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Write three hundred and twenty-four. What period and orders in this number?

Write two hundred thousands and four. What orders are omitted in this last number?

Write two millions and six ? What period omitted ? what orders omitted ?

Write six millions; two hundred and three. Which period and what orders are omitted ?

Write twenty-four millions; three hundred. Which period and what orders are omitted ?

Write the following sums, and mention the periods and orders which are omitted.

EXERCISES. 1. One billion ; twenty-four millions; three thousands and three.

2. Four hundred and sixty-nine billions; forty-four thousands; and seventeen.

3. Fifty billions ; three hundred millions ; four hun. dred and fifty thousands; and nineteen.

4. Fifty billions, and seven.

5. Four hundred and thirteen millions, and two thou. sand.

6. Nineteen billions, and one million.
7. Six trillions; nine thousands, and ten.

8. Seven trillions ; nineteen billions; ten thousands, and four hundred.

9. Four hundred and nine trillions : sixteen millions ; eleven thousands and forty.

10. Fifteen billions; two hundred and four millions ; six thousands, and twenty-one.

11. Sixty-four millions ; four hundred thousands ; three hundreds.

12. Sixteen millions ; five hundred thousands, and six.

13. Three trillions; fourteen millions ; seven thou. sands.

14. Two hundred and sixteen millions.
15. Two billions ; sixteen millions, and sixteen.

16. Three hundred and six trillions ; four thousands, and six.

17. Two quintillions ; six quadrillions and five.

18. Three hundred and sixty-four thousands.
19. Three millions and six.
20. Fourteen trillions ; three hundreds.

21. Sixteen trillions ; four millions ; two hundred and four thousands ; seven hundred and one.

22. Three sextillions; one hundred quadrillions ; four. teen trillions ; two hundred and sixty billions ; four hundred millions; sixteen thousands ; four hundred and one.

23. Five millions; two hundred thousands, and sixtytwo.

24. Two hundred and five millions, and seventy-four.

25. Two hundred and six billions ; four millions, and six thousands.

26. Two hundred sextillions ; four hundred millions ; three hundred and four thousands; two hundred and six.

27. Fifteen quintillions; six quadrillions; one hundred trillions; forty-four billions; two millions, and forty-nine.

28. Fifty quadrillions; six hundred trillions; forty-three millions; two thousands four hundred and six.

29. Two hundred and six trillions ; forty-three billions ; four hundred and nine millions; sixty-four thousands ; four hundred and ninety-six.

30. One hundred and four billions ; six millions ; fortynine thousands ; four hundred and ninety-six.

31. Thirteen millions ; four hundred thousands ; six hun. dred and forty-nine.

32. Six sestillions ; five quintillions; four quadrillions ; three trillions; two billions; and one million.

NUMERATION OF VULGAR FRACTIONS.

The figures used in numeration are of two kinds,Figures for a number of whole things, and figures for a number of parts of things. A unit is a whole thing of any

kind.

How many kinds of figures are used in numeration? What are they? What are units and fractions ? What distinction is made between units and fractions on page 38 ?

A fraction is a part of one thing; or a part of several things.

Figures may therefore be divided into fractional and unit figures.

The following is the mode of showing when the numbers represented are several whole things, and when they are several parts of things.

When there are wo whole things, their number is expressed thus, (2). This is called a unit figure.

But if a whole thing is divided into three parts, and we wish to express two of these, by figures, we write one figure, to show into how many parts the whole thing is di. vided, and then above it, write the number of parts we wish to express; thus, (f).

This is called a fractional figure. The lower figure shows into how many parts the whole thing is divided, and the upper figure shows how many of these parts are expressed.

In , into how many parts is the whole thing divided, and how many of these parts are expressed ?

In , into how many parts is the whole thing divided, and how many parts are expressed?

In q? In ? In 7 ? In ? In is? In ?

Fractional figures show into how many parts one whole thing is divided, and how many of these parts are expressed. Besides this, they can show what part is taken from several whole things. Thus shows that one thing is di vided into four parts, and three of them are taken; or that three whole things, have a fourth taken from each of them. For, three fourths of one whole thing, is the same quantity as one fourth of three whole things.

If you have three apples, and take one fourth out of each, how much will you have, and how will you express it in figures ? If you divide one apple into four parts, and take three of these parts, how do you express the quantity taken ?

If you have two apples, and take one sixth from each,

When two whole things are expressed, what figure is used ? If a thing is divided into three parts, how do you express two of those parts ?

how much will you have, and how will you express it in figures ?

If you divide an apple into six parts, and take two of these parts, how much will you have, and how will you express it in figures ?

If an apple is divided into eight parts, and you take three of them, how much will you have, and how will you express it in figures ?

If the fraction is considered as showing how many parts are taken from one unit, then the lower figure shows into how many parts a unit is divided, and the upper figure shows how many of these parts are taken. But if the frac. tion is considered as showing what part is taken out of several units, then the upper figure shows the number of units, and the lower figure shows what part is taken from each.

Thus the fraction may be considered as expressing, two sixths of one thing, or as one sixth of two things.

iis either one twelfth of three things, or three twelfths of one thing.

is either four fifths of one thing, or one fifth of four things.

either shows that one ninth is taken out of two things ; or that two ninths are taken out of one thing.

If 4 is considered as showing how many parts are taken out of one thing it is four sevenths of one unit.

If it is considered as showing what part is taken out of several things, it is one seventh of four units.

If shows how many parts are taken out of one thing, it is two thirds of one thing. If it shows what part is taken out of several things, it is one third of two things.

If is considered as showing how many parts are taken out of one unit, what does the 8 show, and what does the 7 show?

If it is considered as expressing what part is taken out of several units, what does the 7 show, and what does the 8 show ?

If 4 is considered as expressing how many parts are taIf a fraction shows how many parts are taken from one unit, what does the upper figure and what does the lower figure show? Give an example. If a fraction shows what part is taken from several units, what does the upper and what does the lower figure show ?

ken out of one unit, what does the 6 show, and what does the 4 show? If it is considered as expressing what part is taken out of several units, what does the 4 show, and what does the 6 show ?

Whenever the numerator is larger than the denominator, the fraction is called an improper fraction, and always is to be considered as expressing what part is taken out of seve. ral units. Which of the following are improper fractions ?

f 1 1 ¥ 4f.

RULE FOR READING VULGAR FRACTIONS.

Read the number of parts expressed by the numerator, and then the size of the parts expressed by the denominator ; 07*

Read the part expressed by the denominator, and then the number of units, expressed by the numerator.*

Read the following fractions in both ways, thus : is either three fourths of one thing, or one fourth of three things. is either three fifths of one, or one fifth of three.

4 g 13 na la

RULE FOR WRITING VULGAR FRACTIONS.

Write the number of parts into which a unit is divided, and draw a line above it. Over it write the number of parts which are to be expressed; op*

Write the whole numbers which have a certain part taken from them, and draw a line under. Beneath it write the figure which expresses the part which is to be taken out of each of the units above.*

* The pupils should give an example, when reciting these rules, to show that they understand them.

What is an improper fraction, and what does it express ? What are the rules for reading vulgar fractions? What are the rules for writing vulgar fractions ?

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