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MISCELLANEOUS PROBLEMS.

1. If 63 barrels of flour cost $51%, what will 4 barrels cost? Ans. $35.

2. If 9 pounds of sugar cost $2.25, what will 12 pounds cost? Ans. $3.06. 3. If 5% tons of hay cost $283, how many tons will $853 buy? Ans. 16 tons.

4. The sum of two fractions is 31%, and one is 123; what is the other? Ans. 1.

5. The difference of two fractions is 11, and the greater is 1834; what is the less?

88

43

Ans. 635

635. 1899*

Ans. 144.

6. The multiplicand is 23, and product 3; required the multiplier. 7. The divisor is 1, and quotient 253; what is the dividend? Ans. 3. 8. The dividend is, and quotient ; required the divisor. Ans.. 9. Divide the fraction 17 into two parts, one of which is 2 times the other. 10. The sum of two fractions is §, and difference ; required the fractions.

Ans. 1; 12.

85 63 126*

65

47

Ans. 126; 126

11. One-half of the sum of two fractions is 393, and twice the difference is; required the fractions.

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12. What is the value of (+5—7) × (1+§−1), divided

by 3?

13. What is the value of (57−3+27)÷(31—13+21) multiplied by 21 divided by 13?

of of 73 by 5 of 18 of 54.

Ans. 18.

Ans. 243.

581

14. Divide
15. Multiply of of
16

Ans. 13.

11

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18. Find the value of

18/10 612

+

Ans. 18.

85

3–+

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4× 83

19. Find the value of of11× 2 = 4 × 3 = 4 × 1 = 4

20. Find the value of

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5

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+(2+3)÷3+11+ 7

23

5125

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23. What is the sum of, 3, 4, 5, 3, 10, 11, 14, 15, and 13?

Ans. 81523.

24. A man sold of 10 of his bank stock in a month; how many fifths of 1o remained? Ans. 3 fifths.

25. If I pay $0.623 a cord for sawing wood 4 feet long into 3 pieces, how much more should I pay for sawing wood 8 feet long into pieces of the same length? Ans. $0.15. $6131 at $17 a

26. A dry goods merchant bought silk for 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. $2774

27. A bought of B 171⁄2 tons of hay at $113 a ton, and of C 221 tons at $12 a ton, and then sold D 15 tons at $131 a ton, and the remainder E bought at $133 a ton; what was A's gain? Ans. $5472.

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., another yd., and a hat requiring § yd.

and

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 he 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 $1863 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. 26 min.; 53 rods.

32. A steamboat starts from Memphis, Tenn., to go up the Missouri River to a point 10114 miles from the starting place. Her rate is 10 miles an hour for 12 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 13 miles per hour; how many days did the voyage require? Ans. 818 days.

33. In a piece of machinery there are 3 wheels, A, B, and C, each measuring 113 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 74 feet, and C 9 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. 47142 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 1.

5

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

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264. A Decimal is a fraction expressed by the decimal notation; thus .5 is a decimal, while 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, .343.

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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