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be expressed by a fraction in which the first number, (called the antecedent) is put for numerator, and the second number (called the consequenl) is put for denominator. Thus the ratio of 8 to 4 is . This is an improper fraction, and, changed to whole numbers, is 2 units. The ratio of 8 to 4, then, is 2. That is, 8 is twice 4, or stands to 4 in the relation of a duplicate or double.*

PROPORTION. When quantities have the same ratio, they are said to be proportional to each other. Thus the ratio of 2 to 4 is }, and the ratio of 4 to 8 is į ; 'that is, 1 has the same relation to 2, that 4 has to 8, and therefore these numbers are called proportionals. Again, 4 is the same portion or part of 8, as 10 is of 20, and therefore these numbers are called proportionals. A proportion, then is a combination of equal ratios.

Points are used to indicate that there is a proportion between numbers. Thus 4:8::9:18 is read thus; 4 has the same ratio to 8, that 9 has to 18. Or more briefly, 4 is to 8, as 9 to 18.

It will always be found to be the case in proportionals, that multiplying the two antecedents into the two consequents, produce the same product.

Thus, 2:4::6:12 Here let the consequent 4 be multiplied into the ante.

* The pupil needs to be forewarned that there is a difference be. tween French and English mathematicians in expressing ratio,

The French place the antecedent as denominator, and the conse. quent as numerator. The English, on the contrary, place the ante. cedent as numerator, and the consequent as denominator. It seems desirable that there should be an agreement on this subject, in school books at least. Two of the most popular Arithmetics now in use, have adopted the French method, viz. Colburn and Adams. It seems needful to mention this, that pupils may not be needlessly perplexed, if called upon to use different books.

The method used here, is the English ; as the most common, and as most consonant with perspicuity of language. For there seems to be no propriety in saying that the relation of 2 to 4 is 4-2. The ratio between these two numbers may be eithei 4-2 or 2-4, but the re. lation of 2 to 4, to use language strictly, can be nothing but 2.4.

How is the ratio between numbers expressed ?

cedent 6, and the product is 24; and let the antecedent 2 be multiplied into the consequent 12, and the product also is 24,

If then we have only three terms in a proportion, it is easy to find the fourth. For when we have multiplied one antecedent into one consequent, we know that the term left out is a number that, multiplied into the remaining term, would produce the same product. Thus let one term be left out of this proportion.

8 : 4 :: 12 : Here the consequent is gone from the last ratio. We multiply the antecedent 12 into the consequent 4, and the answer is 48. We now know that the term left out, is a number which, multiplied into the 8, would produce 48. This number is found by dividing 48 by 8, the answer is 6.

Whenever, therefore, a term is wanting to any proportion, it can be found by multiplying one of the antecedents by one of the consequents, and dividing the product by the remaining number. What is the number left out in this proportion ?

3 : 12 : : 24 : What is the number left out in this proportion?

9:8: : 27 : In a proportion, the two middle terms are called the means, and the first and last terms are called the extremes.

RULE FOR FINDING A FOURTH TERM IN A PROPORTION.

Multiply the means together, and divide the product by the remaining number.

It is on this principle, that what is commonly called the “ Rule of Three,” is constructed. By this process, we find a fourth term when three terms of a proportion are given.

Such sums as the following are done by this rule.
If 4 yards of broadcloth cost $12, what cost 9 yards ?

Now the cost is in proportion to the number of yards ; that is, the same ratio exists between the number of yards, as exists between the cost of each.

What is proportion? Having three terms of a proportion given, how can the fourth be found ?

:

:

:

Thus,--as 4 yards is to 9 yards, so is the cost of 4 yards to the cost of 9 yards. The proportion, then, is expressed thus :

yds. yds.
4 : 9

12 Here the term wanting, is the cost of 9 yards; and if we multiply the means together, and divide by the 4, the answer is 27 ; which is the other term of the proportion, and is the cost of 9 yards.

Again, if a family of 10 persons spend 3 bushels of malt a week, how many bushels will serve at the same rate when the family consists of 30 ?

Now there is the same ratio between the number of bushels eaten, as between the numbers in the family. That is, as is the ratio of 10 to 30, so is the ratio of 3 to the nurober of bushels sought.

Thus, 10: 30 :: 3 :

RULE OF PROPORTION; OR RULE OF THREE.

When three numbers are given, place that one as third term, which is of the same kind as the answer sought. If the answer is to be greater than this third term, place the greatest of the remaining number as the second term, and the less number as first term. But if the answer is to be less, place the less number as second term, and the greater as first.

In either case, multiply the middle and third terms together, and divide the product by the first. The quotient is the answer, and is always of the same order as the third term.

NOTE.--This rule may be used both for common, com. pound, and decimal numbers. If the terms are compound, they must be reduced to units of the lowest order men. tioned.

Many of the sums which follow will be better understood if performed by the mode of analysis, which has been explained and illustrated in a former part.

What is the Rule of Three ?

For example, we will take the first sum done by the rule of proportion.

If 4 yards of broadcloth cost $12, what cost 9 yards ?

We reason thus,If 4 yards cost $12, one yard must cost a fourth of $12. Therefore, divide $12 by 4, and we have the cost of one yard. Multiply this by 9, and we have the cost of 9 yards.

(It is usually best to multiply first, and then divide, and it has been shown that this is more convenient, and does not alter the answer.)

Let the following sums be done by the Rule of Proportion, and then explained by analysis.

1. Ifthe wages of 15 weeks come to 64 dols. 19 cts. what is a year's wages at that rate ? Ans. $222, 52 cts. 5 m.

2. A man bought sheep at $1.11 per head, to the amount of $51.6 ; how many sheep did he buy? Ans. 46.

3. Bought 4 pieces of cloth, each piece containing 31 yds, at 16s. 6d. per yard, (New England currency, what does the whole amount to in federal money? Ans. $341. When a tun of wine cost $140, what cost a quart?

Ans. 13 cts. 88 m. 4. A merchant agreed with his debtor, that if he would pay him down 65 cents on a dollar, he would give him up a riote of hand of 249 dollars 88 cts. I demand what the debtor must pay for his note ? Ans. $162.42 cts. 2 m.

5. If 12 horses eat 30 bushels of oats in a week, how many bushels will serve 45 horses the same time?

Ans, 1121 bushels. 6. Bought a piece of cloth for $48.27 cts. at $1.19 cts. per yard; how many yards did it contain ?

Ans. 40 yds. 2 qrs. jvoo 7. Bought 3 hhds. of sugar, each weighing 8 cwt. 1 qr. 12 lb. at $7.26 cts. per cwt. ; what come they to?

Ans. $182.1 ct. 8 m. 8. What is the price of 4 pieces of cloth, the first piece containing 21, the second 23, the third 24, and the fourth 27 yards, at $1.43 cts. a yard ?

Ans. $135.85 cts. 21+23+24+27=95 yds. 9. Bought 3 hhds. of brandy, containing 61, 62, 623

30

What is the method of doing the Rule of Three?

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gals. at $1 38 cts. per gallon. I demand how much they amount to?

Ans. $255.99 cts. 10. Suppose a gentleman's income is $1,836 a year, and he spends $3.49 cts. a day, one day with another, how much will he have saved at the year's end ?

Ans. $562.15 cts. 11. A merch't. bought 14 pipes of wine, and is allowed 6 months credit, but for ready money gets it 8 cents a gal. lon cheaper; how much did he save by paying ready money?

Ans. $141.12 cts. 12. Sold a ship for 5371. and I owned of her; what was my part of the money?

Ans. £201 78. 6d. 13. If te of a ship cost $718.25 cents, what is the whole worth?

5 : 781,25 : : 16 : $2500 -Ans. 14. If I buy 54 yards of cloth for £31. 10s. what did it cost per Ell English ?

Ans. 14s. 7d. 15. Bought of Mr. Grocer 11 cwt. 3 qrs. of sugar, at $8,12 per cwt. and gave him James Paywell's note for £19 7s. (New England currency) the rest I pay in cash; tell me how many dollars will make up the balance.

Ans. $30.91. 16. If a staff 5 feet long casts a shade on level ground 8 feet, what is the height of that steeple whose shade at the same time measure 181 feet?

Ans. 1133 ft. 17. If a gentleman has an income of 300 English guineas a year, how much may he spend, one day with another, to lay up 500 dollars at the year's end?

Ans. $2,46 cts. 5 m. 18. Bought 50 pieces of kerseys, each 34 Ells Fle. mish, at 8s. 4d. per Ell English ; what did the whole cost ?

Ans. £425. 19. Bought 200 yards of cambric for £90, but being damaged, I am willing to lose £7. 10s. by the sale of it; what must I demand

per

Ell English ? Ans. 10s. 3 d. 20. How many pieces of Holland, each 20 Ells Fie. I have for £23 8s. at 6s. 6d. per Eli English ?

Ans. 6

pieces. 21. A merchant bought a bale of cloth containing 240 yds. at the rate of $7} for 5 yards, and sold it again at the rate of $114 for 7 yards ; did he gain or lose by the bargain, and how much?

Ans. He gained $25,71 cts. 4 m. +

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