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90. Two men require 81 days to take account of a stock of goods. Six men would need what time? 91. What fraction is the quotient of 1994 • 3?

92. In sowing a field, one kind of seed is used at the rate of 121 bushels to 5 acres. What will be required to sow 224 acres, using as much to the acre as before ?

93. When oysters yield 11 gallons to the bushel, a 25gallon barrel can be filled from how many bushels in the shell ?

94. 1 of a bushel of berries is picked; } of them are sold to one man, 1 of the remainder to another. What fractional part remains unsold ?

95. Oranges are bought at 3 for $.05 and sold at 4 for $.09. What is gained on a box of 9 dozen, 1 in 12 of which are worthless. 96. 21

e ? 97. Find the G. C. D. and the L. C. Dd. of 45, 90, 100, and 200.

98. A., B., and C. can do a piece of work in 10 days. A. can do it in 25 days, and B. in 30 days. In what time can C. do it?

99. A. and B. together had $5700. of A.'s money was equal to šof B.'s. How much had each ? 100. Define: 1. Fraction.

10. Reduction. 2. Decimal fraction. 11. Higher terms. 3. Common fraction. 12. Lower terms. 4. Fractional unit. 13. Lowest terms. 5. Denominator. 14. Like fractions. 6. Numerator.

15. Unlike fractions. 7. Proper fractions. 16. Common denominator. 8. Improper fractions. 17. Least common denominator. 9. Mixed number. 18. Fractional relation.

101. Repeat :

1. The principles of Addition of Fractions.
2. The rule for Addition of Fractions.
3. The brief directions for finding L.C.D. of Fractions.
4. The principles of Subtraction of Fractions.
5. The rule for Subtraction of Fractions.
6. The principles of Multiplication of Fractions.
7. The rules for Multiplication of Fractions.
8. The principles of Division of Fractions.
9. The rules for Division of Fractions.

10. The principle of Fractional Relation. 102. Invent and solve:

1. Five problems in Reduction of Fractions.
2. Five problems in Addition of Fractions.
3. Five problems in Subtraction of Fractions.
4. Five problems in Multiplication of Fractions.
5. Five problems in Division of Fractions.
6. Five problems in Relation of Fractions.
7. Five miscellaneous problems in Fractions.

DECIMAL FRACTIONS.

DEFINITIONS.

1. A Decimal Fraction denotes one or more of the decimal divisions of a unit.

2. Decimal Fractions are usually called decimals (Latin, decem, “ten").

3. A Pure Decimal consists of decimal figures only, as 4. A Mixed Decimal consists of an integer and a decimal, as 23.005.

5. A Complex Decimal has a common fraction on the right of the decimal, as .063.

NOTATION AND NUMERATION.

1. By placing a mark (.), called the decimal point, after units of the first order, the numeration and notation table is extended to express parts of a unit on the decimal scale.

2. The relation of decimals and integers to each other is clearly shown by the following

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By examining this table we see that:

Tenths are expressed by one figure.
Hundredths are expressed by two figures.
Thousandths are expressed by three figures.
Ten thousandths are expressed by four figures.
Hundred thousandths are expressed by five figures.

Millionths are expressed by six figures. 3. The decimal point is a separatrix, not a period; it is read " and.”

Remember that the name of the 6th decimal order is Millionths, and give orally the names of the following orders : 6th order, 5th order, 4th order, 3d order, 2d order, 1st order, 3d order, 5th order, 4th order, 6th order, 1st order, 5th order, 2d order, 4th order, 3d order, 6th order, 5th order, 4th order, 3d order, 2d order, 1st order, 6th order.

In what decimal place do you find : Millionths? Thousandths ? Tenths ? Hundredths ? Ten-thousandths ? Hundred-thousandths ? Ten-millionths ? Hundredths ? Millionths? Thousandths ?

4. Read the following: 1.2, 1.03, 1.004, 1.0005, 1.00006, 1.000007, 2.008, 3.09, 4.0001, 5.000002, 6.00003, 7.0004, 8.9, 9.10.

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PRINCIPLES.

1. Decimals and integers are subject to the same law of local value.

2. Each cipher inserted between the decimal point and the first figure of a decimal diminishes the value of the decimal ten-fold. 3. Annexing ciphers to a decimal doos not alter its value.

.05 = .050, for 0 thousandths add nothing to 5 hundredths.

4. The denominator of a decimal, when expressed, is 1 with as many ciphers annexed as there are orders, or places, in the decimal. Read 7.039.

ANALYSIS. 7 is an integer representing 7 units, and is read “

The decimal point is read " and " O denotes the absence of tenths, and is not read. 3 hundredths + 9 thousandths is read “ 39 thousandths.” Hence 7.039 is read “7 and 39 thousandths"

seven.

RULE.

Read the decimal as an integral number, and add the decimal name of the right-hand figure.

EXERCISES.

1. Read the following: 1. .7.

18. 29.15625. 35. 6.839.
2. .36.

19. 341.63456. 36. .243
3. .625. 20. 1001.000089. 37. 3.703.
4. .025. 21. .6305.

38. 7.039.
5. .0005. 22. .4461 39. 8.1367.
6. .12345. 23. .00371. 40. 7.03083
7. .789123. 24. .0506.

41. 9.10076
8. .405607. 25. .087345. 42. 146.0302056.
9. .890123. 26. 6.00056. 43. 376.932474.
10. .456789. 27. 11.04735. 44. 2.234006.
11. 8.54. 28. 63.04048. 45. 487.000081035.
12. 85.4. 29. 100.000001. 46. 586.0004003256.
13. 9.213. 30. 734.819181. 47, .5.
14. 7.389. 31. 341.63456. 48. 5.078.
15. 12.3601. 32. .684.

49. 8.008. 16. 19.0032. 33. .084.

50. 6.2040. 17. 25.00081. 34. .004.

51. 37.40253. 2. Write decimally 13 thousandths.

ANALYSIS. 13 thousandths =one hundredth + 3 thousandths. O tenths are given. As the number is a pure decimal, the expression of it must begin with the decimal point. Hence 13 thousandths expressed decimally is .013.

RULE.

Write the number as an integer, and give the right-hand figure the place indicated by the decimal name of the number. 3. Express decimally:

1. Seven tenths. Twelve hundredths.
2. Nine tenths, Seventeen hundredths.

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