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MISCELLANEOUS PROBLEMS. 1. If 63 barrels of flour cost $51%, what will 48 barrels cost?

Ans. $35. 2. If 93 pounds of sugar cost $2.25, what will 12 pounds cost?

Ans. $3.06. 3. If 51 tons of hay cost $283, how many tons will $85} buy?

Ans. 16 tons. 4. The sum of two fractions is jis, and one is 123 ; wbat is the other ?

Ans. 3645 5. The difference of two fractions is 311, and the greater is 1634 ; what is the less ?

Ans. 1935 . 6. The multiplicand is is, and product *; required the multiplier.

Ans. 149. 7. The divisor is , and quotient 25%; what is the dividend ?

Ans. ?? 8. The dividend is 139zand quotient $4 ; required the divisor.

Ans. . 9. Divide the fraction it into two parts, one of which is 21 times the other.

Ans. } ; 1826 10. The sum of two fractions is g, and difference 4; re. quired the fractions.

Ans. Por ai 176 11. One-balf of the sum of two fractions is 328, and twice the difference is zao; required the fractions.

Ans. 3558; 793. 12. What is the value of (+-3)*(4+5-1%), divided by 3 ?

Ans. 12. 13. What is the value of (51–+277)+(34–1}+21) multiplied by 21 divided by 11?

Ans. 2431 14. Divide 4 of 1 of 74 by 5 of 1% of 54. Ans. 131.

15. Multiply it of 5 of 3 by $ of 56,of 544.

Ans. 367864 16. Subtract 1 of 1 from of 2 Ans. $984. 17. Ada 4 of 411 of 3745, 973 of 4 of , and 97

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19. Find the value of 1 of

-1 4-1

4 - 5

. Ans. * 20. Find the value of 5 x 53 x 51 x 51-1

* + 363865 5}x 51 x 54-1

Ans. 51906021 2-1 (33)2 21. Find the value of

-+(2+1)+3+1+;

Ans. 21115%. 22. Find the value of

2+49-ito 44 x 52001,

Ans. 3. 23. What is the sum of }, j, 4, 5, ž, 10, 12, 14, 16, and 13 ?

Ans. 83523 24. A man sold of 19 of his bank stock in a month; how many fifths of 11 remained ?

Ans. 3} fifths. 25. If I pay $0.622 a cord for sawing wood 4 feet longinto 3 pieces, how much more should I pay for sawing wood 8 feet long into pieces of the same length ? Ans. $0.15%.

26. A dry goods merchant bought silk for $613] at $1} a yard, and sold of the quantity bought at a profit of of a dollar a yard; what did he receive for the part sold ?

Ans. $27724 27. A bought of B 17} tons of hay at $11s a ton, and of C 221 tons at $121 a ton, and then sold D 15 tons at $134 a ton, and the remainder E bought at $13; a ton; what was A's gain?

Ans. $541.. 28. Required the least number of yards of velvet, allowing 1 yard for waste, that can be cut up without loss into bonnets and hats, one style of bonnet requiring 14 yd., and another žyd., and a hat requiring á yd. Ans. 36 yd.

29. A grocer bought 25 barrels of apples at $4 a barrel; he sold Mr. Smith of them at $51, but finding they were

beginning to spoil, and wishing to get rid of them, he sold the remainder to Mr. Brown at $4 a barrel; what did be gain or lose by the whole transaction ? Ans. Lost $41.

30. Samuel Jackson agreed to work for a farmer a year, receiving as wages $300 and a suit of clothes. Having worked 8 months, his employer sold his farm, and Jackson received as his pay $186; and the clothes; what was the value of the suit ?

Ans, $40. 31. Three men start at the same time to walk around a circular race-course 80 rods in circumference, the first, walking 26 rods, the second 35 rods, and the third 50 rods a minute; when are they first together after starting, and how far from the starting point ? Ans. 263 min.; 53} rods.

32. A steamboat starts from Memphis, Tenn., to go up the Missouri River to a point 10111 miles from the starting place. Her rate is 10} miles an hour for 123 hours a day, anchoring at night for fear of snags; but when the voyage is half completed, the anchor is lost, and she then drifts back every night at the rate of 1 miles per hour; how many days did the voyage require ? Ans. 87139 days.

33. In a piece of machinery there are 3 wheels, A, B, and C, each measuring 11 feet in circumference, their axles being in a straight line. If these wheels begin to revolve, A at the rate of 6 feet in a second, B 7. feet, and C 93 feet, how long before the given points will again be in a straight line, and how many revolutions will each wheel have made?

Ans. 140 sec.; A, 78 rev.; B, 87; C, 116. 34. Three men were employed to plow a field; the first plowed a furrow in 174 minutes, the second in 234 minutes, and the third in 2611 minutes, and it so happened that they all came to the end of their furrow at the same moment for the first time when the work was finished. How long did they work, how many furrows did they plow, and how much should each receive, if $65.10 was paid for the work ?

Ans. 47144 min.; 651; 1st, $27.50; 2d, $20; 3d, $17.60.

SECTION V.
DECIMAL FRACTIONS.

257. A Decimal Fraction is a number of the decimal divisions of a unit.

258. A Decimal Division of a unit is a tenth, a hundredth, a thousandth, etc. A decimal fraction is thus a number of tenths, hundredths, etc.

259. A Decimal Fraction is usually expressed by placing a point before the numerator and omitting the denominator ; thus .5 expresses .

260. The Symbol of a decimal is the period, called the decimal point, or separatrix. It indicates the decimal and separates decimals and integers.

261. The places at the right of the decimal point are called decimal places. The first place to the right of the point is tenths, the second place is hundredths, etc.

262. This method of expressing decimal fractions arises from the decimal scale used for integers by continuing it to the right of units.

Thus, since tens is 1 tenth of hundreds, and units 1 tenth of tens, if we write a figure to the right of units it will express 1 tenth of units or tenths; two places to the right, 1 tenth of tenths or hundredths, etc.

263. This beautiful law, as applied to the expression of integers and decimal fractions, is exhibited in the following

DECIMAL NOTATION TABLE.

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264. A Decimal is a fraction expressed by the decimal notation; thus .5 is a decimal, while it is a decimal fraction.

265. A Pure Decimal is one which consists of decimal figures only; as, .25 and .345.

266. A Mixed Decimal is one which consists of an integer and a decimal; as, 6.75.

267. A Complex Decimal is one which contains a common fraction at the right of the decimal; as, .34%.

268. A Terminate Decimal is one which ends; an Interminate is one which does not end.

NOTES.-1. Decimals may originate by passing from common fractions to decimals, or by an extension of the decimal scale to the right of units.

2. Decimal fractions appear to have been first used by Regiomontanus, about the year 1464. The first treatise upon the subject was written by Stevinus, published in 1585.

3 The decimal point, Dr. Peacock thinks, was introduced by Napier, the inventor of logarithms, in 1617; though De Morgan says that Richard Witt made as near an approach to it as Napier.

. PRINCIPLES OF DECIMAL NOTATION.

1. Moving the decimal point one place to the right, multiplies the decimal by 10; two places, multiplies by 100, etc.

For, if the point be moved one place to the right, each figure will express ten times as much as before, hence the whole decimal will be ten times as great; etc.

2. Moving the decimal point one place to the left, divides the decimal by 10; two places, divides by 100, etc.

For, if the point be moved one place to the left, each figure will express 1 tenth of its previous value, hence the whole decimal will be only 1 tenth as great; etc.

3. Placing a cipher between the decimal point and the decimal, divides the decimal by 10.

For, this moves each figure one place to the right in the scale, in which case they express 1 tenth as much as before, and hence the decimal is only 1 tenth as great.

4. Annexing ciphers to the right of a decimal, does not change its value.

For, each figure retains the same place as before, and hence expresses the same value as before, and consequently the value of the decimal is unchanged.

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