It follows from what has been stated that: ART. 415.-1. The product of the means equals the product of the extremes. 2. Either extreme equals the product of the means divided by the other extreme. 3. Either mean equals the product of the extremes divided by the other mean. WRITTEN EXERCISES. 1. What is the cost of 5 yards of broadcloth, if 16 yards cost $64 ? Process. yd. yd. $ 16:5 : : 64 : x = $20 Analysis.—The fourth term, which is the one we are seeking, must be a certain number of dollars, for the problem demands the number of dollars that 5 yards of cloth will cost. For convenience, we represent the unknown term by x. Then, as ratio can exist only between like things, the other term of the latter couplet must be dollars; and since $64 is the only number of dollars named in the problem, we place that for the third term. If 16 yards cost $64, 5 yards must cost a less sum. Consequently x, the fourth term, which we are seeking, will be less than the third term, and to preserve the equality of ratios, the first consequent must be less than the first antecedent. Furthermore, 16 yards, the number bought with $64, must bear the same ratio to 5 yards that the cost of 16 yards ($64) bears to the cost of 5 yards. Hence, we make the first consequent less than the first antecedent. Arranged thus we have: yds. yds. $ $ 16 5 64: x. Multiplying the means together and dividing by the first extreme, we obtain $20 for the other extreme, which is the required cost, of 5 yards of broadcloth. ART. 416.-Rule for Simple Proportion.-Write for the third term the number which is of the same kind as the number sought. Then from the nature of the problem, consider whether the fourth term will be greater or less than the third term. If it will be greater, arrange the remaining two terms so that the second will be greater than the first. If the fourth term will be less than the third, make the second less than the first. Multiply the second and third terms together and divide the product by the first: the quotient is the fourth term, or term required. TO TEACHER.-Proportion, formerly known as the Rule of Three, is a form of mathematical logic so admirable that it has been called the Golden Rule. The reason for every step taken is so plain that a young child can readily understand it, and, once fully explained to him, it will never be forgotten. The teacher, therefore, should make certain that by continued drill and exercises the pupils comprehend the fundamental truths of Proportion. 2. What is the cost of 40 bushels of potatoes, if 30 bushels cost $45 ? 3. What is the cost of 60 hhd. of molasses, if 45 hhd. cost $2,500 ? 4. What are 16 horses worth, if 8 are worth $1,800 ? 5. What are 320 acres of land worth, if 60 acres cost $5,000? 6. If $200 gains $14 in 1 year, how much will $500 gain in the same time? 7. If 30 men eat 48 loaves of broad in a certain time, how many loaves will 70 men eat in the same time? 8. If in a certain time 30 men eat 48 loaves, how many men will be required to eat 80 loaves in the same time? 9. If 25 cows cost $1,250, how many cows bought for $3,000. can be 10. If a clerk earns $75 in 5 weeks, how much can he earn in 32 weeks? 11. If a staff 8 feet high casts a shadow 11 feet in length, how many feet of shadow will a pole 23 feet high cast at the same time of day? 12. If 5 lb. 8 oz. of butter cost $1.50, how much will 12 lb. 4 oz. cost? 13. If 5 acres will pasture 10 cows a certain time, how many cows can be pastured on 14 acres for the same time? 14. If of a yard of satin cost $1.75, what will 21 yards cost? 15. If 12 men can build a wall in 7 days, how long would it take 15 men to build the wall? 16. If of the value of a ship is $11,000, what is of its value? 17. If 71⁄2 tons of coal cost $37.50, how many tons can be bought for $75 ? 18. If an express averages 8 miles in 9 minutes, how long will it take to go 48 miles? 19. If there are 96 men in a single company, how many men compose a regiment composed of 10 companies? 20. A bankrupt merchant pays 80 cents on a dollar: if he owes A $1,500, how much should A receive? 21. If of a bushel of potatoes cost $3, what will of a bushel cost? 22. If 14 ounces of silver make 5 tea-spoons, how many will pounds of silver make ? 23. If 48 men build a house in 104 days, how long Iwill it take 12 men to build a similar house? 24. If A can saw a cord of wood in 5 hours, and B in 8 hours, how long will it take both together to saw a cord? 25. A man can dig a ditch in 6 hours, and with the help of a boy he can dig it in 4 hours: how long will it take the boy to dig it? 26. If the interest of $550.75 is $33.045 for 1 year, how much will the interest of $110.15 be for the same time? 27. If 36 yards of carpet of a yard wide will cover my office, how many yards yard wide will be required to cover it? 28. If 10 masons build a wall 40 ft. long, 2 ft. wide and 12 ft. high, in 8 days, how long will it take them to build a wall 18 ft. long, 1 foot wide and 10 ft. high? 29. If a block of marble 8 ft. long, 4 ft. 6 in. broad and 3 feet deep, weighs 1,500 lb., what is the weight in tons of a block 9 ft. long, 5 ft. 4 in. broad and 4 ft. 8 in. deep? 30. Illustrate by an original problem, the method of solving examples in Simple Proportion. Compound Proportion. WRITTEN EXERCISES. ART. 417.-1. If 8 men dig a trench 40 rods long in 12 days, how many men can dig a trench 100 rods long in 20 days? number of Now, if 8 men can dig 40 rods in a certain time, it will longer time to dig 100 rods. question of rods, the fourth take the same number of men a Hence, basing the problem on the term will be greater than the third, and the couplet will be 40 rods: 100 rods. If 8 men can do the work in 12 days, it will take a less number of men to do it in 20 days. Hence, the last term will be less than the third term, and the couplet will be 20 days: 12 days. Multiplying the means together and dividing the product by the product of the first terms, canceling when possible, we obtain 12, as the fourth term, which represents the number of men required. Therefore, etc. ART. 418.-Rule for Compound Proportion.— Write for the third term, the number that is of the same kind as the answer required. Then arrange in turn the remaining ratios as though the answer depended upon each of them and the third term alone. Multiply the second and third terms together, and divide their product by the product of the first terms. The quotient is the fourth term, or answer required. NOTE.-Compound Proportion offers abundant opportunities for cancel lation. 2. If $400 gain $10 in 6 months, how long will it take $300 to gain $5 ? 3. If $500 in 3 yr. 6 mo., at 6%, gain $105 interest, how much will $600 gain in 4 yr. 9 mo., at 8% ? 4. If $250 gain $17.50 in 12 months, what will $450 gain in 8 months ? |