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7×6=42 divisor; therefore, 4 2 ) 1 5 1 2 ( 3 6, the required

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But by the preceding fourth principle, 36÷6=6, and 42÷7= 6, and 6×6=36 Ans. In this example the divisors are, as it were, expunged or lost, since they divide without remainder. This will not, however, always be the case. It will frequently be necessary to assume some number, which will divide some two given numbers, without remainder, agreeably to a rule soon to be given.

But for further illustration, suppose it be required to multiply the numbers 36, 12, 27, and 72 together, and to divide the product successively by 24, 18, and 48. Now it is evidently desirable to arrange these numbers so that they may be conveniently compared with each other. We will adopt the following mode :-We will place the numbers whose product is to form a dividend, above a horizontal line; and those whose product is to form a divisor, below the same line, thus:

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Now by the fourth and last principle laid down, I can divide 36 in the dividend and 18 in the divisor by 18, without a remainder, and obtain 2 in the dividend and 1 in the divisor; thus, 2. 12. 27. 72 I can also divide 72 in the dividend and 24 in

24. 1. 48

the divisor by 24, and obtain 3 in the dividend and 1 in the divisor (it will be remembered that the divisors stand below the

2. 12. 27. 3

line)-thus, 1. 1. 48 Again, I can divide 12 in the divi

2. 1. 27. 3

It is now evi

dend and 48 in the divisor by 12, and obtain 1 in the dividend and 4 in the divisor; thus, 1.4. Again, I can divide 2 in the dividend and 4 in the divisor by 2, and obtain 1 in the 1. 1. 27.3 dividend and 2 in the divisor; thus, 1. 1. 2' dent that the division can be carried no farther without remainder. The next step therefore is to divide the product of the numbers remaining above the line by the product of those below it. The product of those above the line is 27×3=81; and of those below the line, 2; therefore, 81÷2=40}, the

number required. The same result would have been obtained by multiplying the numbers above the line and dividing their product by the product of those below it, previous to canceling. In the above example, as the numbers have been canceled, they have been omitted, and a new statement made. This is by no means necessary. One statement is sufficient.

It will be noticed that in every instance division is effected without a remainder. Such must always be the case.

The following rule will be found a competent guide for the scholar in all operations of canceling.

RULE 1st.-Place all the numbers whose product is to form a dividend, above a horizontal line; and all those whose product is to form a divisor, below the same line.

2d. Notice whether there are cyphers both above and below the line; if so, erase an equal number from each side.

3d. Notice whether the same number stands both above and below the line; if so, erase them both.

4th. Notice again if any number on either side of the line will divide any number on the opposite side, without remainder; if so, divide and erase the two numbers, retaining the quotient figure only on the side of the larger number.

5th. See if any two numbers, one on each side, can be divided by any assumed number, without a remainder; if so, divide them by that number, and retain only their quotients. Proceed in the same manner as far as practicable; then,

6th. Multiply all the numbers remaining above the line for a dividend, and those remaining below, for a divisor.

7th. Divide, and the quotient will be the number required.

Note 1st.-If only one number remain on either side of the line, that number is the dividend or divisor, according as it stands above or below the line.

Note 2d. The figure 1 is not to be regarded in the operation, because it avails nothing either to multiply or divide by 1.

In the two following examples, a separate statement will be made for each step of the rule, as successively taken, with a reference to the rule.

Ex. 2. Multiply together the numbers 100, 16, 24, and 36, and divide their product by the product of 60, 10, and 16.

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no farther reduction, for the divisors or lower numbers are all canceled. The product of the numbers remaining above the line is, therefore, the quotient required; viz. 4× 36=144 Ans. Ex. 3. Divide the product of 96, 18, 110, 5, and 42 by the product of 50, 27, 11, and 28. Statement,

second step of the rule, we obtain

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96. 18. 110. 5. 42
50. 27. 11. 28°

96. 18. 11. 5. 42

By

By the

9 is assumed

5.

27.11.28°

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27.

28'

By the fifth,

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only number remaining above the line, except the 1, and nothing but 1 remains below the line. 96 is therefore the quotient sought.

To show more completely the mode of solution, the canceled numbers will be retained in the following sum; they will, however, be crossed, to show that they are canceled.

Ex. 4. Divide the product of 16, 9, 4, by the product of 3 and 32.

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The following statements are solved without repetition, that the scholar may obtain accurate views relative to the mode of solution here presented.

5. Divide the product of 21, 12, 42, 100, by the product of 7, 35, and 50.

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3×6×2×12=432, and 432-5-862. Ans.

6. Divide the product of 99, 49, 15, 20, 32, 13, 16 by the product of 77, 10, 16, 49, 39, and 12.

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then 3×5×2×8=240, and 240÷7=344. Ans.

7. Divide the product of 164, 88, 4, 28, 3, 2, by the product of 328, 36, 21, 12, 32.

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therefore, 11 divided by the product of 9×3×4=108, is the Ans.; and may be thus expressed,. It will be observed, that, in the upper numbers, the product of 3 and 4 cancels 12 in the lower numbers; and generally, when the product of any two or more numbers on one side equals any number on the opposite side, they may all be erased at the same time. By a careful examination of the preceding seven examples, the scholar will be prepared to solve the remaining examples without further elucidation. He will, however, remember that the only object in view is to prepare him to make a successful application of the principle of canceling, to the solution of useful problems.

Note 3d. Whenever the product of the remaining numbers above the line is less than the product of those below, the answer will be in the form of a fraction, as in the last example. The scholar, when he has learned respecting the nature of fractions, will find that a number standing directly over another, with a line between them, always indicates division; that is, the number above the line is divided by that below it.

8. Divide the product of 27, 111, 32, 40, 42, by the product of 12, 4, 24, 12, 30. Quo. 3881.

9. Divide the product of 132, 8, 49, 4, 84, 81, 42, by the product of 16, 28, 7, 5, 6, 45, and 35. Quo. 399.

10. Divide the product of 32, 18, 72, 81, 7, 56, 63, by the product of 24, 36, 162, 63, 2, 32. Quo. 147.

11. Divide the product of 96, 54, 108, 72, 56, 18, and 21, by the product of 27, 81, 324, 28, 72, 3.

Quo. 1991. 12. Divide the product of 16, 27, 42, 44, 55, and 66, by the product of 48, 88, 22, 33, and 49. Quo. 194.

13. Divide 6, 8, 10, 5, 20, 25, 36, 48, and 60, by the product of 9, 12, 15, 28, 36, 42, and 9. Quo. 2013331.

14. If 12 horses eat 15 bushels of oats, how many would 36 horses eat in the same time? In solving this sum, if I divide 15 bushels by 12, it will evidently give me what one horse will eat, viz. 14 bushels; and if I multiply this by 36, I shall obtain the quantity that 36 horses will eat. This sum

may then be easily solved by the principle here introduced. The statement would be thus:

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15.36

12°

and 15×3=45 bushels:

But we will not anticipate enough has already been done to show that the principle is a practical one. We will leave its application, for future consideration.

QUESTIONS.-What is the object of the rule of canceling? What problems may be solved on this principle? What is the first of the facts on which this principle is founded? Illustrate. What is the second? Illustrate. What is another advantage? What is the third fact? What is the fourth? Illustrate. What is the first step of the rule? What is the second step? The third? The fourth? The fifth? The sixth? The seventh, and last? What is note first? What note second? When the product of the remaining numbers above the line is less than the product of those below, in what form will the answer be? What does one number standing directly over another, with a line between them, always indicate?

COMPOUND NUMBERS.

We have thus far been operating with numbers, of the same denomination, and increasing in the constant ratio of 10.

There is, however, another class of numbers composed of several denominations, increasing in no uniform ratio, and requiring to be separately denoted or expressed. These are called Compound Numbers. Under this head are included all those denominations employed to express measures of any defi

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