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22. Four men, A, B, C, and D, bought a grindstone, the dia meter of which was 40 inches and the place for the shaft 4 inches in diameter. It was agreed that A should grind off his share first, then in turn B, C, and D. Required how many inches each man will grind off from the semidiameter, providing they each paid the same sum.

Ans. A, 2.65lin. ; B, 3.137in. ; C, 4.064in.; and D, 8.148in.

23. I have a board whose surface contains 493 square feet; the board is 14 inches thick, and I wish to make a cubical box of it. Required the length of one of its equal sides.

Ans. 36 inches. 24. A carpenter has a plank 1 foot wide, 2243 feet long, and 21 inches thick ; and he wishes to make a box whose width shall be twice its height, and whose length shall be twice its width. Required the contents of the box.

Ans. 5719 cubic inches. 25. If a ball, 3 inches in diameter, weigh 4 pounds, what will be the weight of a ball that is 6 inches in diameter?

Ans. 32lbs. 26. If a globe of gold, one inch in diameter, be worth $ 120, what is the value of a globe 34 inches in diameter?

27. If the weight of a well-proportioned man, 5 feet 10 inches in height, be 180 pounds, what must have been the weight of Goliath of Gath, who was 10 feet 4 inches in height?

Ans. 1015.1+1b. 28. If a bell, 4 inches in height, 3 inches in width, and į of an inch in thickness, weigh 2 pounds, what should be the dimensions of a similar bell that would weigh 2000 pounds ?

Ans. 3ft. 4in. high, 2ft. 6in. wide, and 2din. thick. 29. What are the two mean proportionals between 56 and 12096 ?

Ans. 336 and 2016. 30. Having a small stack of hay, 5 feet in height, weighing 1cwt., I wish to know the weight of a siinilar stack that is 20 feet in height.

Ans. 64cwt. 31. If a man dig a small square cellar, woich will measure 6 feet each way, in one day, how long would it take him to dig a similar one that measured 10 feet each way?

Ans. 4.629+ days. 32. If an ox, whose girth is 6 feet, weighs 600lb., what is the weight of an ox whose girth is 8 feet? Ans. 1422.2+lb.

33. Four women own a ball of yarn, 5 inches in diameter. It is agreed that each shall wind off her share from the ball. How many inches of its diameter shall each wind off? Ans. First, .45+ inches ; second, .57+ inches ; third, .82+

inches; fourth, 3.149+ inches. 34. John Jones has a stack of hay in the form of a quadrangular pyramid. It is 16 feet in height, and 12 feet wide at its base. It contains 5 tons of hay, worth $ 17.50 per ton. Mr. Jones has sold this hay to Messrs. Pierce, Row, Wells, and Northend. As the upper part of the stack has been injured, it is agreed that Mr. Pierce, who takes the upper part, shall have 10 per cent. more of the hay than Mr. Rowe; and Mr. Rowe, who takes his share next, shall have 8 per cent. more than Mr. Wells; and Mr. Northend, who has the bottom of the stack, that has been much injured, shall have 10 per cent. more than Mr. Wells. Required the quantity of hay, and how many feet of the height of the stack, beginning at the top, each receives. Ans. Pierce receives 27543cwt. and 10.366+ feet in height;

Rowe, 2414cwt. and 2.493 feet ; Wells, 22344cwt. and 1.666 feet; Northend, 252-25cwt. and 1.474 feet.

PROGRESSION, OR SERIES.

554. A SERIES is a succession of numbers that depend on one another by some fixed law.

The numbers constituting a series are called its terms ; of which the first and last are called extremes, and the other terms the means.

ARITHMETICAL PROGRESSION.

555. ARITHMETICAL PROGRESSION, or PROGRESSION BY DIFFERENCE, is a series that increases or decreases by a constant number, called the common DIFFERENCE.

The series is said to be an ascending one when each term

after the first exceeds the one before it ; and a descending one when each term after the first is less than the one before it.

Thus, 1, 5, 9, 13, 17, 21, 25, 29, 33, is an ascending series, in which each term after the first is derived from the one preceding it by the addition of the common difference 4; and 25, 22, 19, 16, 13, 10, 7, 4, 1, is a descending series, in which each term after the first is derived from the one preceding it by the subtraction of the common difference 3.

556. In arithmetical progression, the first term, the last term, the number of terms, the common difference, and the sum of the terms, are so related to each other, that, three of these being given, the two others may be readily determined.

OPERATION.

557. To find the common difference, the extremes and number of terms being given.

· Ex. 1. The extremes of an arithmetical series are 3 and 45, and the number of terms is 22. Required the common difference.

Ans. 2. It is evident that the number of common 45 3

differences in any series must be 1 less than = 2. the number of terms. Therefore, since the 22 1

number of terms in this series is 22, the num

ber of common differences will be 22 - 1 21, and their sum will be equal to the difference of the extremes ; hence the difference of the extremes, 45 — 3 = 42, divided by the number of common differences, 21, gives 2 as the common difference required.

RULE. Divide the difference of the extremes by the number of erms less one, and the quotient will be the common difference.

EXAMPLES.

2. A certain school consists of 19 teachers and scholars, whose ages form an arithmetical series; the youngest is 3 years old, and the oldest 39. What is the common difference of their ages ?

Ans. 2 years. 3. A man is to travel from Albany to a certain place in 11 days, and to go but 5 miles the first day, increasing the distance equally each day, so that the last day's journey may be 45 miles. Required the daily increase.

Ans. 4 miles.

OPERATION.

558. To find the number of terms, the extremes and common difference being given.

Ex. 1. If the extremes of an arithmetical series are 3 and 19, and the common difference is 2, what is the number of terms ?

Ans. 9. It is evident, that, if the difference of the 19 3

extremes be divided by the common dif+1= 9. ference, the result will be the number of 2

common differences; thus 19 — 3 = 16;

16 • 2 = 8. Then, as the number of terms must be 1 more than the number of common differences, 8 + 1= 9 is the number of terms in the series.

RULE. - Divide the difference of the extremes by the common difference, and the quotient increased by 1 will be the requircil number of

terms.

EXAMPLES

2. A man going a journey travelled the first day 7 miles, the last day 51 miles, and each day increased his journey by 4 miles. How many days did he travel?

Ans. 12. 3. In what time can a debt be discharged, supposing the first week's payment to be $1, and the payment of every succeeding week to increase by $ 2, till the last payment shall be $ 103 ?

Ans. 52 weeks.

OPERATION,

X 9

559. To find the sum of all the terms, the extremes and number of terms being given.

Ex. 1. The extremes of an arithmetical series are 3 and 19, and the number of terms 9. Required the sum of the series.

Ans. 99.

In an arithmetical 3 + 19

series the sum of the 99, Ans.

extremes is equal to 2

the sum of two terms Or, 3+ 19 = 22; 22 X 44 = 99, Ans. that are equally dis

tant from them, or to double the middle term, if the number of terms be odd. Thus, in the series, 3, 5, 7, 9, 11, 13, 15, 17, 19, the sum of 3 and 19 is equal to the sum of 5 and 17, or of 7 and 15, and is double the middle term, 11. The reason of this is evident, since 5 and 7 exceed the less extreme by the same quantities by which 17 and 15 are respectively less than the other extreme.

Hence, in this latter series, it is evident that, if each term were made 11, half the sum of the extremes, the sum of the whole would remain the same; therefore the sum of the series must equal half the sum of the extremes multiplied by the number of terms, or the sum of the extremes multiplied by half the number of terms.

RULE. · Multiply half the sum of the extremes by the number of terms. Or,

Multiply the sum of the extremes by half the number of terms.

EXAMPLES.

2. If the least term of a series of numbers in arithmetical progression be 4, the greatest 100, and the number of terms 17, what is the the sum of the terms ?

Ans. 884. 3. Suppose a number of stones were laid a rod distant from each other, for thirty miles, the first stone being a rod from a basket. What distance will that man travel who gathers them up singly, returning with them one by one to the basket ?

Ans. 288090 miles 2 rods. 560. To find the sum of the terms, the extremes and common difference being given.

Ex. 1. If the two extremes are 3 and 19, and the common difference is 2, what is the sum of the series ?

It has been shown (Art. 558) that, 19 - 3

if the difference of the extremes be +1=9; divided by the common difference, 2

the quotient will be the number 19 + 3

of terms less one. Therefore the X 9 99, Ans.

number of terms less one will be 2

8; and 8 +1 will equal the number of terms. It has also been shown (Art. 559) that, if the number of terms be multiplied by the sum of the extremes, and the product divided by 2, the quotient will be the sum of the series; therefore 19 + 3

99, the answer required. RULE. Divide the difference of the extremes by the common difference, and to the quotient add 1; by this sum multiply half the sum of the extremes, and the product will be the sum required.

OPERATION

19-3

2

x 9

2

EXAMPLES

2. If the extremes are 3 and 45, and the common difference 2, what is the sum of the series?

Ans. 528. 3. A owes B a certain surn, to be discharged in a year, by

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