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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.


i Admit a debt be discharged at 16 several payments in arithmetical progression ; the first to be 141, the last sool. 'what is the common difference, and what each payment, and the whole debt?

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14+100 X 8=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 young 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.


of the first person.

of the second. 22 +2=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?

- 3

miles. Distance







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 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. 1. Sold 24 yards of Holland, at ad. 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,
2 4 8 16 leading terms:


256 8th term.

65536 16th term.

16777216 24ih last term

z ratio.


2 first term.

12)33554430 sum of the series. 1,0) 279620,2 6

4.139810,2 6 answet.

2 Bought



2 Bought 30 bushels of wheat; the first bushel for 2d. the second 4d. the third 8d. doubling the price of each preceding bushel for, that of the next; query the amount, and price per bushel at an average ?

$ 89478481 ios 6d. Amount. {

2982611 123 4d. per Bushel. 3 Sold 15 yards of sattin, the first yard for Is, the second for 25. the third for 45. &c. what sum did they amount to?

answer. 16381 75. 4 Admit a goldsmith sold one lb. 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 4l. per ounce ?

5825l 8s 5d. Sold for.

57771 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 4473924/ 55. 3d. 6 Sappose 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 laboųe come to, rating the barley at 2s 6d. per bushel ?

answer 1773! 75 60 Note 7680 wheat, or barley corns, are supposed 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 sol. per yard? 21446992921 ,

. 8 A certain person married his daughter on new year's day, and gave her one guinea towards her portion, promising to double it on the first day of every month for one year s what was her portion in sterling money?

answer 4299! 155

answer {214469971921 13. od Gained.

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Note. The ratio is the Interest of 11. for one year, and is thus found.

£ £o
100w..5 :: I.,05

5,5 :: I.. •,055

,06 &C. Which is only dividing the rate per cent. by 100, by moving the point 2 places to the left.


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A TABLE of Ratios.

| Rate per Cent. Ratio. | Rate per








.055 6 .06



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9 9를

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 interest by decimals, the money should be in

the denominations of pounds, or dollars, and the time in years, with their parts (if any) annexed decimally.

EX AMPLES 1 Required the amount of 5377 1os. at 6 per cent. per annum for 5 years ? Principal 537,5 X 5X,06=161,25 Interest.

537,5 Principal.
£ 698,75=6981 155. answer.

2 What is the interest of 9171. tás. at 5 per cent. per annum for 7 years ?

answer 221/ 4s 7d. 3

If my correspondent be to have 41 per cent. what will his commission on 3911 175. come to ? ans. 171 125 70++

4 What will be the interest and amount of 5671 ios, in 9 years, at 6 per centper annum?

S 3061 9s. Interest. answer

28731 195. Amount. 5

What is the interest of 47261 185 6d. for 3 years, at 7 per cent per amnum ?

answer 11581 is 11d. 6 What will 95261 125 9d. amount to in 12 years and 9 months, at 7 per cent. per annum ? - answer 18029/ 35 2d.


A simples of compound



LLIGATION is a rule for adjusting the prices and


CASE 1: When several simple quantities, and their prices, are givens and a mean price of any part of the compound is required.

As the sum of the several quantities,
Is to their total value ;
So is any part of the composition,
To its value.

EXAMPL E So 1 If 19 bushels of wheat at 6s. the bushel, 40 of rye at 4s. and 12 of barley at 35. be mixt together; what is a bushel of this mixture worth?

19 at 6=114
40 at 4=160
12 at 3= 36

72 0310(4 4 answer.
2 A

2C wt. at 56s. 1С.wt. at 43%. and 2C.wt. at 5 os. per Cowt, what is 3C.wt. of this mixture worth?

answer 71 135.

mixed sugars ;

3 IF

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