again, until each product is called up to your mind just as soon as you look at the two numbers.

6. Pursue the same course in memorizing the products of 3's, 4's, 5's, 6's, 7's, 8's, and 9's.

MULTIPLIER ONE FIGURE.

PREPARATORY STEPS.

83. STEP I.-Find by using the Multiplication Table the product of each of the following:

1. 8 x 6; 8 tens × 6; 8 hundred × 6; 8000 × 6.

2. 9 × 7; 90 × 7; 900 × 7; 9000 × 7.

3. 3 × 5; 30 × 5; 300 × 5; 3000 × 5.

4. 7000 × 4; 500 × 9; 8000 × 4; 4000 × 3.

5. 60000 × 6; 900000 × 9; 5000000 × 7.

= 3500.

84. STEP II.-The orders in a number are independent of each other; hence, to find any number of times a given number, we multiply each order separately, thus:

To find 6 times 748, we regard the 748 = 700 + 40 + 8. We know from memorized results that 6 times 8 are 48, that 6 times 40 are 240, and that 6 times 700 are 4200. Having taken each of the three parts of 748 6 times, the sum of these products must be 6 times 748. Hence, 48 +240 + 4200 = 4488 6 times 748.

EXERCISE FOR PRACTICE.

Multiply and explain, as shown in Step II, each of the following:

85. The method of finding the sum of two or more times a given number by using memorized results is called Multiplication. The number taken is called the Multiplicand, and the number which denotes how many times the multiplicand is taken is called the Multiplier.

86. PROB. I.-To multiply any number by numbers less than 10.

EXPLANATION.-1. The 369 is equal to the three parts, 9, 60, and 300. 2. By taking each of these parts four times, the 369 is taken four times. Hence, to find 4 times 369, the 9 is taken 4 times; then the 60; then the 300, as shown in the analysis.

3. Uniting the 36, the 240, and the 1200 in one number, we have 4 times 369. Hence, 1476 is 4 times 369.

4. In practice, no analysis is made of the number. We commence with the units and multiply thus:

(1.) 4 times 9 units are 36 units or 3 tens and 6 units. We write the 6 units in the units' place and reserve the 4 tens to add to the product of the tens.

(2.) 4 times 6 tens are 24 tens, and the 3 tens reserved are 27 tens or 2 hundred and 7 tens. We write the 7 tens in the tens' place, and reserve the 2 hundred to add to the product of the hundreds.

(3.) We proceed in the same manner with hundreds, thousands, etc. From these illustrations, we obtain the following

87. RULE.-Begin at the right hand and multiply each order of the multiplicand by the multiplier. Write in the product, in each case, the units of the result, and add the tens to the next higher result.

88. Continue the practice with abstract numbers by taking examples from Arithmetical Table No. 1, page 16, in the following order:

Three Figures in the Multiplicand.

1. Use three columns and copy for multiplicands each number in the columns, commencing at the top of the Table.

2. Take as multiplier the figure immediately under the right-hand figure of the multiplicand.

The first six examples taken in this way from columns A, B, C, are

3. Let examples be copied in this way from columns A, B, C; B, C, D; C, D, E; D, E, F; E, F, G; F, G, H; G, H, I; and H, I, J.

Four Figures in the Multiplicand.

1. Use four columns, and copy the multiplicands and multipliers in the same way as with three figures, taking the multipliers from the first column on the right.

2. Copy from columns A, B, C, D; then B, C, D, E; C, D, E, F; D, E, F, G; E, F, G, H; F, G, H, I.

Six Figures in the Multiplicand.

1. Copy, as already directed, examples from columns A, B, C, D, E, F; then B, C, D, E, F, G; C, D, E, F, G, H; D, E, F, G, H, I; and E, F, G, H, I, J. Take the multipliers from the right-hand column used.

2. Let the examples from each of these sets be worked at your seat between recitations or out of school.

89. 1. Bought 3 barrels of flour, at $12 a barrel, and a barrel of crackers for $5; how much did the whole cost?

SOLUTION.-The whole cost three times $12, plus $5, which is $41.

2. If it requires 4 yards of cloth to make a coat, and 1 yard to make a vest, how many yards will make 8 of each? 12 of each? 9 of each ?

3. Bought 12 chairs at $2 each, a sofa at $45, and 5 tables at $9 each; how much did the whole cost?

4. Gave $7 each to 4 men, paid for 9 yards of cloth at $3 a yard, and for a coat $18; how much money have I spent? 5. At 8 dollars a cord, what will 5 cords of wood cost? 7cords? 11 cords? 9 cords? 12 cords?

90. 6. How much will acres of land cost, at $285 an acre? Ans. $1995. SOLUTION.-7 acres will cost 7 times $285. 7 times $285 = 7 times $5 +7 times $80 + 7 times $200 = $1995. Hence, 7 acres cost $1995. 7. What will 647 cords of wood cost at $6 a cord? 8. What will be the cost of building 213 yards of iron fence, at 3 dollars a yard? Ans. 639 dollars. 9. There are 5280 feet in a mile; how many feet in 12 miles? Ans. 63360 feet.

10. I sold 284 acres of land at 9 dollars an acre; how much money did I receive? Ans. $2556. 11. There are 4 farthings in one penny; how many farthings in 379 pennies? Ans. 1516 farthings. 12. James Reed went to market with $485; he paid for 20 barrels of flour at $8 a barrel; 16 boxes of soap at $3 a box; and 3 tubs of butter at $12 a tub; how much money did he have left? Ans. $241. 13. A merchant bought 9 hogsheads of molasses at $52 a hogshead, and sold the whole for $544; how much did he gain by the transaction? Ans. $76. 14. Sold 89 bushels of beans at $2 a bushel, and 7 loads of hay at $19 a load; how much did I receive for both?

91. STEP I. To multiply any number by 10, 100, 1000, and so on.

1. A figure is multiplied by 10 by moving it one place to the left, by 100 by moving it two places, etc. Thus, 4 expresses four, 40 expresses 10 fours, 400 expresses 100 fours, etc.

2. A cipher placed at the right of a number moves each significant figure in it one place to the left; hence, multiplies it by 10.

Thus, in 372 the 2 is in the first place, the 7 in the second, and the 3 in the third; but in 3720 the 2 is in the second, the in the third, and the 3 in the fourth place; hence, annexing the cipher has removed each figure one place to the left, and consequently multiplied each order in the number by 10.

3. In like manner annexing two ciphers, three ciphers, etc., multiplies a number by 100, 1000, etc., respectively.