CASE 2. When the two extremes and number of terms are given, and the common difference of all the terms required; RULE. Divide the difference of the extremes by the number of terms, less one, the quotient will be the common difference. 1 Admit a debt be discharged at 16 several payments in arithmetical progression; the first to be 14, the last 100l. what is the common difference, and what each payment, and the whole debt? 100-14 16- 1 £. s. d. 5 14 8 common difference. 14 0 o the first payment. 19 14 8-second. 25_9_4=third, &c. 14+100X8=9121. the whole debt. 2 A man had 10 sons, whose several ages differed alike; the youngest was 3 years old, and the eldest 48; what was the common difference of their ages? answer 5 years. 3 There are 21 persons, whose ages are equally distant from each other; the youngest is 20 years old, and the eldest 60; what is the common difference of their ages, and the age of each person? answer Common difference 2 years. 20 the age of the first person. 20+2=22 of the second. 2242 24 of the third, &c. 4 A footman is to travel from Philadelphia to a certain place in 19 days, and to go but six miles the first day, in-. creasing every day by an equal excess, so that the last day's journey may be 60 miles; what is the common difference, and distance of the journey? answer Common difference3 miles. GEOMETRICAL GEOMETRICAL PROGRESSION. GEOMETRICAL Progression is a series of numbers, increasing by a common multiplier, or decreasing by a common divisor, called the ratio; as, 2, 4, 8, 16, 32, &c. increase by the multiplier 2 ; and 32, 16, 8, 4, 2, decrease continually by the divisor 2, &c. The last term and sum of the series are found by this RULE. Raise the ratio to the power whose index is one less than the number of terms given, which multiply by the first term, that product is the last term or greater extreme. Multiply the last term by the ratio, from the product subtract the first term; and divide the remainder by the ratio less one; the quotient will be the sum of the series. EXAMPLE S. I Sold 24 yards of Holland, at zd. for the first yard, 4d. the second, 8d. the third, &c. in a duplicate proportion ; how much do they amount to? 1 2 3 * 4 indices, 16 256 8th term. 256 256 16777216 24ih last term, 2 ratio. 33554432 2 first term. 12)33554430 sum of the series. 1,0) 279620,2 6 6.139810,2 6 answers 2 Bought 2 Bought 30 bushels of wheat; the first bushel for 2d. the second 4d. the third 8d. doubling the price of each pre ceding bushel for that of the next; query the amount, and price per bushel at an average? { S89478481 10s 6d. Amount. 298261/ 125 4d. per Bushel. 3 Sold 15 yards of sattin, the first yard for 1s. the second for 25. the third for 4s. &c. what sum did they amount to? answer 16381 75. answer 4 Admit a goldsmith sold one . of gold, at one farthing for the first ounce, a penny for the second, 4d. for the third, &c. in a quadruple proportion; what did it amount to ? and what did he gain by it, supposing it cost him 41. per ounce? 58251 8s 5d. Sold for. answer 157771 8s 5d. Gained. 5 What sum would purchase a horse with 4 shoes, and 8 nails in each shoe, at one farthing for the first nail, a halfpenny for the second, a penny for the third, &c. doubling to the last? answer 44739241 5s. 3d.4 6 Suppose a man wrought 20 days, and received for the first day 4 barley corns, for the second 12, for the third 36, &c. in a triple proportion; what did the twenty days labour come to, rating the barley at 2s 6d. per bushel? answer 1773! 75 6d Note. 7680 wheat, or barley corns, are fuppofed to make a pint. 7 Sold 30 yards of velvet, at 2 pins for the first yard, 6 for the second, 18 for the third, &c. and these disposed of at one farthing per 100, how much did the velvet amount to? And whether did the seller gain or lose, and how much, supposing the prime cost of the velvet at 50l. per yard? S2144699292/ 13s od Amount, answer 2144697792/ 13s od Gained. 8 A certain person married his daughter on new year's day, and gave her one guinea towards her portion, promis ing to double it on the first day of every month for one year; what was her portion in sterling money? answer 4299! 155. SIMPLE SIMPLE INTEREST-BY DECIMALS. Note. The ratio is the Interest of 11. for one year, and is thus found. As £.£. £. £. 100... 5 : 1..,05 100 5,5 1,055 100 6 :: I ,06 &c. Which is only dividing the rate per cent. by 100, by moving the point 2 places to the left. A TABLE of Ratios. Rate per Cent. | Ratio. | Rate per Cent. Ratio. The principal, time, and ratio given, to find the interest, and amount? RULE. Multiply the principal, time, and ratio together, the last product will be the interest, commission, brokage, &c. to which add the principal, and the sum will be the amount. Note. In operations of intereft by decimals, the money fhould be in the denominations of pounds, or dollars, and the time in years, with their parts (if any) annexed decimally. EXAMPLES. I Required the amount of 537 10s. at 6 per cent. per annum for 5 years? Principal 537,5X5 X,06-161,25 Interest. 537.5 Principal. £698,75=6981 15s. answer. 2 What is the interest of 917/ 16s. at 5 per cent. per annum for 7 years? answer 2211 45 7d. 3 If my correspondent be to have 44 per cent. what will his commission on 391/ 175. come to? ans. 17! 12s 7d++ 4 What will be the interest and amount of 567/ 10s, in 9 years, at 6 per cent. per annum ? S306 9s. Interest. answer 873/ 19s. Amount. 5 What is the interest of 4726/ 18s 6d. for 31 years, at 7 per cent per annum ? answer 1158/ Is 1rd. 6 What will 9526/125 `9d. amount to in 12 years and months, at 7 per cent. per annum? answer 180291 3s 2d. 9 A ALLIGATION. LLIGATION is a rule for adjusting the prices and When several simple quantities, and their prices, are given, and a mean price of any part of the compound is required. RULE. If As the sum of the several quantities, Is to their total value; So is any part of the composition, 19 EXAMPLE ST bushels of wheat at 6s. the bushel, 40 of rye at 4s. and 12 of barley at 3s. be mixt together; what is a bushel of this mixture worth? 71 )310(4 44 answer. 2 A grocer mixed sugars; 2C wt. at 56s. 1C.wt. at 435. and 2C.wi. at 50s. per G.wt. what is 3C.wt. of this mixture worth? answer 71 135. |