N. B. The Term, which moves the Question, hath generally some of those Words before it; viz. What will? How many? How long? How far? or How much? &c. Then (carefully observe this; viz.) the first Term in the Supposition must always be of the fame Kind and Denomination with that Term which moves the Question. And the Term fought will always be of the fame Kind and Denomination with the second Term in the Supposition. Thus, {yds shil. yds phil. 3:9::6: Then All Questions in direct Proportion may be answered by three several Theorems. Theorem 1. Theorem { Multiply the second and third Terms together, and divide their Product by the first Term; the Quotient will be the Answer required. 2. { 6 3) 54 (18 Shillings, { because the second Term was Shillings. Divide the fecond Term by the first, then multiply the Quotient into the third Term, and their Produt will be the Answer required. yds. shil. yds. shil. Thus 3) 9 (=3. Then 3 x 6 = 18, as before. Theorem 3. Divide third Term by the first, then multiply the Quotient into the second Term, and their Product will be the Answer. Thus 3) 6 (= 2. And 9 x 2 = 18, as before. Here you fee that all the three Theorems are equally true; but the first is most general, and usually practised. Yet the two last may be readily performed, when either the second or third Term can be divided by the first; and will be found of fingular Ufe in the Rules of Fellowship, &c. as will appear further on. Quest. Quest. 2. If 8 Pounds of Tobacco cost 14 Shillings; what will half a hundred Weight (viz. 56 Pounds) cost at the same Rate? Thus 8lb: 145. :: 56lb : 41. 185. The Answer. 14 224 56 8) 784 (=985. = 41.185. Or thus 8) 56 (=7 Then 14 × 7 = 98 s. as before. Quest. 3. If 14 Shillings will buy 8 Pounds of Tobacco; how much will 41. 18 s. buy after the fame Rate? Stated thus, 14: 8 lb :: 41.185. = 985. : Then 98 x 8 = 784. And 14) 784 (56 lb. The Answer. Quest. 4. If half a hundred Weight of Tobacco be worth 41. 18 s. How much may I buy for 14 Shillings at that Rate? - Stated thus, 41. 185. = 985. : 56 lb :: 14s.: Then 56 x 14 = 784. And 98) 784 (8 lb. The Answer. Quest. 5. Suppose 41. 18 s. will buy 56 Pounds of Tobacco; what will 3 Pounds of the same Tobacco cost? This Question is thus stated, 56 lb: 41.185.985.:: 8 lb :Then 98 x 8 = 784. And 56) 784 (= 145. The Answer. Note, The three last Questions are only the second varied, being proposed purely to give an Instance how any Question in this Rule of Three may be varied, according to page 86. Quest. 6. What will three quarters of a Yard of Velvet cost, when the Price of 21 Yards and a half is worth 221. 105. 6 d. This Question truly stated will stand Thus, 21 yds.: 221. 10s. 6d.:: To the Answer. Which may be found three several Ways; viz. by Reduction; by Vulgar Fractions; and by Decimals. 1. By Reduction. Bring the first and third Terms into one Denomination; viz. into Quarters, and reduce the second Term into it's least Denomination, per Sect. 4. page 42. Thus 2186 Quarters. And 221. 10 s. 6d. = 5406 Pence. Then 86: 5406::3:15 8,88 d. For 5406 x 3 = 16218. And And 86) 162 18 (1880d. Then 188 Pence = 15 s. 8 d. 2 Farthings; the Answer required. 2. The fame Question stated in Vulgar Fractions will stand 21 १० Thus 21:22 4::: (See Sect. 3. page 50.) These Parts of a Pound are brought into Shillings by Multiplying the Numerator with 20, and dividing the Product by it's Denominator, &c. Thus 5406 x 20 =108120. And 6880) 108120 (155. And there remains 4920. Then 6880) 59040 (8 d. Again 4920 × 12 = 59040. 3. The same wrought by Decimal Fractions will be thus; Therefore 21,5:22,525 :: 0,75: to the Answer. And 21,5) 16,89375 (0,7857l. = 15 s. 8 d. 2 far. 272 Quest. 7. If 2 C. 3 qrs. 21 lb. of Sugar cost 61. 1s. 8 d. What will 12 C. 2 qrs. cost at the same Rate? That is, 2 C. 3 qrs. 21 lb : 61. 15. 8d.:: 12C. 2qrs. To what? Viz. 308+21=3291b: 1460 d. :: 1400lb:-Then 1460x1400=2044000. And 329) 2044000 (6212 d. 251.17 5.84 d. the Answer required. The fame Question stated in Decimals will stand Thus 2,9375:6,0833 :: 12,5: To the Antwer. Then 6,0833 x 12,5 = 76,04125, which being divided by 2,9375 will give 25,8863, &c. the Answer in Decimals, which brought into Coin, will be 25 l. 175. 8 d. as before. Note, When the first Term is an Unit or 1, the Question is answered by Multiplication only. Example. Suppose I give 5 Shillings 4 Pence for one Ounce of Silver, What must I pay for 32 Ounces at the same Rate ? That is I Ounce: 5s. 4d.:: 32 Ounces: To, &c. Which is best stated thus 1 : 64d.:: 32,5: Then Then 32,5x64=2080d.=81. 135. 4 d. the Answer required. For I neither multiplies nor divides. When the second or third Term is an Unit or 1, then the Question is answered by Division only. As in this Example. If a Silver Tankard weighing 21 Ounces, cost 51. 195. What is that an Ounce? Thus 21 oz.: 51. 1195. = 195. :: 1:55. 8 d. the Answer. That is 21) 119 (=55.=5 s. 8 d. The Proof of all Questions in the Rule of Three Direct, may be easily conceived from what hath been already faid; viz. That the Product of the first and fourth Terms, must always be be equal to the Product of the second and third Terms. Or otherwise, by varying the Question, as in the second, third, fourth, and fifth Questions. I shall conclude this Section with inferting a few Questions and their Answers; leaving their Work for the Learner's Practice. Quest. 1. What will the Carriage of 17 C. 3 qrs. 11 lb. come to, at the Rate of 7 s. the Hundred? Answer 61. 45. 11 d. Quest. 2. If 61. 45. 11 d. be paid for the Carriage of 17 C. 3 qrs. 11lb; What was paid for the Carriage of 1 lb? Answer 3 Farthings. Queft. 3. A Grocer bought 3 C. 1qr. 14 lb Weight of Cloves, at the Rate of 2 s. 4 d. per Pound, and fold them for 521. 14 5. Whether did he gain or lofe by the Bargain, and how much? Answer, he gained 81. 125. Quest. 4. A Draper bought of a Merchant eight Packs of Cloth; every Pack had four Parcels in it; and each Parcel contained ten Pieces; every Piece was Twenty-fix Yards; he gave after the Rate of four Pounds fixteen Shillings for 6 Yards. What came the eight Packs to, and what were they worth per Yard? Anfw. They came to 66561. And were worth 16 s. per Yard. Quest. 5. A Merchant bought 436 Yards of Broad Cloth for 8 s. 6 d. per Yard; and fold it again for 105. 4 d. per Yard. What did he gain by the 436 Yards? Anfw. he gained 391. 195. 4 d. Quest. Quest. 6. A Goldsmith bought a Wedge of Gold, which weighed 14 lb. 3 oz. 8 pw. for 5141. 45. What did he pay per Ounce? Answ. 3 1. per Ounce. Quest. 7. What will 48 oz. 17 pw. 20 Grains of Silver Plate come to, at the Rate of 5 s. 6d. per Ounce? Answ. 131. 15. 10d. Quest. 8. If in four Weeks one spend 13 s. 4 d. How long will 531. 6s. last at that Rate? Answ. 6 Years, 47 Days, 2 Hours, 24. Quest. 9. What will the one eighth Part of a Ship be worth when the half is valued at 10151. 105. Answ. 2531. 17 s. 6d. Quest. 10. The Sun is faid to perform one entire Revolution, (or 360 Degrees) in the Space of 365 Days, 5 Hours, 48 Minutes, and 57 Seconds of Time, called a Tropical or Solar Year; How much doth it move in one Day? 11 Answ. 59 8 19 &c. Quest. 11. If of a Yard of Velvet cost of a Pound Sterling, What will of a Yard cost of the same Velvet at that Rate? Answ. 1 s. 4 d. Quest. 12. Suppose 21. and of of a Pound Sterling will buy 3 Yards and off of a Yard of Cloth, How much will of a Yard cost at that Rate? Sect. 2. Of Reciprocal Proportion; usually called The Rule of Three Inverse. PEciprocal Proportion is, when of four Numbers the third (viz. that which moves the Question) beareth the fame Ratio to the first: As the second does to the fourth. Therefore, the less the third Term is, in respect to the first; the greater will the fourth Term, be in respect to the second. EXAMPLE 1. If fixteen Men can do a Piece of Work in fix Days; How many Days must eight Men require to do the fame Work, at the fame Rate of working? Here it is plain that eight Men must needs have more Time than 16 Men to do the same Work. Consequently the greater N 2 the |