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 4X5.

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 signifies 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)×3 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.]

[The class should practise similar exercises till they become

familiar.]

What is addition? What is the result of addition called? What is multiplication? What is the multiplicand? the multiplier? the factors? the product? What is subtraction? What is the minuend? the subtrahend? the difference, or remainder? What is division? What is the dividend? the divisor? the quotient? What are the factors of the dividend? What is the dividend a product of? What is the difference between addition and multiplication? between subtraction and division? What is the sign of addition? its name? the signs of multiplication? their names? of subtraction? its name? of division? In how many ways can multiplication be expressed by signs? In how many ways can division be expressed by signs? What is the sign of equality? What is the result of addition called? the result of multiplication? of subtraction? of division? In multiplication, what is the number to be repeated called? What is the number called which shows how many times the multiplicand is to be repeated? What is the general name for both these terms? Why may they be properly called by the same name? In subtraction, what is the number to be diminished called? What is the number called which is to be taken away? In division, what is the number to be divided called? the number by which we divide? What do we call what is left? What is a vinculum? What does it signify? What characters are sometimes used in place of a vinculum? What is the precise meaning of the terminations end, ent, and and? What do they signify in arithmetic? What do the terminations er and or signify? What in arithmetic ?

SECTION XVIII.- Shortened Multiplication, or Multiplication by Easy Numbers.

1. How many are 10 times 4? 10 times 24? 37? 45? 72? 158? 326? [Write a few such numbers on the blackboard, to be multiplied by 10, thus:

24×10= 158x10=, &c.,

and direct the attention of the class to the fact, that the significant figures remain unchanged when multiplied by 10.]

2. How many are 10 times 8? 84? 16? 49? 52? 93? 176? 248?

3. How many are 5 times 8? half of 10 times 8? 5 times 4? half of 10 times 4? [Here direct attention, on the board, to the fact, that a number multiplied by 5 produces the same amount as half the same number multiplied by 10; consequently, the easiest way to multiply a number by 5, is to multiply its half by 10. Thus, 72x5=32×10, or 36×10.

4. How many are 5 times 16? Ans. Half of 16 or 8 ty. 5 times 24? 36? 28? 46? 72? 64? 84? 34? 58? 96? 128? 136? 248? 372?

5. How many are 5 times 17? [Here direct attention, on the board, to the fact, that every odd number of fives may be considered as the next lower even number of fives and one five more. Thus, 73×5=72×10+5=36 ty and 5, or 365; and 27X5=26×10+5, or 135.] How many are 5 times 19? 13? 21? 35? 37? 65? 49? 77? 33? 95? 67? 129? 247? 653? 875? 555 ?

6. How many are 15 times 14? [Show that 15 times any number is 10 times and 5 times that number. Our 14 times 15, then, becomes 14 times 10 and the half of 14 times 10, together 21×10-210. Thus, to multiply by 15, it is only necessary to increase the number to be multiplied by its half, and multiply by 10.] How many are 15 times 22? Ans. 22 and half of 22=33×10=330. How many are 15 times 24?

42? 48? 36? 28? 54? 72? 84? 58? 96? 64? 68? 94? 7. How many are 15 times 17? [Here we have 17×10 and 17X5. By the 5th question above, 17×5 becomes 16×5 and 5. Thus, when an odd number is to be multiplied by 15, we add half the next lower even number, multiply by 10, and add 5. Thus, 17X15 becomes 17+8×10+5, or 255, and 23×15 becomes 23+11×10+5=345.] How many are 15 times 25? 19? 23? 47? 75? 37? 49? 55? 97? 83? 45? 33? 87? 75? 29? 85? 67? 77? 53? 57 ? 95?

8. How many are 15 times 34? 67? 26? 57? 74? 128 ? 39? 156? 159? 234? 562? 325? 628? 473? 654? 637 ? 429? 579? 777 ?

9. How many are 20 times 24? [As 20 times any number is twice ten times that number, we have only to double the number to be multiplied by 20, and then multiply it by 10; or, expressed more briefly, multiply twice the number by 10. Thus, 20×24 2×24×10=480.] How many are 20 times

32? 41? 72? 93? 156? 428? 349? 572? 643? 377? 756? 278? 542? 503? 637 ?

10. How many are 25 times 8? [As every 4 times 25 makes 100, we have only to find how many fours are in any number, to know how many hundreds that number will make when multiplied by 25. Thus, 25X24-100X24-600. And 25×16=100×16-400. [How many are 25 times 36? 44? 28? 52? 60? 32? 56? 40? 72? 128? 436? 372? 116? 348 ?

11. How many are 25 times 37? [Dividing 37 by 4 gives 9 and 1 over; therefore, 37×25-9 hundred, and 1 twentyfive, or 925. In the same manner, 38×25 9 hundred, and 2 twenty-fives, or 950; and 39X25-9 hundred, and 3 twentyfives, 975. Every remainder, then, gives as many twentyfives as it contains units to be added to the hundreds.] How many are 25 times 17? 15? 22? 19? 47? 54? 63? 95? 86? 74? 125? 237? 355? 178? 323? 218? 346?

12. How many are 25 times 20? 120? 55? 84? 173? 267? 348? 133? 87? 195? 388? 193? 327? 136? 113? 125? 239?

13. How many are 30 times 24 (3×24×10)? 30 times 45? 76? 255? 327? 54? 96? 238? 126? 272? 49? 78? 232 ?

14. How many are 35 times 24? (3×24+2+(=84)×10. 840.) 35 times 37? (3×37+37X10.) [Let it always be remembered that the remainder 1, in such cases, is always one 5. Thus, 35 times 37=3×37(=111)+37(18 and 1 over)= 129X10+5=1295.] 35 times 29=87+14×10+5. 35

times 14? 27? 96? 128? 85? 74? 254? 93? 232? 75? 15. How many are 40 times 24 (4×24×10)? 40 times 27? 84? 56? 47? 53? 125? 67? 238? 152? 95? 73? 182? 245 ?

16. How many are 50 times 24 (24×50=2x100)? 50 times 27 (27x100)? [In halving odd numbers, the remainder 1 is one 50. Thus 27x100-1300 and one 50=1350.] 50 times 36? 48? 57? 73? 94? 85? 29? 132? 173? 178? 127? 185? 142? 155? 187? 143? 172? 189?

17. How many are 45 times 24? (45-50-58.) Therefore, 45 times 24 is 50×24-1200, minus the tenth of that number (120)=1080. 45 times 26? (1300-130-1170.) 45 times 29? (1450-145-1305.) 45 times 36? 93? 45? 27? 39?

96? 78? 124? 104? 156? 118? 97? 138? 125? 187? 152? 175? 126 ?

18. How many are 55 times 24? (55=50+58.) Therefore, 55 times 24-50X24=1200, plus the tenth of that number, 120=1320. 55 times 26 (1300+130=1430.) 55 times 34? 75? 82? 53? 72? 96? 28? 29? 73? 125? 146? 132? 165? 184? 173? 105? 115? 123? 136? 145?

19. How many are 60 times 24? (6×24×10.) 60 times 35? 94? 362 52? 65? 72? 39? 46? 53? 76? 89? 37? 48? 20. How many are 90 times 24? (24×100=2400—240°— 2160.) How many are 90 times 36? 45? 58? 73? 92? 84? 42? 35? 27? 29? 57? 99?

The above are the easy methods of multiplying mentally by every fifth number from 5 to 60 inclusive, and also by 90. The intermediate numbers are managed as follows: Consider 8 and 9 as 10-2 and 10-1; 11* and 12 as 10+1 and 10+2; 13, 14 as 15-2, 15-1; 16, 17 as 15+1 and 15+2; and so of all the others. Thus, 17X24=(15X24)+(2×24); and 23×24=(25×24)—(2×24). Thus, the intermediate are solved like the others, excepting that once or twice the multiplier has to be added or subtracted. The following table will make this more clear.]

From merely glancing the eye down this table, it becomes apparent that no multipliers need be used under 60, except the easy numbers 5, 10, 15, 20, 25, &c., the intermediate factors being rectified by the addition or subtraction, as the case may

*The product of 11 and any number between 10 and 99 inclusive is found by placing their sum between the two figures of the latter factor. Thus 11X347 (sum of 3 and 4) between the 3 and 4374; and 11X45 =495. Why? When the sum of the figures exceeds 9, the first figure of course must be increased by 1. Thus, 11X48 = 528.

Why ?