SECTION XVI.- Multiplication by Higher Numbers.

1. How many are 12ty? Why? Because 10ty are a How many are 15ty, then? 18ty? 17ty? 14ty? 19ty?

Ans. A hundred and twenty. hundred, and 2ty are twenty. How many are 13ty? 16ty?

2. How many are 20ty? Ans. Two hundred. Why? Because each of the 10ty make a hundred. How many are 24ty? 27ty? 23ty? 36ty? 11ty? 45ty? 72ty? 69ty? 37ty? 84ty? [Continue and extend similar questions till sufficiently familiar.]

3. How many are 100ty? Ans. A thousand. Why? Because ty means tens, and ten times 100 are a thousand. How many are 160ty? 140ty? 170ty? 240ty? 110ty? 520ty? 370ty? [Continue and extend till familiar.]

4. How many are 124ty? Why? 356ty? Why? 247ty? 563ty? 116ty? 218ty? 311ty? &c.

[In reviewing, these questions should be varied by asking, How many are 10 times 12, 16, 84, 270, &c., in place of 12ty, 16ty, 84ty, 270ty, &c.]

5. How many are 2 times 20? Why? Because, as 2 times 2 are 4, 2 times 2ty are 4ty. How many are 2 times 30? 50? 40? 70? Why? Because, as 2 times 7 are 14, 2 times 7ty must be 14ty. 60? 90 ? 80?

6. How many are 3 times 20? 40? Ans. 12ty or 120. 50? 30? 70? 90? 60? 80?

7. How many are 4 times 20? 50? 30? 60? 40? 90? 70? 80?

8. How many are 5 times 20? 90? 30? 70? 60? 40? 80? 50?

9. How many are 6 times 20? 40? 70? 30? 90? 50? 80? 60?

10. How many are 7 times 20? 80? 60? 40? 50? 30? 90? 70?

11. How many are 8 times 20? 30? 90? 70? 50? 80? 60? 40 ?

12. How many are 9 times 20? 80? 50? 30?

13. How many are 2 times 13? Because 2 times 10 are 20 and 2 arithmetic, the higher order should

90? 60? 70? 40?

How do you know? Ans. times 3 are 6. [In oral always be multiplied first,

because the figures are thus taken in their natural order, but chiefly because in practice it is found more easy and convenient.] 2 times 14? Why? [Repeat why after the questions that follow, till the reasoning is perfectly familiar.] 2 times 15? 16? 24? 27? 34? 45? 17? 47? 28? 19? 39? 14. How many are 3 times 13? 15? 14? 16? 19? 17? 21? 18? 24? 37? Why? Because 3 times 30 are 9ty, and 3 times 7 are 2ty one; together 11ty one, or a hundred and eleven. 3 times 54? Why? 72? Why? 87, &c.? 96? 38? 34? 37?

15. How many are 4 times 13? 72? 87? 96? 38? 34? 87? 88? 56? 49?

19? 53?

26? 54? 18? 62 ? 99? 79?

13?

17? 26? 22?

16. How many are 5 times 15? 23? 32? 47? 73? 31? 54? 27? 85? 96? 74? 17. How many are 6 times 13? 18? 15? 17? 19? 54? 36? 28? 72? 69? 93? 77? 65? 59? 48? 18. How many are 7 times 18? 13? 15? 19? 14? 27? 94? 36? 52? 73? 87? 76? 84? 55? 29? 19. How many are 8 times 13? 19? 16? 14? 17? 94? 85? 22? 73? 87? 54? 45? 95? 17? 57?

144 ? 365? 427?

20. How many are 9 times 27? 35? 13? 18? 72? 81? 58? 62? 73? 95? 46? 32? 17? 29? 55? 84? 21. How many are two times 126? 2 times 524? 2 times 346? 725? 274? 373? 644? 375? 863? 588? 453? 22. How many are 3 times 132? 629? 863 ? 275? 529? 246? 23. How many are 4 times 132? 125? 637? 528? 276? 677? 24. How many are 5 times 132? 724? 452? 671? 346? 248? 25. How many are 6 times 132? 362? 241? 526? 728? 126?

26. How many are 7 times 132? 542? 233? 621? 126? 272? 27. How many are 8 times 132? 725? 637? 256? 428? 572? 28. How many are 9 times 132? 522? 615? 926? 328? 218?

321? 126? 428?

234? 621? 532 ?

342? 254? 524?

241? 324? 246?

244? 166? 342?

214? 326? 148?

SECTION XVII. — Definitions.

1. WHEN two or more unequal numbers are joined together into one, the process is called addition, and the whole number is called the sum, or amount. Thus, joining 2 and 4 to make 6, or 3, 4, and 5, to make 12, is called adding those numbers, and 6 is called the sum, or the amount, of 2 and 4, and 12 the sum or the amount of 3, 4, and 5.

2. When two or more equal numbers are joined into one, the process is called multiplication. The number which is to be repeated is called the multiplicand, and the number which shows how many times the multiplicand is to be repeated is called the multiplier, and the increased number, or the multiplicand repeated as often as is required, is called the product. Thus, the process 4 times 5 are. 20, is multiplication. The 4, which shows the number of times that 5 is to be taken, is the multiplier, 5 is the multiplicand, and 20 is the product. A more convenient name for the multiplicand and multiplier, as it applies equally to both, is that of factor. It is evident that both may always be called by the same name, since 4 times 5 is the same as 5 times 4, a remark applicable to any two numbers whatever. The word factor, in this connection, signifies maker; product signifies the number made, or produced. Multiplication, then, is nothing but a short way of performing addition, when the numbers to be added are equal. For, to say 4 times 5 are 20, is precisely the same as to say 5 and 5 and 5 and 5 are 20.

3. When one number is to be taken away once from another number, the process is called subtraction. The number to be diminished is called the minuend, the number to be taken away the subtrahend, and the number remaining after the subtrahend is taken away is called the difference or remainder. Thus, if we take 5 from 8, 3 will remain. Here 8 is the minuend, or number to be diminished; 5 the subtrahend, or number to be subtracted, or taken away; and 3 the difference, or remainder.

4. When many subtractions of the same number are to be performed, or when we wish to find how many times one number can be taken from another, the process is called division. This is, evidently, nothing more than a short way of performing subtraction, since it comes to precisely the same thing, whether we find, at once, that 5 is contained in 20 4 times,

which is called division, or produced by the slower method called subtraction, taking 5 from 20 as many times as possible, thus changing the 20 to 15, to 10, to 5, and to 0. The number to be divided is called the dividend; the number by which we divide is called the divisor; and the result of the division is called the quotient. Thus, if it be required to find how many times 4 is contained in 20, 4 is the divisor, 20 the dividend, and 5, the number of times that 4 is contained in 20, is the quotient. Sometimes the divisor is not contained an exact number of times in the dividend, and, consequently, there will be a remainder at the close of the operation. Thus, if it be required to find how many times 5 is contained in 22, we find it to be 4 times, and 2 over. The 2 is the remainder, and it forms an undivided part of the dividend.

5. It is evident that the dividend is a product of the divisor and quotient, since, if 4 be contained 5 times in 20, it is plain that 4 times 5 will make 20, and so of any numbers whatever. As the divisor and quotient, then, may be considered factors of the dividend, division may be defined the process for finding one factor when the product and the other factor are given. When a remainder occurs, as this remainder is an undivided portion of the dividend, it must be added in if the divisor and quotient are multiplied to reproduce the dividend. Thus, if there be 4 fives in 23, and 3 over, the dividend evidently consists of 3 more than the 4 fives. [Show this on the blackboard.]

6. The termination end, ent, or and, in several of these terms, is derived from a Latin word signifying being, or thing. In this connection it stands for number. Hence, multiplicand signifies the number to be multiplied; minuend, the number to be diminished; subtrahend, the number to be subtracted; dividend, the number to be divided, and quotient, the number showing how many (the quota) times the divisor is contained in the dividend. The termination er, or or, signifies a man, or thing, that works, as in the words baker, miller, printer, farmer, &c. Hence, the multiplier, the factor, and the divisor, are the numbers by which the work is performed, whether in multiplication or division.

7. Signs, or characters, have been invented to express these different processes. Thus, a vertical cross, +, is the sign of addition, and an inclined cross, in the shape of the letter X, X, is the sign of multiplication, or contracted addition. Thus,

4+5 are 9, and 4×5 are 20. The sign of addition, +, is generally read plus, which is a Latin word signifying more. The sign of multiplication, X, is called multiple. Thus, 4+ 5 is read four plus five, and 4X5 is read four multiple five. A dot is also frequently used as the sign of multiplication. Thus, 4 5 is the same as 4×5.

8. A short horizontal line, —, called minus, or less, is the sign of subtraction. The same sign, with a dot above and below it,, is the sign of contracted subtraction, or division. Thus, 20-5, which is read twenty less five, are 15; and 20 ÷5, read twenty divided by five, gives only 4. Sometimes a part of the sign of division is used in place of the whole. Thus, 20: 5, or 20, is precisely the same as 20÷5, all three of them signifying 4.

9. Two parallel lines,, form the sign of equality. It sig nifies that the numbers placed on each side of it are equal. Thus, 20—5—15, is read twenty less five is equal to fifteen; and 20÷5=4, is read twenty divided by five is equal to four.

10. A line drawn over several numbers is called a vinculum. It signifies that the numbers thus joined are to be considered as one number. Thus, 4+5x3, signifies that the sum of 4 and 5, and not 5 alone, is to be multiplied by 3; and 6—2 2 signifies that the difference between 6 and 2 is to be divided by 2. Two parentheses are sometimes used instead of a vinculum. Thus, (4+5)x3 is the same as 4+5×3.

[Write the following lines on the blackboard, the first five to be read, the rest to be solved by the pupils.]

16+4-20

16×4, or 16 • 4—64

16-4-12

[The class should practise similar exercises till they become familiar.]