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10. If 1 pair of gloves cost 75 cents, what will 1 dozen pair cost?-9. What will 1 doz. ?-1350. What will 2 doz. ?18. What will 2 doz. ?-2250. What will 3 dəz. ?–27.

A. $30.

11. If 3 doz. pair of shoes cost 27 dollars, what will 1 pair cost?-75. What will 2 doz.?-2250. What will 2 doz.? 18. What will 11⁄2 doz. ?-1350. What will 1 doz. ?–9.

A. $63,75. 12. If 5 tons of hay will keep 25 sheep over the winter, how many sheep can be kept on 7 tons, at the same rate ?-35. On 8 tons ?-40. On 15 tons ?-75. On 60 tons ?-300. On 80?-400. A. 850.

13. Boarding at $2,25 a week, how long will $9 last me?-4. How long will $13,50?-6. How long will $18?-8. How long will $20,25?-9. How long will $49,50?-22. A. 49 weeks.

14. If a man receive $50 for 2 months' wages, what is that a year?-300. What will 8 months' come to?-200. 16 months come to?-400. 1 year's come to?-450. 2 years' come to ?600. 10 years' come to ?-3000. A. $4950.

15. What will 6 pieces of cloth, each piece containing 20 yards, come to, at $1,50 a yard ?-180. What will 1 piece come to ?-30. What will 3 pieces ?-90. What will 5 pieces?-150. What will 10 pieces?-300. A. $750.

16. Bought 5 hhds. of rum, each containing 60 gallons, for $2 a gallon; what do they come to ?-600. What will 4 hhds. come to ?-480. What will 20 hhds. ?-2400. A. $3480.

17. William's income is $1500 a year, and his daily expenses are $2,50; how much will he have saved at the year's end? A. $587,50.

18. If William's income had been $2000, how much would he have saved?-108750. If $2500?-158750. If $3600 ?–268750. If $4000?-308750. A. $8450.

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19. If a hhd. of molasses cost $20,16, how much is it a gallon (Divide by the number of gallons in a hhd.)-32. How much is it a quart? (Divide by the number of quarts in a hhd:)8. How much is it a pint ?-4. How much is it a gill ?-1.

A. 45 cents.

The foregoing questions have been solved by a method termed analysis. This method is thought to accord with the natural operations of the human mind. Men in business scarcely recognise any other. The formality of statements is rarely if ever inade by them; and, when it is made, they do it more for the sake of testing the correctness of the other method, than for any prac. tical purpose. They may have adopted a statement in the commencement of their business, from the circumstance of having been taught it at school; but the longer they continue in business, the less occasion they have for it. There is another method, which consists in ascertaining the ratio or relation which one number has to another. This is used more or less by all, but more extensively by scientific men, and those well versed in mathematical principles.

20. If 8 pair of shoes cost 63 cents, what will 24 pair cost? of 6373 cents, the price of 1 pair, which we multiply by 24 to get the price of 24 pair; thus, 24 × 7=$1,89. But since 7 is a fraction, it would be more convenient to multiply

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by 24 first, and divide by 8 after wards, as this cannot make any difference; and that we may make no mistake in the operation, we will make a statement by writing the 63 cents on the right, as a third term (see operation); on the left of which we write the multiplier, 24, as a 2d term, and, as a first term, the divisor, 8: then, 63 X 24= 15128 $1,89, the Answer, as before.

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21. If 3 yards of cloth cost 24 cents, what will 6 yards cost?

OPERATION.

Yds. Yds. Cts. 3, 6, 24

6

3) 144

24 x 6144348, the Ans

Or, as we know that 6 yards cost 2 times as much as 3 yds., that is, §=2, by simply multiplying 24 by 2, it makes 48, the answer, the same as before. This is a much shorter process; and, could we discover the principle, it would oftentimes render operations very simple and short In searching for this, we shall naturally be led to the consideration of ratio, or relation; that is, the relation which necessarily exists between two a more numbers.

Ans. 8,48

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Q. What is the finding what part one number is of another called? A. Finding the ratio, or relation of one number to another.

Q. What is ratio, then?

A. The number of times one number or quan tity is contained in another.

Q. What part of 10 is 9? or, what is the ratio of 10 to 9 ?
A. 1.

9

Q. What is the ratio of 17 to 18?

A. 19.

Q. What is the ratio of 18 to 17?

A. 1.

Q. What part of 3 oz. is 12 oz.? or,what is the ratio of 3 oz. to 12 oz.? A. 12

4, ratio.

Q. What part of 4 yards is 9 yds. ? or, what is the ratio of 4 to 9? A.

21.

Q. Hence, to find the ratio of one number to another, how do you proceed?

A. Make the number which is mentioned last (whether it be the larger or smaller), the numerator of a fraction, and the other number the denominator; that is, always divide the second by the first

1. What part of $1 is 50 cents? or, what is the ratio of $1 to 50 cents?

A. $1100 cents; then, the ratio, Ans. 2. What part of 5 s. is 2 s. 6 d.? or, what is the ratio of 5 s. to 2 s. 6 d.?

2s. 6d.

ratio, Ans.

30 d., and 5s. 60 d.; therefore, 8, the

A. 15, the ratio . Of 4 to 20? A.5.

3. What is the ratio of £1 to 15 s.? 4. What is the ratio of 2 to 3? A. Of 20 to 4? A. . Of 8 to 63? A. 77. Of 200 to 900? A.44 Of 800 to 900? A. 13. Of 2 quarts to 1 gallon? A. 2.

Let us now apply the principle of ratio, which we were in pursuit of, to practical questions.

PROPORTION. 22. If 2 melons cost 8 cts., what will 10 cost? It is evident, that 10 melons will cost 5 times as much as 2; that is, the ratio of 2 to 10 is 5; then, 5 x 8=40, Ans. But by stating the question as before, we have the following proportions :

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Q. When, hen, numbers bear such relations to each other, what are the numbers said to form?

A. A proportion.

Q. How may proportion be defined, then?

A. It is an equality of ratios.

Q. How many numbers must there be to form a ratio?

A. Two.

Q. How many to form a proportion?

A. At least, three.

To show that there is a proportion between three or more numbers, we write them thus:

Melons. Melons. Cents. Cents.

2 : 10: 8: 40,

which is read, 2 is to 10 as 8 is to 40; or, 2 is the same part of 10 that 8 is of 40; or, the ratio of 2 to 10 is the same as that of 8 to 40.

Q. What is the meaning of antecedent?

A. Going before.

Q. What is the meaning of consequent ?

A. Following.

Q. What is the meaning of couplet?

A. Two, or a pair.

Q. What may both terms of a ratio be called?

A. A couplet.

Q. What may each term of a couplet be called, as 3 to 4. A. The 3, being first, may be called the antecedent; and the 4, being after the 3, the consequent.

Q. In the following proportion, viz. 2 : 10 :: 8: 40, which are the antecedents, and which are the consequents?

A. 2 and 8 are the antecedents, and 10 and 40 the consequents.

Q. What are the ratios in 2: 10 :: 8: 40?

Q. In the last proportion, 2 and 40, being the first and last terms, are called extremes; and 10 an: 8, being in the middle, are called the means. Also, in the same proportion, we know that the extremes 2 and 40, multiplied together, are equal to the product of the means, 10 and 8, multiplied together, thus; 2X 40=80, and 10 X 880. Let us try to explain the reason of this. In the foregoing proposition, the first ratio, 2, (= 5,) being equal to the second ratio, 4, (=5,) that is, the fractional ratios being equal, it follows, that reducing these frac tions to a common denominator will make their numerators alike; thus, 8 become and ; in doing which, we multiply the nu

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merator 40 (one extreme) oy the denominator 2 (the other extreme). also the numerator 10 (one mean) by the denominator, 8, (the other mean); hence the reason of this equality. When, then, aay foar numbers are proportional, what may we learn respecting the product of the extremes and means?

A. That the product of the extremes will always be equal to the product of the means.

Hence, with any three terms of a proportion being given, the fourth or absent term may easily be found. Let us take the last example:

Melons. Melons. Cents. Cents.

: 10 :: 8: 40

Multiplying together 8 and 10, the two means, makes 80; then 80 49. the known extreme, gives 2, the other extreme required, or first term.

Ans. 2.

Again, let us suppose the 10 absent; the remaining terms are Melons. Melons. Cents. Cents.

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By multiplying together 40 and 2, the extremes, we have 80; which, divided by 8, the known mean, gives 10, the 2d term, or mean, required. Let us ex emplify this principle more fully by a practical example.

23. If 10 horses consume 30 bushels of oats in a week, how many bushels will serve 40 horses the same time?

In this example, knowing that the number of bushels eaten are in proportion to the number of horses, we write the propor tion thus:

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in this example. The ratio of 10 to 40 is 484, that sume 4 times as many bushels as 10; then 4 X 30 bu. term, or extreme, as before.

By multiplying together 40 and 30, the two means, we have 1200, which, divided by the known extreme, 10, gives 120; that is, 120 bushels, for the other extreme, or 4th term, that was requir ed. Let us apply the principle of ratio in finding the 4th term is, 40 horses will cou120 bushels, the 4th

Q. When any three terms of a proportion are given, what is the process of finding the fourth term called?

A. The Rule of Three.

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