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Ans. 64022 16.. 3

The component parts will answer in any order; it is best, however, when it can be done, to take them in such order as may clear off some of the lower places in the first multiplication, as is done in both the examples. The operation may be proved by taking the component parts in a different order, or by dividing the multiplier in a different manner.

III. If the multiplier be a prime number, multiply first by the composite number next lower, then by the difference, and add the products.

£. S. d.

576. 4.. 94 × 8712 × 7+3

We multiply the given sum by 6 and 6, because 6 × 6=36; the answer is 1431667. 13s. 6d. we then multiply the sum by 2, and subtracting that product from the former we get the true answer.

V. If the multiplier be large, multiply by 10, and multiply the product again by 10; by which means you obtain an hundred times the given number. If the multiplier exceed 1000, multiply by 10 again, and continue it farther if the multiplier require it; then multiply the given number by the unit place of the multiplier; the first product by ten place, the second product by the hundred place, and so on. Add the products thus obtained together.

Ex.

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The following examples will furnish the learner with practice.

1. 21 ells of Holland, at 7s. 34d. per ell. Ans. £8..1.. 101. 2. 35 firkins of butter, at 158 34d. per firkin. Ans. £26.. 15.. 24 3. 75lb. of nutmegs, at 78. 22d per fb. Ans. £27..2.. 24. 4. 37 yards of tabby, at 9s. 7d. per yard. Ans. £17.. 14.. 7. 5. 97 cwt. of cheese, at 11. 5s. 3d. per Ans. £122.. 9.. 3. 6. 43 dozen of candles, at 6s. 4d. per doz. Ans. £13.. 12.. 4.

cwt.

7. 127 lb of bohea tea, at 12s. 3d. per b. Ans. £77.. 15.. 9. 8. 135 gallons of rum, at 7s. 5d. per gallon. Ans. £50..1.. 3. 9. 74 ells of diaper, at 1s. 44d. per ell. Ans. £5..1..9. The use of multiplication is to compute the amount of any number of equal articles, either in respect of measure, weight, value, or any other consideration. The multipli. cand expresses how much is to be reckoned for each article, and the multiplier expresses how many times that is to be reckoned. As the multiplier points out the number of articles to be added, it is always an abstract number, and has no reference to any value or measure whatever. It is therefore quite improper to attempt the multiplication of shillings by shillings, or to consider the multiplier as expressive of any denomination. The most common instances in which the practice of this operation is required, are to find the amount of any number of parcels, to find the value of any number of articles, to find the weight or measure of a number of articles, &c. This computation for changing any sum of money, weight, or measure, into a different kind, is called Reduction. When the quantity given is expressed in different denominations, we reduce the highest to the next lower, and add thereto the given number of that denomination; and proceed in like manner till we have reduced it to the lowest denomination.

Ex. Reduce 581. 4s. 24d, into farthings. 58..4.. 24

20

1164 shillings in £58.. 4. 12

13970 pence in £58.. 4 .. 2.

4

Ans. 55882 farthings in £58.. 4.. 24.

VOL. I.

DIVISION.

In division two numbers are given, and it is required to find how often the former contains the latter. Thus it may be asked how often 21 contains 7, and the answer is exactly 3 times. The former given number (21) is called the dividend; the latter (7) the divisor; and the number required (3) the quotient. It frequently happens that the division cannot be completed exactly without fractions. Thus it may be asked, how often 8 is contained in 19? the answer is twice, and the remainder of 3. This operation consists in subtracting the division from the dividend, and again from the remainder, as often as it can be done, and reckoning the number of subtractions. As this operation, performed at large, would be very tedious, when the quotient is a high number, it is proper to shorten it by every convenient method; and, for this purpose, ber whose product is not greater than the we may multiply the divisor by any numdividend, and so subtract it twice or thrice,

or oftener, at the same time. The best way is to multiply it by the greatest number, that does not raise the product too high, and that number is also the quotient. For example, to divide 45 by 7, we inquire what is the greatest multiplier for 7, that does not give a product above 45; and we shall find that it is 6; and 6 times 7 is 42, which, subtracted from 45, leaves a remainder of 3. Therefore 7 may be subtracted 6 times from 45; or, which is the same thing, 45 divided by 7, gives a quotient of 6, and a remainder of 3. If the divisor do not ex

ceed 12, we readily find the highest multiplier that can be used from the multiplication table. If it exceed 12, we may try any multiplier that we think will answer. If the product be greater than the dividend, the multiplier is too great; and if the remainder, after the product is subtracted from the dividend, be greater than the divisor, the multiplier is too small. In either of these cases, we must try another. But the attentive learner, after some practice, will generally hit on the right multiplier at first. If the divisor be contained oftener than ten times in the dividend, the operation requires as many steps as there are figures in the quotient. For instance, if the quotient be greater than 100, but less than 1000, it requires 3 steps.

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In this example we say the 9's in 48, 5 times and 3 over; put down 5 and carry 3, and say 9's in 37, 4 times and 1 over; put down 4 and carry 1; 9's in 16, 1 and 7 over; and so on to the end; there is 8 over as a remainder. The proof is obtained by multiplying the quotient by the divisor, and taking in the remainder: this is called "Short Division," of which we give for practice the following examples.

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7... 994357971 by 8 8.......... 449246812 by 9

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Count the same number of figures on the left of the dividend as the divisor has in it; try whether the divisor be contained in this number, if not contained therein, take another dividend figure and then try how many times the divisor is contained in it.

To find more easily how many times the divisor is contained in any number; cast away in your mind all the figures in the divisor except the left hand one, and cast away the same number from the dividend figures as you did from the divisor: the two numbers, being thus made small, will be easily estimated.

If the product of the divisor with the quotient figure be greater than the number from which it should be taken, the number thought of was too great, the multiplying must be rubbed out, and a less quotient figure used.

When after the multiplying and subtracting, the remainder is more than the divisor, the quotient figure was too small, the work must be rubbed out, and a larger number supplied.

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49446327 by 796 47324967 by 699 275472734 by 497 43927483 by 586

96543245 by 763 25769782 by 469

A number that divides another without a remainder is said to measure it, and the several numbers that measure another, are called its aliquot parts. Thus 3, 6, 9, 12, 18, are the aliquot parts of 36. As it is frequently necessary to discover numbers which measure others, it may be observed, 1. That every number ending with an even number, that is, with 2, 4, 6, 8, or 0, is measured by 2. Every number ending with 5, or 0, is measured by 5. 3. Every number, whose figures, when added, amount to an even number of 3's or 9's, is measured by 3 or 9 respectively.

2.

In speaking of the contractions and variety in division, we have already seen, that when the divisor does not exceed 12, the whole computation may be performed without setting down any figure except the quotient.

When the divisor is a composite number, we may divide successively by the component parts: thus if 678450 is to be divided by 75, we may either perform the operation by long division, or divide by 5, 5, and 3, because 5 × 5 × 3 =75.

When there are cyphers annexed to the divisor, cut them off, and cut off also an equal number of figures from the dividend; annex these figures to the remainder.

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When the divisor consists of several figures, we may try them separately, by enquiring how often the first figure of the divisor is contained in the first figure of the dividend, and the considering whether the second and following figures of the divisor be contained as often in the corresponding ones of the dividend, with the remainder, if any, prefixed. If not, we must begin again, and make trial of a lower number.

We may form a table of the products of the divisor multiplied by the nine digits, in order to discover more readily how often it is contained in each part of the dividend. This is always useful when the dividend is very long, or when it is required to divide frequently by the same divisor.

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171

148

⚫23

As multiplication supplies the place of frequent additions, and division of frequent subtraction, they are only repetitions and contractions of the simple rules, and when compared together, their tendency is exactly opposite. As numbers increased by addition are diminished and brought back to their orginal quantity by subtraction, in the same manner numbers compounded by multiplication are reduced by division to the parts from which they are compounded. The multiplier shows how many additions are necessary to produce the number, and the quotient shows how many subtractions are necessary to exhaust it. Hence it follows, that the product divided by the multiplicand will give the multiplier; and because either factor may be assumed for the multiplicand, therefore the product divided by either factor gives the other. It also follows, that the dividend is equal to the product of the divisor and quotient multiplied together, and of course these operations mutually prove each other.

To prove Multiplication. Divide the product by either factor: if the operation be right, the quotient is the other factor, and there is no remainder.

To prove Division. Multiply the divisor and quotient together; to the product add the remainder, if any; and if the operation be right it makes up the dividend.-We proceed to

COMPOUND DIVISION,

For the operation of which the rule is: when the dividend only consists of different denominations, divide the higher denomination, and reduce the remainder to the next

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lb.

oz. dwts.

Ans. 282.. 10. 10.

Before we conclude this article we may observe, that in computations which require several steps, it is often immaterial what course we follow. Some methods may be preferable to others, in point of ease and brevity; but they all lead to the same conclusion. In addition, or subtraction, we may take the articles in any order. When several numbers are to be multiplied together, we may take the factors in any order, or we may arrange them into several classes; find the product of each class, and them multiply the products together. When a number is to be divided by several others, we may take the divisors in any order, or we may multiply them into one another, and divide by the product; or we may multiply them into several parcels, and divide by the products successively. Finally, when multiplification and division are both required, we may begin with either; and when both are repeatedly necessary, we may collect the multipliers into one product, and the divisors into another; or we may collect them into parcels, or use them singly; and that in any order. To begin with multiplication is generally the better mode, as this order preserves the account as clear as possible from fractions.

We have hitherto given the most ready and direct method of proving the foregoing examples, but there is another which is very generally used, called "casting out the 9's," which depends on this principle : That if any number be divided by 9, the remainder is equal to the remainder obtained, when that sum is divided by 9. For instance, if 87654 be divided by 9, there is a remainder of 3; and if 8, 7, 6, 5, 4, be added together, and the sum 30 be divided by 9, there will be likewise a remainder of 3.

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