Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

Thus, to multiply 436 by 324 according to the rule, is the same as to multiply separately by 4 units, 2 tens, and 3 hundreds

1.

436

324

1744 872 1308 141264

Here it will be seen that
the result of multiplying
is the same in both cases.
In No. 2, the figures have
their proper position given
to them, according to the
rank of the multiplier, by
means of nothings annexed,

[blocks in formation]

and in No. 1 by putting each line of figures a place further to the left, which serves the same purpose as the nothings.

[ocr errors]

PROOF. To prove that any question in multiplication is correctly performed, we may reverse the operation, by making the multiplicand the multiplier; and if the product is the same, the account is correct.

36

27

252

72

972

For example, if we multiply 36 by 27, the product is 972; and if we multiply 27 by 36, the product will also be 972.

Exercises.

27

36

162

81

972

[blocks in formation]
[blocks in formation]

29. Two factors are 30957 and 839; what is their product?

Ans. 25972923 30. How many oranges are there in 47 boxes, when each box contains 279?

Ans. 13113 31. In a desk there were 6 drawers, each drawer was divided into 8 compartments, and in each compartment were 87 pounds; how many pounds did the desk contain? . Ans. 4176

[ocr errors]

III. WHEN THE MULTIPLIER IS A NUMBER HAVING nothings

ANNEXED TO IT.

RULE.-Extend the nothings beyond the multiplicand, then write them down as part of the answer, and multiply by the other figure or figures of the multiplier, as in Rules I. and II.

Examples.-Multiply 7348 by 70; 47683 by 70600; and 87300

by 760.

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small]

NOTE. WHEN THE MULTIPLIER is 10, 100, 1000, or 1 with any other number of nothings annexed, the multiplication is accomplished merely by annexing as many nothings to the multiplicand as are contained in the multiplier; thus

386 is multiplied by 10, by annexing 1 nothing 736 "1

" 1000,

[ocr errors]

=

3860

"

3 nothings = 736000

IV. WHEN THE MULTIPLIER IS A FRACTION-AS, OR 3.*

RULE.-Multiply the given number by the upper figure of the fraction, and divide† the product by the under figure.

To multiply a number by a fraction, such as, 3, &c., is to find what is the half or three-fourths of the number: thus to multiply 16 by 2, is the same thing as to find how much is of 16.

† For the process of division, see page 20.

Example.-Multiply 16 by 3.

16 3

4)48

12 Answer.

Here 16 is multiplied by 3 and then divided by 4.

When the upper figure of a fraction is 1, such as 3, it is unnecessary to multiply by the 1, as it will obviously make no difference on the number; all that is required is to divide the given number by the under figure; thus-to multiply 24 by, we divide 24 by 3, and the answer is 8-that is, 8 is one-third of 24.

V. WHEN THE MULTIPLIER IS AN INTEGER OR whole NUMBER, WITH A FRACTION ANNEXED-AS 83.

RULE.-Multiply first by the integer, then by the fraction as in Rule IV., and add both the products for the answer.

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

* See explanation of Fractions, under that head, p. 67.

SIMPLE DIVISION.

DIVISION is that process by which we discover how often one number is contained in another, or by which we divide a given number into any proposed number of equal parts. We can

ascertain, by aid of the Multiplication Table, how many times any number is contained in another, as far as 144, or 12 times 12, without writing figures; but the calculations beyond this point are usually written down.

The number to be divided, is called the dividend; the number by which it is divided, is the divisor; and the result is termed the quotient, from a Latin word, which means, literally, How many times ?

Division is denoted by the following character ÷; thus7525, signifies that 75 is to be divided by 25.

RULES FOR DIVISION.

I. WHEN THE DIVISOR DOES NOT EXCEED 12.

RULE.-1. Write down the dividend; draw a curved line on the left side of it, and a straight line below it: then write the divisor on the left of the curved line.

2. Find how often the divisor is contained in the first figure, or (if the divisor is larger than it) in the two first figures of the dividend, and write the quotient below.

3. Multiply the divisor by the quotient, and deduct the product from the figure or figures just divided; then annex to the remainder, if any, the next figure of the dividend, find how often the divisor is contained in this new sum,* and write down the quotient; and so on as before, till all the figures of the dividend have been divided, when the division is completed.

If there is a remainder after the division is finished, it is marked down as part of the answer, with the divisor written below it, forming a fraction.

*NOTE.-If, after annexing a figure to the remainder, the number be less than the divisor, place a nothing in the quotient to express this, then annex another figure to the remainder, and proceed with the division.

The process of division here described, is termed Short Division when part of the process is carried on in the mind, and the results only written down short division is employed when the divisor does not exceed 12. In numbers above 12, it is necessary, for convenience, to write down at length the various steps of the process; and when this is done, it is termed Long Division. The principle is the same in both cases, the sole difference being, that in the one, the operation is 'only partly written down, whilst in the other, all the figures of the process

are written; as will be seen in the following example, in which both methods are given :

Example.-Divide 7958 by 6.

Short Division.

6)7958 13262 Quot.

Long Division. 6)7958(1326 Quot.

6

19

18

15

12

38

36

Here we find that there is one 6 in 7, the first figure of the dividend, and 1 over; we therefore write I in the quotient, and multiplying the divisor, 6, by 1, the quotient, subtract the product, 6, from 7 of the dividend. To the remainder, 1, we bring down 9, the next figure of the dividend, making 19. As there are 3 times 6 in 19, we place 3 in the quotient, and multiplying the divisor, 6, by 3, subtract 18 from 19, which leaves 1. To this I we bring down 5, making 15; and as there are 2 times 6 in 15, we place 2 in the quotient, and multiplying 6 by 2, subtract 12 from 15, leaving 3. To this 3 we bring down 8, making 38, in which there are 6 sixes; therefore, placing 6 in the quotient, we multiply 6 by 6, and subtract 36 from 38, leaving 2 over. Here the account terminates, it being found that there are 1326 sixes in 7958, with a remainder of 2, below which the divisor is written, thus-, and the fraction is annexed to the quotient as part of the answer.

[blocks in formation]

2=%

199327

[blocks in formation]

17512865

14594054 125091899

438218819

[blocks in formation]

97381959

[blocks in formation]

II. WHEN THE DIVISOR EXCEEDS 12.

RULE.-1. Draw a curved line on each side of the dividend, and place the divisor on the left of it.

2. Point off as many figures from the left of the dividend as make a number greater than the divisor: find how often the divisor is contained in this number, and place the result at the right side of the dividend, as the first figure of the quotient. It is to be observed, that 9 is the highest number to be placed in the quotient at any one time.

3. Multiply the divisor by the quotient, and subtract the product from the number pointed off; then bring down and annex to the remainder, if any, the next figure of the dividend.* Proceed

*NOTE.-If, after bringing down a figure to any of the remainders, during the process, the number be less than the divisor, place a nothing in the quotient to express this, then bring down another figure to the remainder, and proceed with the division.

« ΠροηγούμενηΣυνέχεια »