NOTE 13. The expression and all other expressions of division, whether the dividend or the divisor is the larger, in which the dividend is written above the divisor, are called FRACTIONS. A fraction is read by reading first its dividend, and then its divisor with "th" or "ths" added as the divisor is 1 or greater than 1. Thus, is read three-fourths.

In printed matter a fraction may be expressed by writing the dividend before and the divisor after the horizontal line. Thus, five-sixteenths may be written 5-16.

The formation of the name of the fraction is irregular if the denominator is 2 or 3. Thus, is read one-half; 3 is read three-halves; and is read two-thirds.

If the name of the divisor ends in "ve," "th" is added and ve is changed to f. Thus is read one-fifth, is read one-twelfth.

If the name of the divisor ends in "y," "eth" is added and y is changed to i. Thus is read one-twentieth.

A fraction is commonly defined as "One or more of the equal parts of a unit." When thus considered the divisor of the fraction, from the fact that it shows the name, or denomination, of the parts into which the unit is divided, is called a DENOMINATOR, and the dividend, as showing the number of parts, is called a NUMERATOR. Thus 2 is the numerator and 3 the denominator of the fraction 3.

The numerator and denominator of a fraction are called its TERMS. Thus 2 and 3 are the terms of the fraction .

In this text-book all operations in fractions will be based upon the idea that a fraction is an indicated expression of division. The pupil, therefore, should thoroughly master the fundamental principles of division, which will be developed later, as having done this his work in fractions will be but the intelligent application of previously acquired ideas.

24. Products where Neither Factor is Greater than 12. The first step in multiplication and division is to memorize the following table, which shows all products where neither factor is greater than 12, and all quotients where neither divisor nor quotient is greater than 12.

NOTE. The table should be read in the following manner:

'

Twice 2 are 4, 2 in 4 twice. 2 times 3 are 6, 3 times 2 are 6; 3 in 6 twice, two in 6 three times

Twice 6 are 12, 6 times 2 are 12; 6 in 12, twice, 2 in 12, 6 times. times 4 are 12, 4 times 3 are 12; 4 in 12, 3 times, 3 in 12, 4 times.

3

The pupil should so thoroughly memorize the above facts as to be able to give without an instant's hesitation the mis. sing element in each of the following oral exercises.

Separate each of the following numbers into pairs of factors. Let no factor be greater than 12.

Thus, the factors of 28 are 4 and 7; the factors of 36 are 3 and 12, and 4 and 9, and 6 and 6.

25. Composite Numbers and Prime Numbers.

A number that can be separated into factors is called a Composite Number, and a number that cannot be separated into factors a Prime Number. Hence, a factor that cannot itself be separated into factors is called a Prime Factor.

NOTE. By observing the following directions one can at a glance separate into prime factors any of the numbers given in the preceding list:

I. Memorize the prime factors of 8 and 12.

2. To separate into factors numbers like 54, 48, 120, etc., that have more than two factors, proceed as follows:

(1) First separate the numbers into two factors.

(2) Separate each of these factors into its prime factors.

Thus, the factors of 54 are 6 and 9, or 2 and 3, and 3 and 3; the factors of 48 are 6 and 8, or 2 and 3, and 2, 2 and 2; and the factors of 120 are 10 and 12, or 2 and 5, and 2, 2 and 3.

Ex. 32.

Separate into their prime factors each of the numbers under. Ex. 31.

26. In case the dividend is not divisible by the divisor, two courses may be followed.

1. The part left may be written at the right of the quotient and separated from it by a dash. Thus, the result obtained by dividing 27 by 4 may be written 6-3. When so written the part left is called the Remainder.

2. It may be written above the divisor and placed at the right of the quotient. When so written it forms a fraction and is considered a part of the quotient. Thus, the result obtained by dividing 27 by 4 may also be written 6%.

Ex. 33.

Name the quotient and the remainder in each of the following exercises:

26. To Multiply when the Multiplicand is Greater than 12.

A teacher taught three terms of school of 12 weeks each. The first term she received $7 a week, the second term $8, and the third $9.

What was her salary for the first term?

For the second term?

For the third term?

How can we find her salary for the three terms?

How, then, when our multiplicand consists of several parts can we find the total product?

A study of the preceding processes will show us how to multiply when the multiplicand is so large that it must be separated into several parts. We evidently must multiply each of the several parts by the common multiplier, and unite the several products into one total product.

We are to multiply 1157 by 8.

What sign is understood between eaɔh two figures of the multiplicand?

Of how many parts, then, does the multiplicand consist?

8 times 7 units are how many units?

56 units are how many units and how many tens? What do we do with the 6 units?

What shall we do with the 5 tens?

8 times 5 tens are how many tens?

What do we do with the 5 tens of the first product?

40 tens plus 5 tens are how many tens?

45 tens are how many tens and how many hundreds? What do we do with the 5 tens?

What shall we do with the 4 hundreds?

8 times 1 hundred are how many hundreds?

1157

8

9256

What do we do with the 4 hundreds of the second product? 8 hundreds plus 4 hundreds are how many hundreds?

12 hundreds are how many hundreds and how many thousands?

What do we do with the 2 hundreds?

What shall we do with the 1 thousand?

8 times one thousand are how many thousands?

What do we do with the 1 thousand of the third product? 8 thousands plus 1 thousand are how many thousands? What, then, is the total product of 1157 and 8?

Give, then, a rule for multiplying when the multiplicand is greater than 12 and the multiplier is not greater.

NOTE. It is evident that each of the products by 8 might have been written entire. In that case the final step of the solution would have been to unite the several products into one total product.