OPERATION. Pair. Pair. Cents. 8 > 24, 63 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 73 is a fraction, it would be more convenient to multiply by 24 first, and divide by 8 afterwards, 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. 24 252 126 8) 1512 Ans. $1,89 21. If 3 yards of cloth cost 24 cents, what will 6 yards cost? OPERATION. Yds. Yds. Cts. 24 × 6=144+3=48, the Ans. 3, 6, 24 6 3) 144 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 saine as before. This is a much shorter process; and, could we discover the principle, it would oftentimes render operations very simple and short naturally be led to the consideration of ratio, or relation; that is, the relation which necessarily exists between two or more numbers. Ans. 8,48 In searching for this, we shall 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. 17. oz. to 12 oz.? Q. What part of 3 oz. is 12 oz.? or,what is the ratio of 3 oz. A. 12: 4, ratio. Q. What part of 4 yards is 9 yds. ? or, what is the ratio of 4 to 9? A. 24. 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 58.60 d.; therefore, 88, the 3. What is the ratio of £1 to 15 s.? A. 1, the ratio. 4. What is the ratio of 2 to 3? A. 3. Of 4 to 20? A.5. Of 20 to 4? A. . Of 8 to 63? A. 73. Of 200 to 900? A. 41. Of 800 to 900? A. 1. 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 105; then, 5 x 840, Ans. But by stating the question us before, we have the following proportions : Q. When, then, 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 and 8, being in the middle, are cailed 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 3, multiplied together, thus; 2 X 4080, and 10 × 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,, (=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 and become ff and f; in doing which, we multiply the nu merator 40 (one extreme) by the denominator 2 (the other extreme), also the nuinerator 10 (one mean) by the denominator, 8, (the other mean); hence the reason of this equality. When, then, any four num bers 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 ast example: Melons. Melons. Cents. Cents. : 10 :: 8: 40 Multiplying together 8 and 10, the two means, makes 80; then 80 ÷ 40, 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. 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 exemplify 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 proportion thus: in this example. The ratio of 10 to 40 is 484, that is, 40 horses will consume 4 times as many bushels as 10; then 4 × 30 bu. = 120 bushels, the 4th term, or extreme, as before. 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. Q. How, then, may it be defined? A. It is the process of finding, by the help of three given terms, a fourth term, between which and the third term there is the same ratio or proportion as between the second and first terms. It will sometimes be necessary to change the order of the terms; but this may be determined very easily by the nature of the question, as will appear by the following example:24. If 8 yards of cloth cost $4, what will 2 yards cost? OPERATION. Yds. Yds. $ 8:2:4 2 8)8 $1 In this example, since 2 yards will cost a less sum than 8 yards, we write 2 yards for one mean, which thus becomes the multiplier, and 8 yards, the known extreme, for the divisor; for the less the multiplier and the greater the divisor, the less will be the quo tient; then, 2X4=8+8= $1, Ans. But multiplying by the ratio will be much easier, thus; the ratio of 8 to 2 is =; then, 4 × = $1, Ans., as before. From these illustrations we derive the following RULE. Q. Which of the three given terms do you write for a third term? A. That which is of the same kind with the answer. Q. How do you write the other two numbers, when the answer ought to be greater than the third term? A. I write the greater for a second term, and the less for a first term. Q. How do you write them when the answer ought to be less than the third term? A. The less for a second term, and the greater for a first term. Q. What do you do when the first and second terms are not of the same denomination? A. Bring them to the same by Reduction Ascending or Descending. Q. What is to be done when the third term consists of more than one denomination? A. Reduce it to the lowest denomination mentioned, by Reduction. |