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8X4 X5 X 400 10×3×3

Canceling the 10 of the denominator against

a part of the 400 of the numerator, we get

8X4 X5 X 40

3x3

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3. If a pile of wood 8 feet long, 4 feet wide, and 4 feet high, contain one cord of wood, how many cords are there in a pile 26 feet long, 8 feet wide, and 12 feet high?

In this example one cord is the odd term. The first couplet consists of lengths; and since 26 feet long will give more wood than 8 feet, we shall have 26 for the first ratio.

The second couplet consists of widths; and since 8 feet wide will give more than 4 feet, we get & for the second ratio.

The third couplet consists of heights; and since 12 feet high will give more wood than 4 feet, we get for the third ratio.

Multiplying these ratios and the odd term, 1 cord, together, Canceling the 8 of the numerator

we get

26 × 8 × 12
8X4X 4

against the 8 of the denominator; also one of the 4's of the denominator against a part of the 12 of the numerator, and the factor 2, of the remaining 4 of the denominator, against the factor 2 of 26, in the numerator; our expression by this means becomes 19 cords, for the answer.

13×3

2

4. If a man perform a journey of 1250 miles in 17 days, by traveling 13 hours a day, how many days will it take him to perform a journey of 1007 miles, by traveling 10 hours each day? Ans. 1710947 days.

5. If 10 cows eat 8 tons of hay in 6 weeks, how many cows will eat 56 tons in 21 weeks? Ans. 20 cows.

6. If 8 men will mow 36 acres of grass in 9 days, by work. ing 9 hours each day, how many men will be required to mow 48 acres in 12 days, by working 12 hours each day?

Ans. 6 men.

7. If 11 men can cut 49 cords of wood in 7 days, when they work 14 hours per day, how many men will it take to cut 140 cords in 28 days, by working 10 hours each day?

Ans. 11 men.

8. If 12 ounces of wool make 2 yards of cloth, that is 6 quarters wide, how many pounds of wool will it take to 150 yards of cloth, 4 quarters wide?

Ans. 30 pounds.

9. If 12 men can build a wall 26 feet long, 7 feet high, and 5 feet thick, in 20 days, in how many days will 28 men build a wall 156 feet long, 10 feet high, and 3 feet thick? Ans. 44.52 days.

10. If the wages of 6 men for 14 days be 84 dollars, what will be the wages of 9 men for 16 days?

Ans. 144 dollars.

11. If 100 men in 40 days of 10 hours each, build a wall 30 feet long, 8 feet high, and 2 feet thick, how many men must be employed to build a wall 40 feet in length, 6 feet high, and 4 feet thick, in 20 days, by working 8 hours each day? Ans. 500 men.

12. If a pile of wood 30 feet long, 4 feet wide, and 6 feet high, is worth 25 dollars, how much is a pile 60 feet in length, 3 feet wide, and 4 feet high, worth?

Ans. $24.

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13. If 176 bushels of corn, when corn is worth 60 cents a bushel, be given for the carriage of 120 barrels of flour 60 miles, how many bushels of corn, when corn is worth 70 cents a bushel, must be given for the carriage of 80 barrels of flour 230 miles?

14. A wall was to be built 700 yards long in 29 days; after 12 men had been employed on it for 11 days, it was found they had built only 220 yards. How many more men must be put on to finish it in the given time?

15. In how many days, working 9 hours a day, will 24 men dig a trench 420 yards long, 5 yards wide, and 3 yards deep, if 248 men, working 11 hours a day, in 5 days dig a trench 230 yards long, 3 yards wide, and 2 yards deep?

16. If 25 pears can be bought for 10 lemons, and 28 lemons for 18 pomegranates, and 1 pomegranate for 48 almonds, and 50 almonds for 70 chestnuts, and 108 chestnuts for 2 cents, how many pears can I buy for $1.35?

17. Suppose that 50 men, by working 5 hours each day, can dig, in 55 days, 24 cellars, which are each 36 feet long, 21 feet wide, and 10 feet deep, hów many men would be required to dig, in 27 days, 18 cellars, which are each 48 feet long, 28 feet wide, and 8 feet deep, provided they work only 3 hours each day?

18. If 15 men eat 13 shillings' worth of bread in 6 days, when wheat is sold at 12 shillings per bushel, in how many days will 30 men eat 43 shillings' worth, when wheat is 10 shillings per bushel ?

CHAPTER VI.

ARITHMETICAL PROGRESSION.

62. A series of numbers, which succeed each other regularly, by a common difference, are said to be in arithmetical progression.

When the terms are constantly increasing, the series is an arithmetical progression ascending.

When the terms are constantly decreasing, the series is an arithmetical progression descending.

Thus, 1, 3, 5, 7, 9, &c., is an ascending arithmetical progression; and 10, 8, 6, 4, 2, is a descending arithmetical progression.

In arithmetical progression there are five things to be considered:

1. The first term.

2. The last term.

3. The common difference.

4. The number of terms.

5. The sum of all the terms.

These quantities are so related to each other, that any three of them being given, the remaining two can be found.

Hence, there must be 20 distinct cases, arising from the different combinations of these five quantities.

To give a demonstration to all the rules of these 20 cases

would be a very difficult task for the ordinary rules of arithmetic: we will therefore content ourselves with demonstrating a few of the most important of them.

CASE I.

By our definition of an ascending arithmetical progression, it follows, that the second term is equal to the first, increased by the common difference; the third is equal to the first, increased by twice the common difference; the fourth is equal to the first, increased by three times the common difference; and so on for the succeeding terms.

Hence, when we have given the first term, the common dif. ference, and the number of terms, to find the last term, we have this

RULE.

To the first term, add the product of the common difference into the number of terms, less one.

Examples.

1. What is the 100th term of an arithmetical progression, whose first term is 2, and common difference 3?

In this example the number of terms, less one, is 99, which, multiplied by the common difference, 3, gives 297, which added to the first term, 2, makes 299 for the 100th term.

2. What is the 50th term of the arithmetical progression, whose first term is 1, the common difference? Ans. 25.

3. A man buys 10 sheep, giving 1 dollar for the first, 3 dollars for the second, 5 dollars for the third, and so increasing in arithmetical progression. What did the last sheep cost at that rate? Ans. 19 dollars.

4. The first term of an arithmetical progression is, the

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