PROPORTION. 321. PROPORTION is an equality of ratios. 8:46:3, which is read, 8 is to 4 as 6 is to 3; or, as 8 is to 4 so is 6 to 3; or it may be indicated thus, 8:4 = 6:3, which is read, the ratio of 8 to 4 equals the ratio of 6 to 3. Any 4 numbers are in proportion, and may be written and read in like manner, if the quotient of the 1st divided by the 2d is equal to the quotient of the 3d divided by the 4th. 323. The 1st and 4th terms are called extremes, and the 2d and 3d, means. The 1st and 3d are the antecedents of the two ratios, and the 2d and 4th are the consequents. The product of the extremes is always equal to the product of the means; thus, in the proportion 8: 4 :: 6:3, we have 8 X 3 = 4 × 6. 324. Since the product of the extremes is equal to the product of the means, any one term may be found when the other three are given; for the product of the extremes divided by either mean will give the other mean, and the product of the means divided by either extreme will give the other extreme. Fill the blank in each of the following proportions: 321. What is Proportion? 322. How indicated? Proportion, how read? When are four numbers in proportion? 323. What are the 1st and 4th terms called? 2d and 3d? 1st and 3d? 2d and 4th? The product of the extremes equals what? 324. How many terms must be given? How can the other be found? 325. It follows from Art. 317, that if the 1st and 2d, or 3d and 4th, or 1st and 3d, or 2d and 4th, or all four terms of a proportion are multiplied or divided by the same number, the resulting numbers will be in proportion. 326. If 4 numbers are proportional they will be in proportion in 8 different orders; thus, (1) Given (2) Alternating (1) (3) Inverting (1) (4) Alternating (3) 8:46:3 8: 6:4:3 4 : 8:36 4:38:6 (5) Inverting (1) and transposing couplets 3: 6:4:8 NOTE. These 4 numbers may be written in 16 other orders, but none of them will be in proportion. 327. When the means of a proportion are alike, the term repeated is a mean proportional between the other two, and the last term is a third proportional to the 1st and 2d; thus, in 4: 6 :: 6 : 9, 6 is a mean proportional between 4 and 9, and 9 is a third proportional to 4 and 6. 328. A mean proportional between two numbers may be found by multiplying the two given numbers together, and then resolving the product into two equal factors; thus, the mean proportional between 2 and 8 is 4, for 2 × 8 = 16 = 4 × 4; .. 2:4:4:8. 329. A third proportional to two numbers may be found by dividing the square of the 2d by the 1st. The third proportional to 5 and 10 is 20; for 102 ÷ 5 = 20; .. 5 : 10 :: 10:20. 330. In all examples in Simple Proportion there are three 325. What terms may be multiplied without destroying the proportion? What divided? 326. In how many orders may four proportional numbers be in proportion? In how many not in proportion? 327. What is a mean proportional? A third proportional? 328. How is a mean proportional found? 329. A third proportional? numbers given to find a fourth; .. Proportion is often called the Rule of Three. Two of the three given numbers must be of the same kind, and the other is of the same kind as the answer. Ex. 1. If 3 men build 6 rods of wall in a day, how many rods will 5 men build? This example may be analyzed as follows: If 3 men build 6 rods, 1 man will build of 6 rods, i. e. 2 rods; and if one man build 2 rods, 5 men will build 5 times 2 rods, i. e. 10 rods, Ans.; but to solve it by proportion, we say that 3 men have to 5 men the same ratio that the given number of rods has to the required number of rods; thus, 3 men : 5 men :: 6 rods: required number of rods. Now, since the means and 1st extreme are given, we find the 2d extreme by dividing the product of the means by the given extreme (Art. 324); thus, 6530 and 30 ÷ 3 = 10, Ans. as before. Hence, 331. To solve an example in Simple Proportion, RULE. Write that given number which is of the same kind as the required answer for the third term; consider whether the nature of the question requires the answer to be greater or less than the third term; if greater, write the greater of the two remaining numbers for the second term and the less for the first; but if less, write the less for the second and the greater for the first; in either case, divide the product of the second and third terms by the first, and the quotient will be the term sought. NOTE 1. If the first and second terms are in different denominations, they should be reduced to the same before stating the question. REMARK. Every one who intelligently solves an example by proportion, does, in effect, solve it by analysis; but the teacher should use much care on this point, since the scholar learns much faster when he analyzes a question than when he merely follows 330. Of what kind must two of the three given numbers be? What the other? 331. Rule for solving an example in proportion? Note 1? Remark? a rule. Let the following examples be solved by analysis and by proportion. 2. If a man earn $24 in 2 months how much will he earn in 9 months? 2:9:24: 4th term. 9 2)216 $108, Ans. Since we are seeking for dollars, we make $24 the 3d term, and then, as a man will earn more in 9 months than he will in 2 months, we make 9 the 2d term and 2 the 1st. To analyze the above, we say, If a man earn $24 in 2 months, then in 1 month he will earn of $24, i. e. $12; and if he earn $12 in 1 month, then in 9 months he will earn 9 times $12, i. e. $108, Ans. 3. If 15 bush. of wheat make 3 bbl. of flour, how many bushels of wheat will be required to make 7 bbl. of flour? Ans. 35. 4. If 40 bush. of wheat make 8 bbl. of flour, how many barrels of flour will 75 bush. of wheat make? Ans. 15. 5. If a man can walk 75 miles in 3 days, how far can he walk in 8 days? Ans. 200 miles. 6. If a man travel 64 miles in 2 days, how long will it take him to travel 160 miles? Ans. 5 days. 7. If a locomotive run 39000 miles in 13 weeks, how far, at that rate, would it run in 52 weeks? 8. If 20 men perform a piece of work in 8 days, in how many days will 4 men perform the same? Ans. 40. 9. If 24 cords of wood cost $60, what will 18 cords cost. 10. If $30 pay for 5 cords of wood, how many dollars will pay for 12 cords? Ans. 72. 11. If 4 cords of wood cost $20, how many cords may be bought for $45? Ans. 9. 12. If 6 horses eat 42 bushels of oats in 5 weeks, how many bushels will 11 horses eat in the same time? 13. What cost 7 tons of coal when 4 tons cost $24? 14. In how many days can 6 men build a house, if 10 men can build it in 72 days? 15. If 72 lb. of cheese are worth as much as 30 lb. of butter, how many pounds of cheese will pay for 20lb. of butter? 16. How many tons of coal can be bought for $84, when 3 tons cost $18? 17. If 9 horses eat a ton of hay in 20 days, how will eat a ton in 30 days? Ans. 14. many horses Ans. 6. 18. How many tons of hay will 6 horses eat in 25 weeks, if 8 horses eat 20 tons in the same time? 19. If I pay 2s. 8d. per week for pasturing 2 cows, what shall I pay for pasturing 11 cows? 20. If I pay 2s. 8d. for pasturing 2 cows, how many cows can be pastured the same time for 14s. 8d.? 21. If 8 acres of land cost 75£ 6s. 4d., how many acres may be bought for 131£ 16s. 1d.? 22. If 14 acres of land cost 131£ 16s. 1d., what will 8 acres cost? 23. If g of a ship cost $9875, what are of her worth? 24. If of a barrel of flour cost $3.20, what will 6 bbl. cost? 25. If a man walk 192 miles in 6 days of 8 hours each, in how many days of 12 hours each will he walk 192 miles? 26. Lent a friend $400 for 6 months; afterwards he lent me $300. How long may I keep it to balance the favor? 27. How many yards of cloth of a yard wide are equal to 20 yards 14 yard wide? 28. If when flour is worth $9 per bbl., a penny loaf weighs 4oz., what will it weigh when flour is worth $6 per bbl.? 29. If 10 horses eat 45 bushels of oats in 3 weeks, how many bushels will 12 horses eat in the same time? 30. Three men can do a piece of work in 12 days; how many men must be added to the number to do the same in 4 days? |