per acre. 64 MULTIPLICATION AND PRACTICE. 4. Required the value of 65 bushels and 3 pecks of wheat, at $1.16 per bushel ? Ans. $76.27. 5. Calculate the value of 20 acres, 1 rood, and 20 poles, at $10 Ans. $203.75. 6. Required the value of 5} cords of wood, at$4.25 per cord. Ans. $23.375. 7. What is the value of 224 oranges, at $.015, or 1} cents ? Ans. $3.36. 8. What will 264 pounds of butter cost, at 122 cents per lb. ? Ans. $33. 9. What is the value of 56 hats, at $2.25 a piece? Ans. $126. 10. What is the value of 284 pounds of cheese, at 8 cts. per lb.? 11. Required the value of 12 cords, 72 feet, of bark, at $8.25 per cord. Ans. $103.640625. 12. When port wine is sold at $2.50 per gal. what is it per qt. ? 13. Required the value of 3 quarters 2 nails of velvet, at the rate of 60 cents a yard. 14. What is the value of 184 yards of tape, at 5 mills per yd. ? 15. What is the cost of 3 yards of cloth, at $1.25 per yard? 16. What is the value of 525} bushels of oats, at 30 cents per bushel ? Ans. $157.65. 17. What is the value of 121 pounds of sugar, at 9 cents per pound? Ans. 1.125. 18. Required the value of 45 books, at $4.50 per dozen? Ans. $16.875. 19. Required the value of 18 sheets of paper, at 25 cents per quire? Ans. $.1875. 20. What is the value of 112 feet, or of a cord of wood or bark, at $6. per cord ? Bridgeton, August 22d, 1828. Mr. David Wood, Bought of N. L. Stratton & Co. 101 yards of satin, at $1.25 a yard, black silk, at .75 .85 .65 at .625 $28.55 Pottsville, April 10th, 1834. Mr. George Careful, Bought of Peter Trusty, coffee, at 12, cents Samuel Paywell . $4 32 8 66 at at 23. Farmer's Bill. Dr. 16, 2 barrels cider, at $1.25 per barrel, $12.63c. 24. Required the value of 57 slates, (or 4 doz.) at $1.75 per doz. 25. ired the value of 12cwt. of iron, at $5.25 per cwt. 26. What is the value of 100 pounds of tea, at 90 cents per lb.? NOTE. It is presumed that the scholar remembers note 1, on page 21. 27. What is the value of 4 of a dozen combs, at $1.68 a dozen? 28. What will 14, or 1.75 yds. of broadcloth cost, at $6 per yd.? Questions.-Multiplication and Practice comprises what? What is the rule of multiplying Federal Money? What is Practice? What is an aliquot part of a number? In this compound rule, what is the direction given ? To find the value of articles sold by the 100, or 1000, (either.) RULE.-Divide the articles or quantity given, by the quantity they are sold at per 100 or 1000; and this quotient multiplied by the price (per 100 or 1000) will be the answer required. EXAMPLES. 1. What is the value of 425 cedar rails, at $4. per 100 ? Ans. $17. 2. What is the value of 6425 feet of boards, at $8 per 1000 ? Ans. 51.40. 3. Required the value of 8 feet of heart pine, at $4. a 100. Ans. $.32. 4. Required the value of 75 feet timber, at $30. a 1000. Ans. $2.25. Question. How do you find the value of an article sold by the 100 or 1000 ? COMMON MEASURE. To find the largest common measure, or divisor of two or more numbers. RULE. If there be two numbers only, divide the larger by the less, and this divisor by the remainder, and so on as directed in a vulgar fraction; and the last divisor will be the largest common divisor required. When there are more than two numbers, find the common divisor of two of them as before, then that common divisor and one of the other numbers, and so on, through all the numbers; and the last divisor will be the answer. EXAMPLES. 1. Required the largest common divisor of 45 and 60. Thus, 45)60(1 15)=4. lowest term. 45 F2 66 RATIO AND PROPORTION. 2. Required the largest common divisor of 323 and 425. Ans. 17. 3. What is the largest common divisor of 2310 and 4626; and their lowest terms. Ans. 6 the largest common divisor, and 9 lowest terms. 4. Required the largest common measure of 135, 165, and 235? Ans. 5. 5. What is the largest common divisor of 1092, 1428, 1197, and 805? Ans. 7. 6. What is the largest common measure of 135, 180, and 285? Ans. 15. Question. How is the largest common divisor of two numbers found? When three or more? MULTIPLE Is a term given to a number that can be measured or divided exactly, by another number; as 9 is the multiple of 3; and 48 is a common multiple of 6 and 8, but its least common multiple is 24, and admits to be measured by the same measures, or divisors, as the common multiple, without remainders. To find the least common multiple of two, or more numbers. RULE.--1. Divide by any number that will divide two or more of the given numbers, without a remainder, and set the quotients, together with the undivided numbers, in a line below. 2. Divide the second line as before, and so on, till there are no two numbers that can be divided; then the continued product of the divisors and quotients will give the multiple required. EXAMPLES. * 1. What is the least common multiple of 6, 10, 16 and 20 ? *5)6 10 16 20 We inspect the given numbers, and find that 5 will divide two of them, viz. 10 *4)6 2 16 4 and 20, which we divide by 5, bringing into a line with the quotients, the num*2)6 2 4 1 bers which 5 will not measure: again we view the numbers in the second line 3 1 2 1 and find that 4 will measure 2 of them, which we divide and bring down as be5x4x2x3x2=240 ans. fore, and so proceed. 2. What is the least multiple of 16 and 24 ? Ans. 48. 3. What is the least number that 3, 5, 8 and 10 will measure? Ans. 120. 4. What is the least number that can be divided by the 9 digits separately, without a remainder ? Ans. 2520. RATIO AND PROPORTION. 1. Ratio can only exist between quantities of the same kind, We may speak of one weight as being twice as large as another; but we cannot say that one body weighs twice as much as another cost. The numbers given are called terms, and may be considered as divisor and dividend; and the ratio (required term) as a quotient in a division sum. The terms are written thus, 4 : 12, their ratio is 3. So any two numbers of the same kind, form a ratio. Ex.-2. What is the ratio of 84 : 336? Ans. 4. 3. If the ratio of two terms be 12, and the larger term 96, required the less term. 4. If the less term be 8, and the ratio 12, what is the larger term ? Questions. What is ratio? What are the given numbers called? What does the operation represent? 2. PROPORTION is a comparative relation of one thing, or number, to another ratio. Proportion (complete) consists of four numbers, or terms, and is a combination of two equal ratios; and the product of the first and fourth terms (extremes,) is equal to the product of the second and third (middle terms.) Therefore it is evident, that any one of the four terms is readily obtained when we have the other three given. To obtain either the first or fourth term, divide the product of the second and third (middle terms) by the given extreme, and the quotient will be the other extreme, or term sought. And, to obtain either of the middle terms, divide the product of the extremes, by the given middle term, and the quotient will be the middle term required. EXAMPLE. milcs. Questions. What is proportion? Proportion consists of how many terms? How many ratios? Which terms are called extremes ? Which the middle ? Either extremes, how obtained ? yde. milcs. : THE SINGLE RULE OF THREE, OR, SINGLE PROPORTION. It is called the Rule of Three, from its having three numbers given to find a fourth, which will have the same proportion to the third, that the second has to the first, for which reason it is termed the Rule of Proportion. RULE FOR STATING. -Write that number for the third term, which is of the same kind with the number sought. Then consider, from the nature of the question, whether the answer will be larger, or less than (this) the third term. If the answer is to be larger, place the larger of the (two) other given numbers (on the left) for the second term, and the less number for the first term. But if the answer is to be less than the third term, set the less number on the left of the third term, for the second, and the other (number) for the first term. RULE.Reduce the first and second terms to the same denomination or lowest mentioned in either; and the third term to its lowest denomination mentioned. THE SINGLE RULE OF THREE. 68 Multiply the second and tnird terms together, and divide the product by the first; the quotient will be the answer, of the same denomination with the third term. PROOF. 1. Multiply the first and fourth terms together, (extremes.) Multiply the second and third terms together, (means.) If the four numbers are proportional, these products will be equal. 2. Divide the larger of the first and second terms by the less. Divide the larger of the third and fourth terms by the less; and the two ratios will be equal. 3. Invert the question. NOTE 2.-The pulse of a person in health beats seventy-five times in a minute. EXAMPLES. 1. If 2lbs. of coffee cost 30 cents, what will 8lbs. cost at the same rate ? Operation. Cancelled. 16. 16. 1b. 16. As 2 : 8 :: 30* 2)2 :8:: 30 8 1 : 4 :: 30=120c. 2)240 $1.20c. C. I first write down 30 cents for the third term, like that of the answer ; I then say, if two pounds cost 30 cents, eight will cost more, requiring the answer to be larger than the third term (30 ;) I therefore place the larger number (8) for the second term, and the 2 for the first, as you see. Proof 1st. Proof 2d. Proof 3d. 120 X 2=240 8; 2=4 If 120 : 30 ::8:2 30 X 8=240 120-30=4 2. If 5 pounds of butter cost 75 cents, how much will 13 pounds cost? Ans. $1.95. 3. What is cheese per cwt. at 3} cents, or 35 mills per pound? Ans. $3.92. 4. If indigo be sold for 8 cents per ounce, what is 6cwt. orth? Ans. $860.16. |