CASE XI. To find what part one number is of another, which is called finding their RATIO. RULE. 1. Make that number which is to become the part, the numerator of a fraction, and the other number, the denominator, that is, always divide the second by the first. 2. What part of 20 is 5? 3. What part of 5 is 20? LIV. A. 490 6 9. What part of 625 is the sum of 175 and 225? 10. A and B bought a barrel of flour for $11; A paid $6 and B $5; what part of the whole did each pay? A&; B. 11. Suppose A puts into trade $100, B $200, and C $500; what part of the whole capital did each man advance? A. A; B; CI. 12. Three men form a copartnership, A advancing $1,000, B $1,500, and C $2,500; what is each one's part of the capital? A. A's B's; C's. 13. Suppose in the last question, that they gained the first year $2,000, which they are to share in proportion to the sum each advanced; how many dollars did each gain? ( of $2,000 $400, &c.) A. A's $400; B's $600; C's $1,000. 14. Next suppose that the same company lost the second year $1,700; how many dollars did each lose? A. A's $340; B's $510; C's $850. 15. A, B and C traded together and gained $3,000, which they agreed to share in proportion to each one's capital. A advanced $2,000, B $3,000, and C $1,000; how many dollars did each man receive for his profits? A. A's $1,000; B's $1,500; C's $500. 16. What part of 33 gallons is 21 gallons? Reduce the complex fraction to a single one by Case x. A. 3. 17. A gentleman having 205 barrels of flour, sold 7613 barrels ; what part of the whole did he sell? A. 3. CASE XII. To find what fraction or part, one quantity is of another of the same kind, but of different denominations. RULE 1. Reduce the given quantities to the lowes. denomination mentioned in either quantity; then proceed to find the part as in the last case. CASE XI. Q. What part of 12 is 8? What part of 45 is 15? rule? 1. What is the What is the ratio of 20 to 10?-of 10 to 20?-of 40 to 120? 2 What part of $5 is 50 cents? ($5=500ct.) 3. What part of £1 is 2s. 6d. ? 4. What part of £2. 3s. 4d. is 4s. 2d.? 5. Reduce 16s. 8d. to the fraction of £1. 6. Reduce 18s. to the fraction of a guinea (28s). 7. What part of £1 is 1s. 8d.?-is 2s. 6d.?—is 3s. 4d ?-is.5s.?1s 6s. 8d.?-is 10s.?—is 12s. 6d.?-is 15s.? is 17s. 6d.? 10. What part of 7 miles 4 furlongs is 1m. 2fur.? 11 What part of $1 (=6s.) is 2s. 6d.?—is 4s. 6d.? 12. What part of $1 is 1d.?—is lct.?—is lm.? A. 15 13. What part of 1 month is 1 day?-is 5d.?-is 10d.?-is 15d.?— is 20d.?-is 25d.? mo.; mo.; mo.; mo.; mo.; mo. 14. What part of 2 yards is 3qr. 3na. 15. What part of 5 bushels is 3pk. 7qt. 1 pt.? 16. What part of £3. 15s. 6d. is £1. 5s. 2d.? 17. Reduce 3qr. 18lb. 12oz. to the fraction of lewt. 18. What part of 15Y. 10mo. 1wk. 5d. 2h. 30m. 30sec. is 6Y 4mo. 4d. 20h. 12m. 12sec.? A. 32. 63 A. 320 A.. A. 15 19. Reduce 7cwt. 2qr. 12lb. 8oz. to the fraction of a ton. 16. A. . A. 3. 20. What part of 4d. 23qr. is 1d. 1ğqr.? 21. A merchant bought 2cwt. 2qr. 18lb. 9 dr. of sugar, and sold 2qr. 17lb. 223dr.; what part of the whole did he sell? CASE XIII. A. . To reduce the fraction of any given quantity to whole or compound numbers. RULE. 1. Multiply the given quantity or its equivalent in the next lower denomination, by the numerator, and divide the product by the denominator; proceeding with the remainder, if there be any, as in Compound Division. 2. For dividing by the denominator first would show the value of one equal part, and multiplying the quotient by the numerator would show the value of all the parts meant; a process the same in effect as that described in the rule. What CASE XII. Q. What part of 1 dollar is 75 cents?-is 50 cents?-is 40 cents?is 30 cents?—is 25 cents?-is 163 cents ?-is 12 cents?-is 61 cents? is the rule? 1. What part of £1 is 5s.?-is 2s. 6d.?-is 6s. 8d.?-is 13s. 4d.?— is 15s.? CASE XIII. Q. How many cents are there in of a dollar?-shillings in of £1. What is the rule? 1. Why multiply by the numerator and divide by the denominator? 2. How many seconds are there in of a minute ?-fur. ongs in of a mile ?-feet in of a yard? 3 How much is 3 of a day? 1 day 24 hours; then 24 hours 3÷4=18. A. 18 hours. A. 12 shillings. A. 13s. 4d. 7. Suppose a railroad car goes 10 miles in of an hour, how many minutes is it in going that distance? 8. What is the value of of 5 dollars? 25 A. 24 minutes. A. 20 cents. 9. How much is £,,?—£}?—£}—£}?—£}?—£}? Answers: 1s. 8d.; 2s. 6d.; 3s. 4d.; 5s.; 6s. 8d.; 17s. 6d. 10. How many shillings and how many cents are equal to of a dollar? A. 2s. 6d. 413 cents. 11. How many days are equal to of a month ?-to of a month? to of a month? 12. How much is of 7 miles 4 furlongs? 13. How much is 15 of 2 yards? 32 A. 6d.; 25d.; 221⁄2d. A. 1m. 2fur. A. 3qr. 3na. A. 3pk. 7qt. 1pt. of a hogshead of molasses; how A. 39gal. 1qt. 1pt. many gallons, quarts, &c. does he buy? 17. What is the value of of a day? A. 11oz.6 dr. A. 16h. 36m. 55,5 sec. 18. Suppose you resided in one place 4Y. 3mo. 3d., and in another place as long; what period of time did you spend in the latter place? A. 7mo. 9d. 19. What is the value of of 2hhd. 27gal. 2qt. 1pt. 3gi.? A 43gal. 3qt. 1pt. 19gi. CASE XIV. To reduce a fraction of one denomination to an equivalent fraction of another denomination. RULE. 1. Multiply and divide the fraction according to the principles of Reduction of whole numbers, but recollect 2. That a fraction is multiplied by multiplying its numerator, or by dividing its denominator. LVII. 17. 3. Also, that a fraction is divided by dividing its numerator, or by multiplying its denominator. LVII. 18. 4. Reduce of a pound to the fraction of a farthing. CASE XIV. Q. What part of £1 is is of a quart? What is the rule? 1. A. qr. A. qr. a shilling? What part of a gallon How is the fraction multiplied or divided? 2. 3 How is 1 of a f reduced to the fraction of a farthing? 4 5. Reduce 960 (=) of a farthing to the fraction of a pound. 1920 12d. ÷ 20s. 1920 × 4qr. × 12d. ÷ 20s. 960 1 1920 1843200-£1920. A. £1920 A. £1920 1920 00 5 10-52 1 9. Reduce of a mill to the fraction of an eagle. 10. Reduce of a guinea to the fraction of a farthing. 11. Reduce of a farthing to the fraction of a guinea. 12. Reduce of a pound to the fraction of a guinea. 13. Reduce of a guinea to the fraction of a pound. 14. Reduce of a day to the fraction of a minute. 15. Reduce 1440 of a minute to the fraction of a day. 16. Reduce of a bushel to the fraction of a quart. 17. Reduce of a quart to the fraction of a bushel. 18. Suppose one man has of a pound, another and another of a penny; what fraction of a dollar has each. 1441 1441 of a shilling, A. $. 19. Reduce of a pound to the fraction of a crown at 6s. 8d. each A. 81 crown 20. What fraction of a pound is of 3 of a hundred weight? A. 30lb. 3 pounds. 63 of the debt, A. £3. 21. Suppose you owe of a guinea, (28s.) and pay what part of a pound do you still owe? 22. If A buys of 5 hogsheads of molasses, and sells B. of it, what fraction of a gallon does B buy? A. of a gallon. 23. Suppose of of a pound is the numerator of a fraction, and 3 of £2 the denominator; what fraction of an eagle will express the same value? To find the integer from having a fractional part given. RULE. 1. Multiply the integer by the denominator, and divide the product by the numerator. LII. 10. 2. Find that number of which is 205. 3. What number is that of which is 20,000? A. 55,000. 4. Suppose a merchant sells exactly 11hhd. 49gal. 3qt. 1pt. 3gi. of molasses, it being of all he has on hand; what quantity had he at first? A. 78hhd. 39gal. 1pt. 1920 Q. How is 960 (=) of a farthing reduced to the fraction of a pound? 8,9. CASE XV. Q. 40 is of what number? If 12 be of a certain number, what is that number? What is the rule? 1. When a man pays 8 dollars for of a barrel of flour, what is a barrel worth at that rate? 3 16 5. Find that sum of which is £3. 5s. 6d. A. £5. 17s. 11d. 6. The postage of a letter for 400 miles is of a dollar; how far, at that rate, can a letter be carried for one dollar? A. 2,133 miles. 7. 503 is of of what number? A. 14757. 8. It is estimated that nearly 25,000,000 persons die annually being of the population of the earth; what, according to that esti mate, must be the whole population of the globe? A. 800,000,000. ADDITION OF FRACTIONS. GENERAL RULE. LXIII. 1. Reduce complex and compound fractions to single ones, and all to a common or least common denominator; over which write the sum of the numerators. LV. 1, 2. 2. This sum may often be reduced either to lower terms, or to a whole or mixed number. A. ££2. 3. Add together £3, £ £3 and £}. 4. Add together 111, 347, 417, and 11. A. 135. 5. To add mixed numbers.-Find the sum of the fractional parts first, and if it contain a whole number, add it to the sum of the other whole numbers. 6. Or, reduce the mixed numbers first to improper fractions, then add them by the general rule. 7. Add together $1,20317, $9,40613, and $8,9061. 17 $1203 8 $9406 13 $8906 1 8 $195 1 7 2 9. Find the sum of 67531, 89133, and 91533. A. 2,48317. 10. Suppose that one pile of wood contains 136¦¦ cords, another 956 cords, and another 45231 cords; how many cords are there in all the piles? A. 1,545 cords. 11. Add together 3, 3, 1, and §. Reduce them first to common denominator. 12. Find the sum of 463, 893 and 40 7. 43 their least A. 2. A. 1770. A. $50. 13. Add together $182, $51, $7,5%, $83, and $93. 14. What is the sum of 7894 years, 8173 years, 316 years, and 216 years? A. 2,1403 15. Find the sum of 2, 3 of %, and 3 of 51. Reduce the fractions to single ones first. 09 A. 4188. 11, 77, and LXIII. Q. What is the general rule for adding fractions? 1. What may oftentimes be done with the result? 2. What is the sum of 2 8 What is the sum of 8, 92, and 101? What is the rule for the last example? 5, 6 What is the sum of 3 and? See 11. |