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PROPORTION is the likeness or equalities of ratios. Thus, because 4 has the same ratio to 8, that 6 has to 12, we say such numbers are proportionals.
If, therefore, any four numbers whatever be taken, the first is said to have the same ratio or relation to the second, that the third has to the fourth, when the first number, or term, contains the second, as many times, as the third contains the fourth ; or, when the second contains the first, as many times, as the fourth does the third. Thus, 9 has the same ratio to 3, that 12 has to 4, because 9 contains 3, as many times, as 12 does 4. And 10 has the same ratio to 5, that 12 has to 6, because 10 contains 5, as many times, as 12 does 6. Ratios are represented by colons ; and equalities of ratios by double colons.
The first and third terms are called antecedents, and the second and fourth are called consequents ; also, the first and fourth terms are called extremes, and the second and third are called means.
Whatever four numbers are proportionals, if their antecedents and consequents be multiplied or divided by the same numbers, they are still proportionals; and, if the terms of one proportion be multiplied or divided by the corresponding terms of another proportion, their products and quotients are still proportionals.
If the product of the extremes be equal to the product of the means, it is evident, that if any three of the four proportionals be given, the other may be obtained ; for, if the product of the means be divided by one of the extremes, the quotient will be the other extreme ; and, if the product of the extremes be divided by one of the means, the quotient will be the other mean. Hence the following
State the question by making that number, which is of the same name or quality as the answer required, the third term;
then, if the answer required is to be greater than the third term, make the second term greater than the first ; but if the answer is to be less than the third term, make the second less than the first.
Reduce the first and second terms to the lowest denomination mentioned in either, and the third term to the lowest denomination mentioned in it.
Multiply the second and third terms together, and divide their product by the first, and the quotient is the answer in the same denomination to which the third is reduced.
If any thing remains, after division, reduce it to the next lower denomination, and divide as before.
If either of the terms consists of fractions, state the question as in whole numbers, and reduce the mixed numbers to improper fractions, compound fractions to simple ones, and invert the first term, and then multiply the three terms continually together, and the product is the answer to the ques. tion.
Or, the fractions may be reduced to a common des nominator ; and their numerators may be used as whole numbers. For when fractions are reduced to a common denominator, their value is as their numerators.
Note 1. It may be observed in Proportion, that the third term is the quantity, whose price or value is wanted, and that the second term is the value of the first; when, therefore, the second term is multiplied by the third, the product is as much more than the answer, as the first term is greater than unity; therefore, by dividing the product by the first term, we have the value of the quantity required.
Note 2. The pupil should perform every question by analysis, previous to his performing it by Proportion. 1. If 7lbs. of sugar cost 56 cents, what cost 36lbs. ?
In stating this question, we make 7 : 36 :: 56
56 cents the third term, because the 5 6
answer will be in cents. And, as 216
we perceive from the nature of the 180
question, that the answer or fourth
term will be more than 56 cents, we 7) 2016
know, that of the other two terms, $2.88 Ans.
the second must be larger than the
first, we therefore make 36lbs. the second term, and 71bs. the first term.
To perform this question by analysis, we say, If 7lbs. cost 56 cents, one lb. will cost 4 of 56 cents, which are 8 cents. Then, if llb. cost 8 cents, 36lbs. will cost 36 times as much ; that is, 36 times 8 cents, which are $ 2.88 Ans. as before. 2. If 76 barrels of flour cost $ 456, what cost 12 barrels?
As the answer to this ques76 : 1 2 :: 4 5 6 tion will be in dollars, we place 456
$ 456 in the third term ; and, 72
as the answer or fourth term 60
must be less than $ 456, be48
cause 12 barrels will cost less 765472(72 Ans.
than 76 barrels, we must, of 5 32
the other two terms, make the
less the second term, and the 152 152
larger the first term ; that is, 12 barrels must be the second term,
and 76 barrels the first term. We analyze this question by saying, if 76 barrels cost $ 456, 1 barrel will cost 7 of $ 456, which is $ 6. Then, if I barrel cost $ 6, 12 barrels will cost 12 times as much, that is, $ 72 Ans. as before. 3. If 3 men can dig a well in 20 days, how long would it take 12 men ? days.
As the answer will 1 2 : 3 :: 20
be in days, so the 3
third term will be 12 ) 60 ( 5 days, Ans. days. As 12 men will 60
dig the well in less time than 3
men, therefore, the second term will be less than the first.
By analysis. If 3 men dig the well in 20 days, it will take one man 3 times as long, that is, 60 days. Again, we say, If one man dig the well in 69 days, 12 men would dig it in of 60 days, that is, 5 days, Ans. as before. 4. If Albs. of beef cost 36 cents, what cost 87lbs. ?
Ans. $ 7.83. 5. What cost 9 gallons of molasses; if 63 gallons cost $ 14.49 ?
Ans. $ 2.07
6. What cost 97 acres of land, if 19 acres can be obtained for $ 337.25 ?
Ans. $ 1721.75. 7. If a man travel 319 miles in 11 days, how far will he travel in 47 days?
Ans. 1363 miles. 8. If 7lbs. of beef will buy 4lbs. of pork, how much beef
will be sufficient to buy 48lbs. of pork ? Ans. 84lbs. 9. Paid for 87 tons of iron $ 5437.50, how many tons will $ 7687.50 buy ?
Ans. 123 tons. 10. When $ 120 are paid for 15 barrels of mackerel,
what will be the cost of 79 barrels ? Ans. $ 632. 11. If 9 horses eat a load of hay in 12 days, how many horses would it require to eat the hay in 3 days ?
Ans. 36 horses. 12. When $ 5.88 are paid for 7 gallons of oil, what cost 27 gallons ?
Ans. $ 22.68. 13. When $ 10.80 are paid for 9lbs. of tea, what cost 147lbs. ?
Ans. $ 176.40. 14. What cost 27 tons of coal, when 9 tons can be purchased for $ 85.95 ?
Ans. $ 257.85. 15. If 15 tons of lead cost $ 105, what cost 765 tons ?
Ans. $ 5355.00. 16. If 16hhd. of molasses cost $ 320, what cost 176bhd ?
Ans. $ 3520.00. 17. If 15cwt. 3qr. 17lb. of sugar cost $ 124.67, what cost 76cwt. 2qr. 191b. ?
Ans. $ 601.09.
Note. When any of the terms is a compound number, it must be reduced to the lowest denomination mentioned in it; therefore, the hundred weights, quarters, &c., must be reduced to pounds, before the terms are multiplied and divided by each other. 18. If 7s.6d. of the old Pennsylvania currency are equal to $ 1, what is the value of £76. 19s. 11d. ?
Ang. $ 205.327. 19. If 8s. of the old currency of New York are equal to $ 1, what is the value of £ 19. 19s. 8d.
Ans. $ 49.95+. 20. If 4s. 8d. of the old currency of South Carolina and Georgia are equal to $1, what is the value of £176. 18s. 4d. ?
Ans. $ 758.21+ 21. As 4s. 6d. sterling of the English currency are equal to one dollar in the United Sates, how many dollars are there in £769. 18s. 9d. ?
Ans. $ 3421.94+.
22. If the cars on the Boston and Portland Railroad go one mile in 2 minutes and 8 seconds, how long will they be in passing from Haverhill to Boston, the distance being 32 miles ?
Ans. lh. Amin. 16sec. 23. If one acre of land cost $ 37.86, what cost 144A. 3R. 17p. ?
Ans. $ 5484.25+: 24. If a man travels 3m. 7fur. 18rd. in one hour, how far will he travel in 9h. 45min. 19sec. ?
Ans. 38m. 2fur. 32+rd. 25. A fox is 96 rods before a greyhound, and, while the fox is running 15 rods the greyhound will run 21 rods ; how far will the dog run before he can catch the fox ?
Ans. 336 rods. 26. If 5 men can reap a field in 12 hours, how long would it take them if 4 men were added to their number?
Ans. 6 hours. 27. Ten men engage to build a house in 63 days, but 3 of their number being taken sick, how long will it take the rest to complete the house ? Ans. 90 days. 28. If a 4 cent loaf weighs 5 oz. when flour is $ 5 per barrel, what should it weigh when flour is $ 7.50 per barrel ?
Ans. 34 oz. 29. If 7 men can mow a field in ten days, when the days are 14 hours long, how long would it take the same men to mow the field, when the days are 13 hours long?
Ans. 101% days. 30. If 291bs. of butter will purchase 40lbs. of cheese, how many pounds of butter will buy 79lbs. of cheese
Ans. 571/b. 31. If of a yard cost of a dollar, what will it of a yard cost?
$. f: 11 :: $; $X 11 X = = $0.764 Ans. Note. Let the pupil explain, why the first term is inverted in the operation. 32. If It of a gallon of oil cost of a dollar, what cost of a gallon ?
Ans. $ 1.123. STATEMENT.
CANCELLED. gal. gal. $. a 7 9 1:}::; Х Х == $ 1.127 Ans.
7 8 XX