Multiplying each ratio by 27, (the product of the denominators,) 2X27_6×27 The proportion becomes 3 9 (Art. 295. Ax. 6.) Dividing both the numerator and the denominator of the first couplet by 3; (Art. 116;) or canceling the denominator 3 and the same factor in 27; (Art. 136;) also canceling the 9, and the same factor in 27, we have 2×9=6X3. But 2 and 9 are the extremes of the given proportion, and 3 and 6 are the means; and the product of the extremes 2X9-6X3, 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. 325. It follows from the last Article, that if the product of the means is divided by one of the extremes, the quotient will be the other extreme; and if the prodduct of the extremes is divided by one of the means, the quotient wil be the other mean. For, if the product is divided by one of the factors, the quotient will be the other factor. (Art. 291.) Take the proportion 8:4::6: 3. Hence, 8x34-6, one of the means; If any three terms of a proportion are given, the fourth may be found by dividing the product of two of them by the other term. SIMPLE PROPORTION. 326. Proportion in arithmetic is usually divided into Simple and Compound. Simple Proportion is an equality between two ratios. Its principal object is to find the fourth term of a proportion, when the first three terms are given. (Art. 325.) The QUEST.-225. 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? 226. What is simple proportion? What is its chief object? resolution of this problem is the most important result in the theory of proportion. 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 first three terms of a proportion are 4, 6, 8, what is the fourth term? Solution. 6x8=48 and 48÷4-12, which is the number required; that is, 4:6:8:12. PROOF. 4×12 is equal to 6 × 8. (Art. 324. Obs. 2.) 2. If 12 bls. of flour cost $72; what will 4 bls. cost, at the same rate? Solution. It is evident 12 bls. has the same ratio to 4 bls. as the cost of 12 bls. ($72) has to the cost of 4 bls., which is required. That is, 12 bls. : 4 bls. : : $72 is to the cost of 4 bls. Now, 72 ×4=288; and 288÷12 =$24, the cost required; that is, 12 bls. 4 bls, : : $72 : $24. 3. If 6 men can dig a cellar in 12 days, how many men will it take to dig it in 4 days? Note. Since the answer is men, we put the given number of men for the third term. Then, as it will require more men to dig it in 4 days than it will to dig it in twelve days, we put the larger number of days for the second term, and the smaller for the first term. Solution. 4d.: 12d. : : 6 men to the number of men required. Now, 12×6=72; and 72÷4-18. Ans. 18 men. QUEST. Obs. What is simple proportion often called? Do these terms convey an idea of the nature or object of the rule? 327. 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 this 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, multiplying 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 or fourth term together, and if the product is equal to the product of the second and third terms, the work is right. (Art. 324.) 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, that is, putting it into the form of a proportion, is called stating the question. 3. It has been shown that all questions in Simple Proportion may easily be solved by Analysis. (Art. 296.) After solving the following examples by proportion, it will be an excellent exercise for the pupil to solve each by analysis. 4. If 6 yards of broadcloth cost 30 dollars, how much will 20 yards cost? 5. If 8 bls. of flour cost $40, what will 15 bls. cost? 6. If 16 lbs. of tea cost $12, what will 41 lbs. cost? QUEST.-327. What is the rule for Simple Proportion? Having stated the question, how is the answer found? Of what denomination is the answer? How is Simple Proportion proved? Obs. If the first and second terms contain different denominations, how proceed? When the third term contains different denominations, what is to be done? What is meant by stating the question? 7. If 12 acres of land produce 240 bushels of wheat, how much will 57 acres produce? 8. If a man travels 400 miles in 15 days, how far can he travel in 9 days? 9. If 63 barrels of beef cost $504, how much will 7 barrels cost? Statement. bls. bls. dolls. 637: 504 is to the answer. That is, the product of 504×7÷63 will be the answer. (Art. 227.) But this process may be shortened by canceling the factors common to the dividend and divisor. (Art. 91.a.) Thus, the proportion 63 7: 504, by canceling the common factor 7, becomes 637: 504; and 504÷9=56. Ans. $56. 9 Or thus, since 504 × 7 is to be divided by 63, the answer is equal to which is common both to the numerator and denomina 504X7 Now, canceling the factor 7, 326. If the first term has factors common to either of the other two terms of a proportion, the operation may be shortened by canceling these factors; then proceed as before. (Arts. 91.a, 136.) Note. The question should be stated before canceling the common factors. 10. If 16 sheep cost $48, how much will 6 cost? Statement. 16s.: 6s. :: $48 is to the answer. 3 48X6 Then by cancelation, =3X6, or $18 Ans. 16 11. If 6 men can build a wall in 36 days, how long will it take 18 men to build it? QUEST.-328. When the first term has factors common to either of the other two terms, how may the operation be shortened? 12. If 10 quintals of fish cost $35, how much will 17 quintals cost? 13. If a ship has water sufficient to last a crew of 25 men for 8 months, how long will it last 15 men? 14. If 12 lbs. of sugar cost $1, how much will 84 lbs. cost? 15. If 15 lbs. of lard cost $1.15, how much will 80 lbs. cost? 16. If of an acre of land cost £3, how much will of an acre cost? 17. If of a hogshead of molasses cost $28, how much will 16 hogsheads cost? 18. If 21 yds. of broadcloth cost $18, how much will 27 yds. cost? 19. If 6 acres and 40 rods of land cost $125, how much will 25 acres and 120 rods cost? 20. If 15 yds. of silk cost £4, 10s., how much will 75 yds. cost? 21. If a railroad car goes 35 m. in 1 hr. 45 min., how far will it go in 3 days? 22. If 41 lbs. of chocolate cost 9s., how much will 22 lbs. cost? 23. If 35 lbs. of butter cost $4, how much will 15% lbs. cost? 24. If 84 lbs. of cheese cost $53, how much will 60 lbs. cost? 5 25. If of a ship is worth $6000, how much is 16 of her worth? 26. If 41 bu. of wheat make 1 barrel of flour, how many barrels will 84 bu, make? 27. If the interest of $1500 for 12 mo. is $90, what will be the interest of the same sum for 8 mo. ? |