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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.

EXAMPLES. 1 Admit a debt be discharged at 16 several payments in arithmetical progression ; the first to be 141. the last 1001. what is the common difference, and what each payment, and the whole debt!

£o. s. d.
100-14 5 14 8 common difference.
16-1 14 0 0 the first payment.

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ages

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 ? answer 5 years.

3 There are 21 persons, whose ages are equally distant from each other; the youngest 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.

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, increasing 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?

s Common difference 3 answer

miles. Distance

627 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.

EXAMPLES. 1 Sold 24 yards of Holland, at 2d. for the first yard, 4d. the second, šd. the third, &c. in a duplicate proportion ; how much do they amount to ?

1 2 4 indices,
2 4 8 16 leading terms.

16

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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 10s 6d. `Amount. 2982611 12s 4d.

per

Bushel. 3 Sold 15 yards of sattin, the first yard for 1s, the se. cond for 2s. the third for 4s. &c. what sum did they amount to ?

answer 16381 78. 4 Admit a goldsınith 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 ?

$ 58251 8s 5d. Sold for.

answer 5 What sum would purchase a horse with 4 shoes, and 8 nails in each shoe, at one farthing for the first nail, a half-penny for the second, a penny for the third, &c. doubling to the last ?

answer 44739241 5s 3d. 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 17731 78 6d. 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 50l. per yard?

521446992921 138 Od; Amount. answer

21446977921 13s Od Gained. 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; what was her portion in sterling money?

unswer 42991 15s.

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SIMPLE INTEREST-By DECIMALS.

&

Note. The ratio is the Interest of 1l, for one year, and is thus found.

£ &

1010 : 5 :: 1 ,05 As 100 : 5,5 :: 1 ,055

100 : 6 :: 1 ,06 &c. Which is only dividing the rate per cent. by 100, by mov. ing the point two places to the left.

A TABLE OF Ratios. Rate per Cent. Ratio. Rate per Cent. Ratio. .02

65

065 3

7

.07
31
.035

71

.075 .04

8

.08
41
.045

81

:.085 .05

9

.09
52
055

.095
6

10

.1

03

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.06

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 de.

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

EXAMPLES. 1 Required the amount of 5371 -10s. at 6 per cent, per annum for 5 years ? Principal 537,5X5X06=161,25. Interest.

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

2 What is the interest of 9171 16s. at 5 per cent. per annum for 7 years ?

answer 2211 4s 7d. 3 If my correspondent be to have 41 per cent. what will his commission on 3911 17s. come to ?

answer 171 12s 7d.1t 4 What will be the interest and amount of 5671 10s. in 9 years, at 6 per cent. per annum ?

3061 9s. Interest.

answer

{8981 198. Amount

:

5 What is the interest of 47261 18s 6d. for 3 years, at 7 per cent. per annum ?

answer 11587 is ild. 6 What will 95261 12s 9d. amount to in 12 years and 9 months, at 7 per cent. per annum ? answer 180291 3s 2d.

A

ALLIGATION. LLIGATION is a rule for adjusting the prices and simples of compound quantities.

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

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

EXAMPLES. i If 19 bushels of wheat at 6s. the bushel, 40 of rye.at 4s. and 12 of barley at Ss. be mixt together; what is a bushel of this mixture worth?

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

d. 71 )310(4 44 answer. 2 A grocer mixed sugars ; 2C.wt. at 56$. 1С.wt. at 43s. and 2C.wt. at 50s. per C.wt. what is 3 C.wt. of this mixture worth?

S.

S.

answer 71 13s.

3 If

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