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4. A general has an army composed of three divisions; the first division consists of 450 men; the second of 375; the third of 560; he wishes for a reinforcement equal to the first and third divisions; what was the strength of his army before and after the reinforcement? Ans. before, 1385; after, 2395.

5. The sum of two numbers is 64892; the greater number is 46234; what is the smaller number? Ans. 18658. 6. A, holds a note against B, to the amount of $1200; B pays $675; what was then due ? Ans.. $525.

7. A owed B $4850; at one time he paid $200; at another time $475; at another time $40; at another time $1200; and at another time 156; what was then due ? Ans. $2779.

8.

From the creation to the departure of the Israelites from Egypt, was 2513 years; to the siege of Troy, 307 years more; to the building of Solomon's temple, 180 years; to the building of Rome, 251 years; to the expulsion of the kings from Rome, 244 years; to the destruction of Carthage, 363 years; to the death of Julius Cæsar, 102 years; to the Christian era, 44 years; how long must the world stand after 1823, to make another number of years just equal to the number from the creation to the Christian era? Ans. 2181 years.

CONVERSATION II.

SIMPLE MULTIPLICATION AND SIMPLE DIVISION.

Tut. Can you tell what simple multiplication is? Pup. I cannot, unless it is what the name implies, a method of multiplying or repeating a number.

Tut. Simple multiplication is a rule by which a num. ber is repeated a given number of times. This may be done by addition; if it were required to multiply 12 by 4, we might add 12 to itself three times, and obtain the answer thus:

C

122

12 once,
1 2 twice,
1 2 three times,

48

From this you see that a number may be multiplied any number of times by means of addition. But if it were required to multiply 12 by 16, this would be a tedious manner of doing it. To avoid this we have a rule for repeating a number a given number of times; but before you learn this rule you must know the names by which the several numbers are called, and commit the Multiplication Table to memory.

Multiplication implies three numbers, viz. the number to be repeated, which is called the Multiplicand; the number which shows how many times it is to be repeated, which is called the Multiplier; and the result of the operation, which is called the Product.

The Multiplicand and Multiplier, considered together, are called the factors of the product.

You must now commit the following table to memory; without it you will not be able to make any progress in multiplication.

MULTIPLICATION TABLE.

1 2 3 4 5 6 7 8 9 10
2 4 6 8 10 12 14 16 18

36

4 8 12 16 20 24 28 32 36

51015

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20

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9 | 12 | 15 | 18

21 24 27

30

33 36

20 25

233+

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30 85 40 45

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6 12 18 24 30 36 7 14 21 28 35 42 49 56 816 24 32 40 48 918 27 36 45 54 10 | 20 | 30 | 40 | 50 60 11 22 33 | 44 | 55 | 66 12 24 | 36 | 48

132 | 144

The invention of this table is ascribed to Pythagoras, and is formed in the following manner: The numbers

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, are written first on the same line, then each of these numbers is added to itself and written in the second line, which contains each number of the first line doubled. Each number of the second line is then added to the number over it, in the first line, which gives the triple of the first, and is the third line; in this manner the table is formed throughout.

To commit this table to memory, begin with the 2 in the left hand column, and multiply each figure in the top line by the 2, and you will find the product in the second line from the top, directly under the figure you multiply. Then take the three in the left hand column, and multiply as before; and in this manner learn the whole table. It is not necessary, however, to learn at first, any further than to 9 in the left hand column.

You must now learn the rule for

SIMPLE MULTIPLICATION.

Place the numbers under each other, the largest uppermost, so that units may stand under units, tens under tens, &c. and draw a line beneath.

Begin with the unit figure of the multiplier, and multiply each figure of the multiplicand by it, setting the excess of tens in the product of the two figures, under the line, till you multiply the last figure of the multiplicand, when you write the whole product. Then multiply the multiplicand by the next figure of the multiplier, setting the product one place further to the left than the preceding product. Proceed in this manner with all the figures of the multiplier, setting each successive product one place further to the left than the preceding; then the sum of all these products will be the product of the whole.

If it be required to multiply 38 by 24, the numbers must stand thus:

Factors, 24 multiplier,
3 8 multiplicand,

5 2

76

9 1 2 product.

Pup. I think I understand the reason for carrying for every ten, but do not see why each successive product* is placed further to the left than the preceding; I should like to have you give the reasons.

Tut. In order that you may understand every part, and see the reasons plainly, I will take the following numbers, and multiply them together, explaining every part.

1348
324

5392

Here I begin with the figure 4 in the multiplier, and first multiply the 8 in the multiplicand by it, the product of which is 32; in 32 there are 3 tens and 2 over, therefore set down 2 and reserve 3 for the product of the next figure in the multiplicand. 4 times 4 are 16, and 3 to carry are 19, which contains ten and 9 over; and in this manner each figure of the multiplicand is multiplied by the 4.

1 3 4 8

324

5392

2696

2. When I multiply by 2, I do not obtain the product in units, as with the 4, but in tens, for 2 is in the place of tens, and therefore the product will be in tens; and this is the reason of placing the product by 2 further to the left than the preceding, as in the above example.

To make this appear more plainly, I will decompound the multiplier. In the multiplier there are 300, 20 & 4 ; in the first place I multiply by 4, which gives a product in units; I then multiply by 2, which you see is in reality 20. Now as 2 is in the place of tens, the product of the multiplicand by it will be in tens, and must be put in the place of tens. When I multiply by 3 I get the product in hundreds, because 3 is in the place of hundreds, conse

By product here is meant the product of each figure of the multiplier by the multiplicand.

quently the product of this figure must be carried one place further to the left than the preceding. I will now give you the work in its decompounded state as follows:

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Here you will see the reason of placing each successive product one place further to the left than the preceding. You must make this familiar, for it is important to understand it thoroughly before you proceed further.

Pup. Does it never happen that the multiplier is larg. er than the multiplicand?

Tut. This never happens, because either of the numbers may be used for a multiplier, and the largest number is always placed uppermost. In multiplying one number by another, the product will be the same, whether you use one or the other number for a multiplier, the truth of which you can see in the following numbers:

.

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It does indeed happen that the largest number is used for a multiplier. When there are ciphers at the right of either number, they may be omitted in multiplying, and annexed to the product; therefore when the largest

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