2. A farmer sold 15 cows for 24 dollars apiece, and took his pay in sheep at 5 dollars apiece; how many sheep did he receive?

SOLUTION. We see that 24 is to be multiplied by the composite number 153 X 5, and the product divided by 5. Using the component parts of the multiplier, we multiply 24 by 3. Now the product of 24 X 3 is to be multiplied and the result divided by 5, which operations we may omit, as follows:

OPERATION. 3

24 X X5 5

=72

Writing the numbers as already described, we strike out 5 below, and 153 X5 above the line, and above 15 set the factor 3, by which we multiply 24. Since there is no number by which to di·vide this product, it is the result required. Ans. 72 sheep.

3. Multiply 165 by 33, and divide the product by 31; multiply the quotient by 16 and divide the product by 99; multiply the quotient by 62 and divide the product by 55; multiply the quotient by 3 and divide the product by 20.

tiplied together; and that all the factors below the line are canceled except 5, by which the product of the remaining factors above the line is to be divided.

NOTE 1. It is plain that 16 above and 20 below the line have the factor 4 common, for 16 4 X 4 and 204 X 5; we therefore cancel the factor 4 from 16 and 20; this we do if we erase the two num.bers, and write 4 the other factor of 16 over it, and 5 the other factor of 20 under it. We see also that 3, the reserved factor of 165, cancels 3, the reserved factor of 99.

NOTE 2.-If the pupil will perform the operations at length, of multiplying and dividing, in this example, he will see how much is saved by cancelation.

Cancelation, then, is the method of erasing, or rejecting, a factor or factors, from any number or numbers. It may be applied for shortening the operation where both multiplication and division are required, by rejecting equal factors from the numbers to be multiplied and the divisors.

RULE.

I. Write down the numbers to be multiplied together above, and the divisors below, a horizontal line.

II. Cancel all the factors common to the numbers to be multiplied and the divisors.

III. Proceed with the remaining numbers as required by the question.

NOTE. One factor on one side of the line will cancel only one like factor on the other side.

EXAMPLES FOR PRACTICE.

4. A man sold 35 barrels of flour at 5 dollars per barrel, and took his pay in salt at 3 dollars per barrel; he sold the salt at 4 dollars per barrel, and took his pay in broadcloth at 7 dollars per yard; he sold the broadcloth at 8 dollars per yard, and took his pay in sheep at 2 dollars a head; he sold the sheep at 3 dollars a head, and took his pay in land at 15 dollars per acre; how many acres of land did he purchase?

If like factors be canceled from the numbers to be multiplied and the divisors, there will remain of the numbers to be multiplied 5 X 4 X 480, and of the divisors 3; and = 26. Ans. 26 acres. 5. What is the quotient of 36 X 8 X 4X8X2 divided by 6 × 5 × 3 × 4 × 2?

NOTE.The remaining factors of the numbers to be multiplied are 2, and 8, and of the divisors, 5.

6. In a certain operation the numbers to be multiplied are 27, 14, 40, 8 and 6, and the divisors are 7, 10, 12 and 15; what is the quotient?

9X2 X2X8=288, and 2885-573, Ans.

7. What is the quotient of 4 X 7 X 18 X 10 × 8 × 9, divided by 24 × 72 × 3 ?

NOTE. All the divisors cancel. Ans. 70.

8. If the numbers to be multiplied are 14, 5, 3 and 28, and the divisors 15 and 9; what is the quotient?

NOTE.-The remaining factor of the divisors is 9.

Ans. 435.

Questions.¶ 60. If a number be multiplied and the product divided by the same number, what is the result? When such operations are to be performed, how may they be contracted? What is this proWhat cess called? How do you indicate that a number is canceled? is cancelation? When may it be applied? Repeat the rule. Explain the operation in Ex. 5; in Ex. 6, &c.*

T61. To find a common divisor of two or more numbers.

1. Find a common divisor of 6, 9 and 12.

6=3X2

The factor 3, which is common to the 9 3X3 several numbers, must be a common divisor of them. Hence the

12=3X4

RULE.

Separate each number into two factors, one of which shall be common to all the numbers.

The common factor will be their common divisor.

EXAMPLES FOR PRACTICE.

2. Find a common divisor of 4, 16, 24, 36 and 8.

Ans. 4.

3. Find a common divisor, or common measure, (which terms mean the same thing,) of 22, 44, 66, and 88.

Ans. 11.

4. Required the length of a rod which will be a common measure of two pieces of cloth, one of them 25 feet, the other 30 feet long. Ans. 5 feet.

T62. To find the greatest common divisor of two or more numbers.

The greatest common divisor of several numbers is the greatest factor common to them, and may be found by a sort of trial. Let it be required to find the greatest common divisor of 128 and 160. The greatest common divisor cannot exceed the less number, for it must measure it. We will try, therefore, if the less number, 128, which measures itself, will also divide or measure 160.

128) 160 (1 128

32) 128 (4
128.

128 in 160, 1 time, and 32 remain; 128, therefore, is not a divisor of 160. We will now try whether this remainder be not the divisor sought; for if 32 be a divisor of 128, the former divisor, it must also be a divisor of 160, which consists of 128 +32. 32 in 128, 4 times, without any remainder. Consequently it is contained in 160 128+32, just 5 times; that is, once more than in 128. And as no number greater than 32, the difference of the two numbers, is contained once more in the greater, it is the greatest common divisor. Hence

Questions.-T61. What is a common divisor of two or more numbers? Repeat the rule for finding it.

To find the greatest common measure of two numbers,

RULE.

Divide the greater number by the less, and that divisor by the remainder, and so on, always dividing the last divisor by the last remainder, till nothing remain. The last divisor will be the greatest common divisor required.

NOTE 1. When we would find the greatest common divisor of more than two numbers, we may first find the greatest common divisor of two numbers, and then of that common divisor and one of the other numbers, and so on to the last number. Then will the greatest common divisor last found be the answer.

NOTE 2. - Two numbers which are prime to each other, of course, can have no common divisor greater than 1.

EXAMPLES FOR PRACTICE.

1. Apply the foregoing rule to find the greatest common divisor of 21 and 35.

2. Find the greatest common divisor of 96 and 544.

Ans. 32.

3. Find the greatest common divisor of 468 and 1184.

Ans. 4.

4. What is the greatest common divisor of 32, 80, and 256? Ans. 16.

5. What is the greatest common divisor of 75, 200, 625, and 150? Ans. 25.

6. A certain tract of land containing 100 acres, is 160 rods long and 100 wide; what is the length of the longest chain that will exactly measure both its length and breadth ?

Ans. 20 rods.

7. A has 2640 dollars, B 1680 dollars, and C 756 dollars, which they agree to lay out for land at the greatest price per acre that will allow each to expend the whole of his money, what was the price per acre, and how many acres did each man buy?

Ans. A bought 220 acres, B 140 acres, and C 63 acres, at 12 dollars per acre.

Questions.¶ 62.

What is the greatest common divisor of two or more numbers? Describe the process of finding it for two numbers? rule? How found when the numbers are more than two? What is the greatest common measure of numbers that are prime to each other?

COMMON FRACTIONS.

¶ 63. When whole numbers, which are called integers, (¶ 36,) are subjects of calculations in arithmetic, the operations are called operations in whole numbers. But it is often necessary to make calculations in regard to parts of a thing or unit. We may not only have occasion to calculate the price of 3 barrels, 5 barrels, or 8 barrels of flour, but of one third of a barrel, two fifths of a barrel, or seven eighths of a barrel.

When a unit or whole thing is divided or broken into any number of equal parts, the parts are called fractions, or broken numbers, (from the Latin word, frango, I break.) If it be divided into 3 equal parts, the parts are called thirds; if into 7 equal parts, sevenths; if into 12 equal parts, twelfths. The fraction takes its name, or denomination, from the number of parts into which the unit or whole thing is divided.

If the unit or whole thing be divided into 16 equal parts, the parts are called sixteenths, and 5 of these parts would be 5 sixteenths.

Fractions are of three kinds, Common, (sometimes called Vulgar,) Decimal, and Duodecimal.

Common fractions are always expressed by two numbers, one above the other, with a horizontal line between them; thus,,, .

The number below the line is called the Denominator, because it gives name to the parts.

The number above the line is called the Numerator, because it numbers the parts.

The denominator shows into how many parts a thing or unit is divided; and

The numerator shows how many of these parts are contained in the fraction. Thus, in the fraction g, the denominator, 8, shows that the unit or whole thing is divided into 8 equal parts, and the numerator, 3, shows that 3 of these parts are contained in the fraction. The numerator, 3, numbers the parts; the denominator, 8, gives them their denomination or

Questions. ¶ 63. What are integers? What fractions, and whence their necessity? Whence do fractions take their name? How many kinds of fractions? Name them. How are common fractions written? What is the lower number called, and why? What does it show? What is the upper number called, and why? What determines the size of the parts, and why? What are the terms of a fraction? What are the terms of the fraction o? ? &c.