In this example, it will be seen, that there is an error in the solution by the Connecticut rule, of $605.567-; and in that by the Massachusetts rule, of $643.363+. By the Massachusetts rule, likewise, the error is $37.797-great est. This comparison of results is sufficient to show which of the rules is preferable. If we examine the rules themselves, we see that this difference arises from the fact that, in certain cases, interest is allowed on payments for a time less than a year, by the Connecticut rule, whereas, it is never allowed at all by that of Massachusetts. Besides the erro The subject of endorsements has been an unfortunate one. neous reasoning, of which we have been speaking on the subject of the common rule, a writer has asserted of the Connecticut mode, that when the payment is smaller than the interest due, the excess of interest becomes united with the principal, and draws interest as a part of it. To illustrate which objection, the following example has been instanced. "Messrs. Brown and Ives, Providence, loan to government $200,000 at 6 pr. ct. interest. The avails of the post-office, in said town, being about $1,000, are to be annually applied to the payment of the debt, and to operate as endorsements. The interest of $200,000 for one year is $12,000, and the amount, $212,000. Then, by the Connecticut mode, the payment of $1,000 is to be deducted from $212,000, which leaves $211,000 on interest for the second year; that is, $11,000 MORE THAN THE ORIGINAL DEBT IMMEDIATELY ACCUMULATING INTEREST, besides the $1,000, which is at their disposal." A slight inspection of the rule will convince any one that this objection is groundless. It is there particularly directed that the $200,000 shall be the contin ued principal, and that the $11,000 shall not draw interest. A clause of the same import, though in different language, and the language, we believe of the Superior Court of the State, may be found attached to this rule in the very book from which the above example is taken, which renders the writer's error the more remarkable. Messrs. Brown and Ives in the above example only derive advantage from the $1,000, the remaining $11,000 of interest, lying unproductive. We have been thus full on this subject, because, by the generality of people, very incorrect notions are entertained with respect to it. § XC. is 9 ? RATIO. MENTAL EXERCISES. 1. What part of 2 is 1? (See § XXVIII.) is 3? is 5? is 7? 2. What part of 3 is 2? of 7 is 5? of 17 is 16? of 19 is 18? 3. What part of 4 is 5? of 7 is 8? of 6 is 9? is 12? is 18? We have already mentioned (§ xxvII.) that there is another name for the Fraction expressing what part of one number, another is; and that this name is RATIO. The Fraction is said to express the Ratio, which exists between its denominator and its numerator; er the Ratio of its denominator to its numerator. The meaning of the word ratio, is RELATION. a ratio bet een two numbers, we mean, the tween them, as it respects quantity, or size. When we speak of relation existing beRATIO, then. IS A MEAS URE OF THE COMPARATIVE SIZE OF TWO NUMBERS, OR QUANTITIES. In the Fraction expressing ratio, one of the numbers is the denominator, and the other the numerator. These two numbers in. stead of being written as a Fraction, are often writte one after the other, with a colon between them, thus;, which expresses the ratio of 3 to 4, is written 3: 4. When ratio is expressed thus, the denominator of the Fraction is written first, and the numerator last. On this account, THE DENOMINATOr is called the ANTECEDENT OF THE RATIO, and THE NUMERATOR IS CALLED THE CONSEQUENT OF THE RATIO. EITHER NUMBER TAKEN SEPARATELY IS CALLED A TERM OF THE RATIO, AND BOTH TOGETHER ARE CALLED A COUPLET. Form ratios from the following questions, 4. What part of 8 is 4? Fractional ratio =8:4 Frac. ratio 2 of 7 is 5 5. What part of 6 is 12? 6. What part of 3 is 18? 14? of 6 is 24? of 9 is 36? of 15 is 30 6 is 3 of 9 is 12? of 3 is 2? of 4 is 7? 8 is 64 of 15 is 75? of 5 is 30? 6:121 of 9 is 37 of 9 is 18? of 7 is of 11 is 33 of 8 is 2? of of 16 is 32 of 2 is 16? of Many of the Fractional ratios obtained above, may be reduced to lower terins, or to whole or mixed numbers. This may evidently be done, without affecting the ratio, since ratio is only a measure of comparative size, and has nothing to do with the actual magnitude of the quantities, between which it exists. Thus a bullet may have the same ratio to a small particle of matter, which the planet Jupiter has to our Earth, notwithstanding that the actual size of these quantities is so different. Hence, THE RATIO BETWEEN TWO NUMBERS, OR QUANTITIES, MAY BE EXPRESSED BY THE QUOTIENT OF ONE DIVIDED BY THE OTHER. Perhaps the pupil need not be informed, that, RATIO CAN ONLY For if things be not of the same kind, we cannot have a measure of comparative size be- . tween them. Thus, though we may speak of one weight as being twice as great as another, we cannot say that one body weighs twice as much as another costs. EXIST BETWEEN QUANTITIES OF THE SAME KIND. It may be here remarked that English mathematicians have usually made the Antecedent the Numerator, and the Consequent the Denominator of the Frac. tional ratio, while the French have employed the mode given above. The English mode is obviously most in accordance with reason, but the popularity of the French mathematical writings in this country, renders it almost necessary for us to follow them. It is not of essential importance to our calculations, which is used; but of considerable consequence, as it respects the clearness of our conceptions, that the principles of ratio should be thoroughly understood. Now when 2 is compared with 5 in respect to quantity, a relation of 2 to 5 is discovered; and this is, that the quantity 2, considered in relation to the quantity 5, is only two fifths as great. then expresses the true ratio, between 2 and 5, considering ratio to be, as defined, a measure of the comparative size of quantities. This is the English mode. By the French mode 2 is to 5 in the ratio. No argument is necessary to show that 2 does not bear the relation, (in respect to quantity,) of five halves, to 5; or, in other words, that 2 is not five halves of 5. But this argument is derived from the meaning of the word ratio. When this is clearly understood, it matters not which term is put first, nor which last. The French mode has some advantages in practice, which will, perhaps, make it universally prevalent. PROPORTION. MENTAL EXERCISES. § XCI. 1. What is the ratio of 2 to 4? (Reduce to its lowest terms.) 2. What is the ratio of 7 to 14 ? Ans. or 1:2. 3. Which is the greatest, the ratio 2:4 or the ratio 7: 14? 8. Which is greatest, 2:3 or 200: 300? 5:6 or 125: 150? In the preceding examples, we have several instances of equal ratios, between very different numbers. When two ratios, thus brought together, are equal, they are said to constitute a PROPORTION. Hence, A PROPORTION IS AN EXPRESSION DENOTING THE EQUALITY BETWEEN RATIOS. THE NUMBERS, OR QUANTITIES, WHICH MAKE UP A PROPORTION, ARE CALLED PROPORTIO ALS. 1 THE FIRST AND LAST TERMS ARE CALLED THE EXTREME TERMS, OR EXTREMES; AND THE SECOND AND THIRD, OR MIDDLE TERMS, are calLED MEAN TERMS, OR MEANS. As two numbers are necessary to a ratio, and two ratios to a proportion, it is evident, that, under ordinary circumstances, there must be FOUR numbers, or proportionals, in every proportion. As, however, the consequent of the first couplet may sometimes be the same with the antecedent of the second, a proportion may exist with only THREE numbers. Thus, 3:6 9:18 and 3:6 6:12 are two proportions, in the first of which, we have four proportionals, and in the second, three, because the middle term, 6, is used twice. Three numbers thus forming a proportion, are said to be in CONTINUED PROPORTION. Instead of = the usual sign of equality, two colons, or four points, thus, are used to signify equality of ratios. Thus, instead of 3: 69: 18, we write 36:9: 18. This proportion is read thus; 3 has the same ratio to 6, that 9 has to 18; or more briefly, 3 is to 6 as 9 to 18. These ratios expressed fractionally are. As these two Fractions are equal to one another, it is evident that if I were to reduce them to a common denominator, their numerators would become equal. Now (§ XLIV.) their numerators would be 6×9 and 3x18. For "each numerator is multiplied into every denominator, except its own," in reducing Fractions to a common denominator. Then 6×9 ought to equal 3×18. On trial, we find this the case, for 6X9 54, and 3X18=54. And, as this will always be true, we may conclude that, IN A PROPORTION, THE PRODUCT OF THE EXTREMES, IS EQUAL TO THE PRODUCT OF THE MEANS. This principle, we shall see, is of great use in solving many questions. Thus, 11. If 2 yards of cloth cost 5 dollars, what will 6 yards cost? 6 yards are 3 times as much as 2 yards. Therefore the price of 6 yards will be 3 times as much as the price of 2 yards. That is, the ratio between the prices, is the same as that between the quantities, or numbers of yards. If we knew both prices, then, we should have a complete proportion. As it is, one term is wanting. The proportion is, 2 yards: 6 yards :: price of 2=$5: price of 6=$........ or 2 6: 5:... The term wanting is an extreme. Now, by the principle just laid down, we know that the two means multiplied together, thus 6×5, produce just as great a product as the two extremes multiplied together, thus, 2 the wanting term. But 6X5-30. Therefore, .2×the wanting term=30. In other words 30 is twice the wanting Hence 3015 the price of 6 yards. The proportion is now rendered complete, by writing the 15 with it, thus, term. yds. yds. $ $ 2 .: 6 :: 5 : 15 The same method might be pursued to find any other term that should be wanting. Thus, 12. If $5 will buy 2 yards, what will $15 buy? As 15 is 3 times 5, it will buy 3 times as many yards. That is, there is the same ratio between the yards bought as between the prices. Thus, $ $ yds' yds. : 15 :: 2 : But, by the principle above, 15×2=5× the wanting term. 15X2=30. Therefore, 5× the wanting term=30. But In other words, 30 is five times the wanting term. Hence 36=the number of yards required. Then the completed proportion is $3 5 $ yds. yds. 15: 2: 6 which is the same as in the last example, except that the ratios are differently arranged. So, if the 5 were not known, in the same proportion, I should multiply 2 into 15, and divide by 6 to obtain it; and if the 2 were wanting, I should multiply 5 into 6 and divide by 15. Hence, to find an unknown term in a proportion, IF IT BE AN EXTREME DIVIDE THE PRODUCT OF THE MEANS BY THE OTHER EXTREME; IF A MEAN, divide the PRODUCT OF THE EXTREMES BY THE OTHER MEAN. This rule, for finding, on the principles of proportion, a fourth number, from three given numbers, is commonly called the RULE OF THREE. It is distinguished into two kinds, both comprehended in the above general principle. The difference will be best understood after comparing the following examples. 13. If 3 men dig 15 rods of ditch in a day, how much would 6 men dig in the same time? It is evident that the rods will increase in the same ratio as the men; that is, that 16 rods has to the answer the same ratio, that 3 has to 6, or 3: 6:: 16: 6×16÷3-32. Ans. In this question the more men there are, the more rods are dug. It is therefore called the case in which more requires more, and belongs to the RULE OF THREE DIRECT. WHEN ONE RATIO.INCREASES AS ANOTHER INCREASES, OR DIMINISHES AS ANOTHER DIMINISHES, THE TWO ARE SAID TO CONSTITUTE A DIRECT PROPORTION. 14. If 6 men dig 32 rods of ditch in a day, how much would 3 men dig in the same time? The rods diminish in the same ratio as the men. Therefore 32 has to the answer the same ratio as 6 to 3, or 6:3:32: 3×32÷6=16, Ans. In this question, the fewer men, the fewer rods. It is therefore called the case in which less requires less, and belongs to the RULE OF THREE DIRECT. 15. If 3 men can dig a ditch in 16 days, how long will it take 6 men? The more men the quicker the work will be done; that is, the more men the less time. Then if the number of men be doubled, the number of days will be halved, or if the number of men te increased in any ratio, the number of days will be diminished in the same ratio; or, as the ratio of the men increases, the ratio of the days diminishThe ratio, then, of 3 to 6, is directly the reverse of the ratio of 16 to the answer. Then, if I turn about, or invert this ratio of 3 to 6, making it 6:3, it will be the same which 16 has to the answer. Thus, 6:3: 16:3×16--6-8 Ans. es. In this case, the more men the fewer days. It is therefore, called the case in which more requires less, and on account of the inversion above mentioned, the rule by which it is solved is called the RULE OF THREE INVERSE. WHEN ONE RATIO INCREASES AS ANOTHER DIMINISHES, OR DIMINISHES AS ANOTHER INCREASES, THE TWO ARE SAID TO CONSTITUTE AN INVERSE PROPORTION. |