, 42 cents, 3 mills, and of another mill. The is the remainder, after the last division, written over the divisor, and expresses such fractional part of another mill. For all purposes of business, it will be sufficiently exact to carry the quotient only to mills, as the parts of a mill are of so little value as to be disregarded. Sometimes the sign of addition (+) is annexed, to show that there is a remainder, thus, $7423 +
From the foregoing examples, it appears, that division of federal money does not differ from division of simple numbers. The quotient will be the answer in the lowest denomination in the given sum, which may then be reduced to dollars.
Note. If the sum to be divided contain only dollars, or dollars and cents, it may be reduced to mills, by annexing ciphers before dividing; or, we may first divide, annexing ciphers to the remainder, if there shall be any, till it shall be reduced to mills, and the result will be the same.
EXAMPLES FOR PRACTICE. 5. If I pay $468'75 for 750 pounds of wool, what is the value of 1 pound? Ans. $0'625; or thus, $0'621. 6. If a piece of cloth, measuring 125 yards, cost $181'25, what is that a yard? Ans. 1'45. 7. If 536 quintals of fish cost $1913'52, how much is that a quintal? Ans. $3'57. 8. Bought a farm, containing 84 acres, for $3213; what did it cost me per acre? Ans. $38'25. 9. At $954 for 3816 yards of flannel, what is that a yard
Ans. $0'25. 10. Bought 72 pounds of raisins for $8; what was that a pound? how much?
Ans. $0'111; or, $0'111+. 11. Divide $12 into 200 equal parts; how much is one of the parts? 2120 how much?
12. Divide $30 by 750. = how much? 13. Divide $60 by 1200.98 = how much?
14. Divide $215 into 86 equal parts; how much will one of the parts be? how much?
15. Divide $176 equally among 250 men; how much will each man receive? how much?
SUPPLEMENT TO FEDERAL MONEY.
1. What is understood by simple numbers? by compound numbers? 3.
by different denomina5. What are the de
tions? 4. What is federal money? nominations used in federal money ? 6. How are dollars distinguished from cents? 7. Why are two places assigned for cents, while only one place is assigned for mills? 8. To what does the relative value of mills, cents, and dollare correspond? 9. How are mills reduced to dollars? 10. -to cents? 11. Why? 12. How are dollars reduced to cents? 13. to miils? 14. Why? 15. How is the addition of federal money performed? 16. subtraction? 17. multiplication?
division? 19. Of what name is the product in multiplication, and the quotient in division? 20. In case dollars only are given to be divided, what is to be done? 21. When is one number or quantity said to be an aliquot part of another? 22. What are some of the aliquot parts of a dollar? 23. When the price is an aliquot part of a dollar, how may the cost be found? 24. What is this manner of operating called? 25. How do you find the cost of articles, sold by the 100 or 1000?
1. Bought 23 firkins of butter, each containing 42 pounds, for 16 cents a pound; what would that be a firkin, and how much for the whole ? Ans. 15939 for the whole.
2. A man killed a beef, which he sold as follows, viz. the hind quarters, weighing 129 pounds each, for 5 cents a pound; the fore quarters, one weighing 123 pounds, and the other 125 pounds, for 4 cents a pound; the hide and tallow, weighing 163 pounds, for 7 cents a pound; to what did the whole amount? Ans. $35'47.
3. A farmer bought 25 pounds of clover seed at 11 cents a pound, 3 pecks of herds grass seed for $2'25, a barrel of flour for $650, 13 pounds of sugar at 124 cents a pound; for which he paid 3 cheeses, each weighing 27 pounds, at 84 cents a pound, and 5 barrels of cider at $1'25 a barrel. The balance between the articles bought and sold is 1 cent is it for, or against the farmer'