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How many bushels at $1.,20 -50. How many at 80 cents ?-75. At 50 cents ?-120. At 40 cents ?-150. At 30 cents ?-200.
A. 635 bushele 47. How much in length that is 6 inches in breadth will make a square foot ? (12 inches in length and 12 in breadth make 1 square foot; then, 6 inches in breadth will require more in length; that is, 6 : 12 :; 12.)–24. How many 4 inches in breadth ?-36. How many 8 inches in breadth ?-18. How many 16 inches in breadth ?-9. A. 87 inches.
48. If a man's income be $1750: a year, how much may he spend each day to lay up $400 a year? A. $3,7
49. If 6 shillings make $1, New England currency, how much will 4 s. 6 d. make, in fëderal money ?-75. Will 2 s. 6 d. ?,413. Will 1 s. 6 d.?-,25. Will 3 s. 9 d. ?-,621. A. $2,045.
50. A merchant bought 26 pipes of wine on 6 months' credit, but, by paying ready money, he got it 3 cents a gallon cheaper; how much did he save by paying ready money ?-A. $98,28.
51. Bought 400 yards 2 qrs. of plaid for $406,80, but could sell it for no more than $300; what was my loss per ell French ? A. $,40.
52. If 120 gallons of water, in 1 hour, fall into a cistern containing 600 gallons, from which, by 1 pipe, 20 gallons run out in 1 hour, and by another 50 gallons, in what time will the cistern be filled ? A. 12 hours.
53. A merchant bought 40 pieces of broadcloth, each piece containing 45 yards, at the rate of $6 for 9 yards, and sold it again at the rate of $15 for 18 yards”; how much did he make in trading? A. $300.
54. A borrowed of B $600 for 3 years; how long ought A to lend B $800 to requite the favor ?-2-3. How long ought he to lend him $900 ?-2. How long $500 ?-3-7-6. How long $120021-6. A. 9 years, 4 mo. 6 days.
55. A gentleman bought 3 yards of broadcloth 1} yards wide ; how many yards of flannel, which is only i yd. wide, will line the same?
It is evident it will take more cloth which is only & yd. wide, than if it were lo yd. wide; hence 1} must be the iddle term.
A. 6 yds. Ratio, 2. 56. A regiment of soldiers, consisting of 800 men, are to be clothed, each suit containing 43 yds. of cloth, which is 14 yd. wide, and lined with flannel & yd. wide; how many yards of flannel will be sufficient to line all the suits ?
A. 8633 yds. 1 gr. 1na.
FRACTIONS. 57. If } of a barrel of flour cost Bo of a dollar, what will į of a barrel cost ?
By analysis. It is plain that, if we knew the price of 1 barrel,
of a barrel would cost $ as much. If f of a barrel cost at of a dollar, |, or 1 barrel, will cost 8 times as much, that is,
8 x 5 __ 40 X 3
16 X 4 Or, as is more than $, we may make the 2d, or multiply. ing term, as in the foregoing examples, thus :
5 X 3 15 $:$:: 16 Then,
64 1 XLVII., then, axi)== $15, Ans. Or, multiplying by the ratio, thus; the ratio of } to 4 is ;
6 X 5 f=*=6, ratio ; then,
$13, Ans. as before.
16 Or, which is obviously the same, having inverted the 1st, or dividing term, multiply all the fractions together; that is, proceed as in Division of Fractions, (TXLVII.) thus, 8 X 3 X 5
: $13, Ans., as before. 1 X 4 X 16 The pupil may perform the following examples by either of the preceding methods, but the one by analysis is recommended, it being the best exercise for the mind.
-h pier sold
58. If 3 lbs. of butter cost g of a dollar, what cost } lb.? A. $16.
59. If of a bushel of wheat cost is of a dollar, what will 1 bushel cost? A. $16.
60. If 14 yds. of cloth cost $12, what will 1 yd. cost? A. $3.
64. If of } of Pt of $1 buy 20 apples, how many apples
A. $1,80. 66. If 16% yds. will make 8 coats, how many yards will it take for 1 coat? A. 26 yds. 67. If } of of a gallon cost $3, what will 55 gallons cost?
A. $95. 68. If 6 yds. cost $53, what will 14 yds. cost ? A. $138. 69. If of} cwt. of sugar cost $10, what will 40 cwt. cost?
A. $824. 70. If & yd. of silk cost of $}, what is the price of 50 yds.?
71. If 1 cwt. of four cost $16, what will ifa cwt cost?
A. $1752 72. If 3 yds. of cloth, that is 25 yds. wide, will make a cloak, how much cloth, that is only ya. wide, will make the same garment?
The narrower the cloth, the more yards it will take; hence we make the greater the second term, thus ; & yd. : 24 yds. :: 3 yds. : 10 yds., Ans.
73. If I lend my friend $960 for of a year, how much ought he to lend me į of a year to requite the favor?
He ought not to lend me so much as I lend him, because I am to keep the money longer than he ; therefore, make ź the middle term. A. $8533.
74. If 12 men do a piece of work in 124 days, how many men will do the same in 65 days? A. 24 men. Ratio, 2.
75. A merchant, owning of a vessel, sells of his share for $500; what was the whole vessel worth?
of =6=; then, as Žof the vessel is $500, } is $250, and , or the whole vessel, is 5 X 250 =$1250.
Or thus; off : 1 :: 500 : $1250, Ans., as before.
76. lf 1.3 lb. indigo cost $3,84, what will 49,2 lbs. cost at the same rate ? A. $125,952. 77. If $299 buy 593 yds. of cloth, what will $60 buy?
A. 120 yds. 78. How many yds. of cloth can I buy for $753, if 2674 yds cost $373 ? A. 535. yds. Ratio, 2.
I LXXIV. 1. If 40 men, in 10 days, can reap 200 acres of grain, how many acres can 14 men reap in 24 days?
By analysis. If 40 men, in 10 days, reap 200 acres, 1 man, in the same time, will reap 6 of 200 acres, that is, in 10 days; and in 1 day he will reap 1 of 5 acres=T=1 an acre a day; then 14 men 1 day will reap 14 times as much, which is 14 x =7 acres, and in 24 days, 24 times 7 acres, 168 acres, Ans.
Perform the following sums in the same manner.
2. If 4 men mow 96 acres in 12 days, how 'many acres can 8 men mow in 16 clays ? First find how many acres 1 man will mow in 12 days; then, in 1 day.
A. 256 acres.
3. If a family of 8 persons, in 24 months, spend $480, how much would 16 persons spend in 8 months? A. $320.
4. If a man travel 60 miles in 5 days, travelling 3 hours each day, how far will he travel in 10 days, travelling 9 hours each day?
of 60 =12, and of 12=4 miles, the distance which he travels in 1 hour; then, 4 miles X 9 hours=36 X 10 days= 360 miles, the Ans.
It will oftentimes be found convenient to make a statement, as in Simple Proportion. Take the last example.-In solving this question, we found the answer, which is miles, depended on two circumstances; the number of days which the man travels, and the number of hours he travels each day.
Let us, in the first place, find how far he would go in 5 days, supposing he travelled the same number of hours each day. The question will then be,
If a man travel 60 miles in 5 days, how many miles will he travel in 10 days? This will give the following proportion, to which, and the next following proportion, the answers, or fourth terms, are to be found by the Rule of Three; thus,
5 days : 10 days :: 60 miles : which gives, for the fourth term, or answer, 120 miles. In the next place, we will consider the difference in hours; then the question will be,
If a man, by travelling 3 hours a day for a certain number of days, travel 120 miles, how many miles will he travel, in the same number of days, if he travel 9 hours a day; which will give the following proportion :
3 hours : 9 hours :: 120 miles : which gives for the fourth term, or answer, 360 miles.
In performing the foregoing oxamples, we, in the first operation, multiplied 60 by 10, and divided the product by 5, making 120. In the next operation, we multiplied 120 by 9, and divided the product by 3, making 360, the answer. But, which is precisely the same thing, we may multiply the ho by the product of the multipliers, and divide this result by the product of the divisors; by which process the two statements may be reduced to one ; thus,
5 days : 10 days
:: 60 miles : miles. In this example, the procluct of the multipliers, or second terms, is 9 X 10 = 90; and the product of the divisors, or first terms, is 3 x 5=15; then, 60 X 90 =5400 - 15= 360 miles, the Ans., as before.
Note.--It will be recollected, that the ratio of any two terms is the second divided by the first, expressed either as a fraction, or by its equal whole number. Or,
comparing the different terms, we see that 60 miles has the same proportion to the fourth term, or answer, that 5 days has to 10 days, and that 3 hours has to 9 hours; hence we may abbreviate the process, as in Simple Proportion, by multiplying the third terns by the ratio of the other terms, thus:
The ratio of 5 to 10 is 2, and of 3 to 9 is g=3. But multiplying 60 miles by the product of the ratios 2 and 3, that is, 6, is the same as multiplying 60 by them separately; then, 6X 60=360 miles, Ans., as before.
Note.—This method, in most cases, will shorten the process very materially, and in no case will it be any longer; for, when the ratios are fractious, multiplying the third term by them (according to the rule for the multiplication of fractions) will, in fact, be the same process as by the other method.
Q. From the preceding remarks, wnat aoes Compound Proportion, or Double Rule of Three, appear to be ?
A. It is finding the answer to such questions as would require two or more statements in Simple Proportion; or, in other words, it is when the relation of the quantity required, to the given quantity of the same kind, depends on several circumstances combined.
Q. The last question was solved by multiplying the third term hy the product of the ralios of the other terms; what, then, may the proda
be called, which results from multiplying two or more ratios together? A. Compound Ratio.
From the preceding remarks we derive the following
Q. What number do you make the third term ?
A. That which is of the same kind or denomination with the answer.
Q. How do you arrange all the remaining terms ?
A. Take any two which are of the same kind, and, if the answer ought to be greater than the third term, make the greater the second term, and the smaller the first; but, if not, make the less the second term, and the greater the first ; then take any other two terms of the same kind, and arrange them in like manner, and so on till all the terms are used ; that is, proceed according to the directions for stating in Simple Proportion.
Q. How do you proceed next?
A. Multiply the third term by the continued product of the second terms, and divide the result by the continued product of the first terms; the quotient will be the fourth term, or answer.
Q. How may the operation, in most cases, be materially shortened?
A. By multiplying the third term by the continued product of the ratios of the other terms.