Again, 12:6: is a proportion. (Art. 496.) 12X6X. OBS. 1. The truth of this proposition may also be illustrated in the following manner; The numbers 2:3::6:9 are obviously proportional. (Art. 492.) Multiplying each ratio by 27, (the product of the denominators,) 2×27 6×27 The proportion becomes (Art. 21. Ax. 6.) 3 9 Dividing both the numerator and the denominator of the first couplet by 3, (Art. 191,) or canceling the denominator 3 and the same factor in 27, (Art. 221,) also canceling the 9, and the same factor in 27, we have 2×9=6×3. But 2 and 9 are the extremes of the given proportion, and 3 and 6 are the means; hence, the product of the extremes is equal to the product of the means. 2. Conversely, if the product of the extremes is equal to the product of the means, the four numbers are proportional; and if the products are not equal, the numbers are not proportional. 499. Proportion, in arithmetic, is usually divided into Simple and Compound. SIMPLE PROPORTION. 500. SIMPLE PROPORTION is an equality between two simple ratios. It may be either direct or inverse. (Arts. 479, 496, 497.) The most important application of simplé proportion is the solution of that class of examples in which three terms are given to find a fourth. 501. We have seen that, if four numbers are in proportion, the product of the extremes is equal to the product of the means. (Art. 498.) Hence, If the product of the means is divided by one of the extremes, the quotient will be the other extreme; and if the product of the extremes is divided by one of the means, the quotient will be the QUEST. Obs. If the product of the extremes is equal to the product of the means, what is true of the four numbers? If the products are not equal, what is true of them? 499 How is proportion usually divided? 500. What is simple proportion? What is the most Important application of it? 501. If the product of the means is divided by one of the extremes, what will the quotient be? If the product of the extremes is divided by one of the means, what will the quotient be? other mean. For, if the product of two factors is divided by one of them, the quotient will be the other factor. (Art. 156.) Take the proportion 8:4:6:3. So the product 8×346, one of the means; Now the product 8×364, the other mean. Again, the product 4×6-8=3, one of the extremes; And the product 502. If, therefore, any three terms of a proportion are given, the fourth may be found by dividing the product of two of thems by the other term. OBS. Simple Proportion is often called the Rule of Three, from the circumstance that three terms are given to find a fourth. In the older arithmetics, it is also called the Golden Rule. But the fact that these names convey no idea of the nature or object of the rule, seems to be a strong objection to their use, not to say a sufficient reason for discarding them. Ex. 1. If the product of the means is 84, and one of the extremes is 7, what is the other extreme, or term of the proportion? 2. If the product of the means is 54, and one of the extremes is 18, what is the other extreme? 3. If the product of the means is 720, and one of the extremes is 45, what is the other extreme? 4. If the product of the means is 639, and one of the extremes is 213, what is the other extreme? 5. If the first three terms of a proportion are 8, 12, and 16, what is the fourth term? Solution.-12×16=192, and 1928-24, the fourth term, or number required; that is, 8:12::16:24. 6. It is required to find the fourth term of the proportion, the first three terms of which are 36, 30, and 24. 7. Required the fourth term of the proportion, the first three terms of which are 15, 27, and 31. 8. Required the fourth term of the proportion whose first threa terms are 45, 60, and 90. QUEST. Obs. What is simple proportion often called? Do these terms convey an ea of the nature or object of the rule? 9. If 8 yds. of broadcloth cost $96, how much will 20 yds. cost at the same rate? Solution. It is plain that 8 yds. has the same ratio to 20 yds. as the cost of 8 yds., viz: $96, has to the cost of 20 yds. That is, 8 yds.: 20 yds.:: $96: to the cost of 20 yds. Now $96X20-$1920; and $1920-8-$240. Ans. 10. If 35 men will consume a certain quantity of flour in 20 lays, how long will it take 50 men to consume it ? Note. Since the answer is days, we put the given days for the third term. Then, as the flour will not last 50 men so long as it will 35 men, we put the smaller number of men for the second term, and the larger for the first. 50: 35: 20 to the number of days required. 20 50) 700 Multiply the second and third terms to gether, and divide the product by the first 14 days. Ans. term, as in the last example. PROOF.-50X14-35 X 20. (Art. 498.) 503. From the preceding illustrations and principles, we deduce the following general RULE FOR SIMPLE PROPORTION. I. Place that number for the third term, which is of the same kind as the answer or number required. II. Then, if by the nature of the question the answer must be greater than the third term, place the greater of the other two numbers for the second term; but if it is to be less, place the less of the other two numbers for the second term, and the other for the first. III. Finally, multiply the second and third terms together, divide the product by the first, and the quotient will be the answer in the same denomination as the third term. PROOF.--Multiply the first term and the answer ̈ together, and if the product is equal to the product of the second and third terms, the work is right. (Art. 500.) QUEST.-503. In arranging the terms in simple proportion, which number is put for the third term? How arrange the other two numbers? Having stated the question how is the answer found? Of what denomination is the answer? How is simple proportion proved? Demonstration. -If four numbers are proportional, we have seen that the product of the means is equal to the product of the extremes; (Art. 498;) therefore the product of the second and third terms must be equal to that of the first and fourth. But if the product of two factors is divided by one of them, the quotient will be the other; (Art. 156;) consequently, when the first three terms of a proportion are given, the product of the second and third terms divided by the first, must give the fourth term or answer. The object of placing that number, which is of the same kind as the answer, for the third term, instead of the second, as is sometimes done, is twofold: 1st, it avoids the necessity of the Rule of Three Inverse; 2d, the third term, in many cases, has no ratio to the first; consequently it is inconsistent with the principles of proportion to put it for the second term. Thus, in the ninth example, if we put $96 for the second term, it would read, 8 yds. : $96 :: 20 yds.: $240, the answer. But a yard can have no ratio to a dollar; for one cannot be said to be greater nor less than the other. (Art. 476. Obs. 2.) OBS. 1. If the first and second terms are compound numbers, reduce them to the lowest denomination mentioned in either, before the multiplication or division is performed. When the third term contains different denominations, it must also be reduced to the lowest denomination mentioned in it. 2. The process of arranging the terms of a question for solution, or putting it into the form of a proportion, is called stating the question. 3. Questions in Simple Proportion, we have seen, may be solved by Analysis. After solving the following examples by proportion, it will be an excellent exercise for the student to solve them by analysis. (Art. 462. Obs. 2.) 11. If 16 barrels of flour cost $112, what will 129 barrels cost? 12. If 40 acres of land cost $540, what will 97 acres cost? 13. If 641 sheep cost $1923, what will 75 sheep cost? 14. At the rate of 155 miles in 12 days, how far can a man travel in 60 days? 15. How much hay, at $17.50 per ton, can you buy for $350 ? 16. If $45 buy 63 lbs. of tea, how much will $1540 buy? 17. If 90 lbs. of pepper are worth 72 lbs. of ginger, how many lbs. of ginger are 64 lbs. of pepper worth? 18. A bankrupt compromised with his creditors, at 64 cts. on a dollar; how much will be received on a debt of $2563.50 ? 19. An emigrant has a draft for £1460 sterling: how much is it worth, allowing $4.84 to a pound? QUEST. Obs. If the first and second terms contain different denominations, how pro ceed? When the third term contains different denominations, what is to be one? What is meant by stating a question? SIMPLE PROPORTION BY CANCELATION. 20. If 72 tons of coal cost $648, how much will 9 tons cost? Operation. Tons. Tons. Dolls. 72: 9: 648 : Ans. 8:1 Now $6488=$81. Ans. 9X648 72 Gr thus, But 72 Having stated the question as before, we perceive the factor 9 is common to the first two terms, and therefore may be canceled. (Art. 151.) =the answer. (Art. 503.) 9X648 9X648 $81, the same as before. Hence, 504. When the first term has factors common to either of the other two terms. Cancel the factors which are common, then proceed according to the rule above. (Arts. 151, 221.) PROOF.—Place the answer in the denominator, or on the left of the perpendicular line, as the case may be, and if the factors of the divisor exactly cancel those of the dividend, the work is right. OBS. 1. The question should be stated, before attempting to cancel the common factors. When the terms are of different denominations, the reduction of them may sometimes be shortened by Cancelation. 2. Instead of points, it may sometimes be more convenient to place a perpendicular line between the first and second terms, as in division of fractions. (Art. 231.) In this case the third term should be placed under the second, with the sign of proportion (::) before it to denote its origin, and its relation the fourth term or the answer. 3. It will be perceived that cancelation is applicable in Simple Proportion tơ all those examples, whose first term has one or more factors common to either of the other terms. 21. If 24 yds. of cloth cost $63, what will 32 yds. cost? 22. If 20 bu. of oats cost £1, how much will 2 quarts cost? |