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25. If a small slate cost 7 cents, how many slates will 14 cents buy? Will 28? Will 35? Will 56? Will 63?

26. If a writing-book cost 8 cents, how many writingbooks will 16 cents buy? 24 cents ? 40 cents? 56 cents ? 80 cents ? 96 cents ?

27. How many spelling-books will 18 cents buy, if 1 cost 9 cents? Will 27? Will 36? Will 45? Will 54 Will 72?

28. How many fish can you buy for 20 cents, if 1 cost 10 cents? How many for 40 cents ? For 60 cents ? For 100 cents ? For 110 cents ? For 120 cents ?

29. If you pay 11 cents for an inkstand, how many can you buy for 22 cents ? For 33 cents? For 55 cents ? For 88 cents ? For 110 cents ? For 132 cents?

30. How many pounds of butter can you buy for 24 cents, when the price is 12 cents for 1 pound? How many pounds for 36 cents ? For 60 cents ? For 108 cents ? For 132 cents ? For 144 cents ?

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Practical Questions on the foregoing. 1. A boy, having 18 apples, gave them to his companions, as follows; to William 4, to Rufus 6, and to Thomas 5; how many did he give away in all, and how many had he left?

2. Thomas gave to one of his companions 6 peaches, to another 3, to another 2, and sold 3 ; how many had he at first?

3. A man bought a wagon for 17 dollars, and gave 5 dollars to have it repaired, then sold it for 26 dollars; how much did he make by the bargain ?

4. A man bought a horse for 25 dollars, and, to pay for it, gave 6 bushels of rye, worth 6 dollars, and the rest in money; how much money did he pay?

5. Rufus, having 20 cents, bought a book for 12 cents, and a knife for 6 cents ; how much more did the book cost than the knife ? and how many cents had he left ?

6. What is the cost of 5 yards of cloth, at 4 dollars a yard? At 3 dollars ? At 7 dollars ? At 2 dollars? At 8 dollars ? At 9 dollars? At 12 dollars ?

7. If 1 lemon be worth 3 apples, how many lemons are 6 apples worth? Are 12 apples worth? Are 18 apples worth? Are 24 apples worth ? Are 36 apples worth?

8. How many barrels of flour, at 8 dollars a barrel, can you buy for 16 dollars ? For 48 dollars ? For 96 dollars ? For 80 dollars ?

9. How many are 2, 3, and 5? Are 4, 2, and 6? Are 33, and 2? Are 9, 3, and 4? Are 10, 8, and 2? Are 5,

would

A.

. A. Q. A.

!

4, 3, and 2? Are 4, 3, 2, and 1? Are 7, 6, 3, and 2? Are 8, 9, and 10? Are 12, 11, 10, and 9 ?

10. How many are 6 times 3 ? 6 times 4? 6 times 7? 7 times 8? 9 times 7? 12 times 7? 9 times 5? 8 times 7? 7 times 6? 7 times 9 ? 12 times 11 ? 8 times 5? 3 times 7 ? 12 times 12?

11. How many times 2 in 12? 2 in 18? 2 in 24? 3 in 6? 3 in 12? 3 in 36? 4 in 20 ? 4 in 32? 4 in 48? 5 in 25? 5 in 35? 5 in 60 ? 6 in 36 ? 6 in 48? 6 in 72 ? 7 in 14? 7 in 56 ? 7 in 84 ? 8 in 40 ? 8 in 96 ? 9 in 36? 9 in 108? 11 in 22 ? 11 in 55 ? 11 in 132. 12 in 144 ?

Note. Younger pupils slıould be required to review and dwell on the preceding questions for illustration, and ine tables, till their sulutions be made perfectly familiar.

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NUMERATION.

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Q. When I say to you, “ Give me that book," do I mean one book, or more than one ?

Q. When we speak of a single thing, then, what is it called ?
A. A unit, or one.
Q. What are one unit and one more, or one and one, called ?
Q. What are two units and one more, or two and one, called ?
Q. What are three units and one inore, or three and one, called ?
Q. What are four units and one more, or four and one, called ?
Q. What are five units and one more, or five and one, called ?
Q. What are six units and one inore, or six and one, called?
Q. What are seven units and one more, or seven and one, called ?
Q. What are eight units and one more, or eight and one, called ?
Q. What are nine units and one more, or nine and one, called

Q. Now, to be obliged always to write these numbers out in words, would be very troublesome: to prevent this, how do we sometimes ex press the numbers one, two, &c. up to thousands millions, &c.?

A. By letters.
Q. What does the letter I stand for?
A. One.
Q. What does the letter V stand for?
A. Five.
Q. What does the letter X stand for?
A. Ten.
Q. What does the letter L stand for?
A. Fifty.
Q. What does the letter C stand for?
A. One hundred.

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Q. What does the letter D stand for?
A. Five hundred.
Q. What does the letter M stand for?
A. One thousand.

Q. You said that V stands for five ; suppose you place the letter I before the V, thus, IV, what will both these letters stand for then ?

A. Only four.

Q. What, ihen, may be considered as a rule for determining the value of these letters ?

A. A letter standing for a smaller number, and before a larger, takes out its value from the larger.

Q. One X stands for ten; what do two X's stand for?
A. Twenty:
Q. What, then, is the value of a letter repeated ?
A. It repeats the value as often as it is used.
Q. How many letters do we use for expressing numbers ?
A. Seven.
Q. Will you name them?
A. I, V, X, L, C, D, M.
Q. What is this method of expressing numbers by letters called ?
A. The Roman method.
Q. Why called Roman?
A. Because the Romans invented and used it.
Repeat the

ROMAN TABLE.
One,
...I. Thirty,

.XXX.
Two,

.II.
Forty,

..XL.
Three,
III. Fifty,

...L.
Four,..
IV. Sixty,

.LX.
Five,
..V. Seventy,

.LXX.
Six,

Eighty,

LXXX.
Seven,
VII. Ninety,

.XC.
Eigh,

VIII. One hundred, ........ Nine,

Two hundred,

...CC.
Ten,
...X. Three hundred,

CCC.
Eleven,
.XI. Four hundred,,

.CCCC.
Twelve,...
.XII. Five hundred,

.D.
Thirteen,
XIII. Six hundred,

.DC.
Fourteen,
.XIV. Seven hundred,

DCC.
Fifteen,..

.XV. Eight hundred, ..DCCC.
Sixteen,..
XVI. Nine hundred,

.DCCCC.
Seventeen,

.XVII.
One thousand,

..M.
Eighteen,

.XVIII.
Fifteen hundred,

.MD.
Nineteen,

XIX.
Sixteen hundred,

.MDC,
Twenly,.....

.XX.

Two thousand,..... MM. Eighteen hundred and thirty-four,......

.MDCCCXXXIV.

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...VI.

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...7.

I VI. We have a shorter method still, which is in very general use, as will appear by observing what follows :A unit, or one, is written...

1. Two,...

..2. Three,

.3, Four,...

.4. Five,

..5.
Six,

..6.
Seven,
Eight,

.8.
Nine,

..9.
Q. What are these characters called ?
A. Figures.
Q. By what other name are they sometimes called ?
A. The 9 digits.
Q. What is this method of expressing numbers called ?
A. The Arabic method.
Q. Why so called ?

A. Because the Arabs are supposed to have invented it.*

Let me see you write down on the slate, in figures, the numbers one, two, three, four, five, six, seven, eight, nine.

Q. To express ten, as we have no one character that will do it, what two characters do we make use of to represent this number?

A. The first character, 1, and 0 or cipher; thus, 10.

Q. What place does the 0, or cipher, in this case take ?
A. The units' place.
Q. What place does the figure 1 take ?
A. A new place.
Q. What is this new place called ?
A. The tens' place.

Q. Write down in figures, on the slate, the number ten; now take away the 1, and what will be left?

A. Nothing but 0, or cipher.
Q. What is the value of this 0, or cipher, thus standing alone ?
A. No value.
Q. Now place the 0 at the right of the figure 1, and what will it become?

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Q: How was it obtained from the Arabs?
A. The Moois communicated it to the Spaniards, and John of Basingstoke,
Archdeacon of Leicester, introduced it into England; hence its introduction
into our own country.

Q. About what time was it introduced into England ?
A. About the middle of the eleventh century.
Q. How extensively is it now used ?
A. All over the civilized worid.

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Two tens, .

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A. Ten, (10.)
Q. How many times is the figure 1 increased by the 0, or cipher ?
A. Ten times.

Q. What effect, then, has a cipher, in all cases, when placed at the right of figures ?

A. It increases the value ten times.
Q. In what proportion is this increase said to be ?
A. Tenfold proportion.

As you have probably learned by this time how to write down ten in figures, by the help of a cipher, and learned also the value of this .cipher, we will now proceed to higher numbers; and to begin : let me see you write down in figures, on the slate, the following numbers, viz One ten and one unit, or eleven,...

. 11 One ten and two units, or twelve,...

12 One ten and three units, or thirteen...

13 One ten and four units, or fourteen,

14 One ten and five units, or fifteen,.

15 One ten and six units, or sixteen,

.. 16 One ten and seven units, or seventeen,..

.17 One ten and eight units, or eighteen,

18 One ten and nine units, or nineteen,

.19
....or twenty,

.20
Three tens,
...or thirty,

30
Four tens,...
...or forty,..

.40
.....or fifty,

.50 .....or sixty,

.60 Seven tens, ........or seventy,

.70 Eight tens,

...or eighty, Nine tens,

.........or ninety,....... Ten tens,

.....or'one hundred, ...100 Q. Here we see the value of the cipher again; for, by placing a cipher at the right of ten, it becomes one hundred, (100,) that is, ten tens: should we place another cipher still at the right of the 100, (thus, 1000,) what would it become ?

A. One thousand, (1000.)

Q. From what you have now seen of the value of figures, what may 2 and 5 be made to stand for?

A. 25 or 52.

Q. What is this different value called, which arises from the figures being placed or located differently?

A. Their local value.
Q. What would be the value of the five written alone ?
A. Simply 5.
Q. What is the value, then, of a figure standing alone ?
A. The simple value.
Q. How inany values do figures appear to have ?
A. Two.

Five tens,..
Six tens,

have Since tinue

...80

........90

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