plier is hundreds, the right-hand figure must be placed in the column of hundreds; and, in general, whatever the order of the multiplier is, the right-hand figure must be in the column of that order. COR. 4.-If there be one or more zeros in the multiplier, the product of the next figure will be put back one figure for every zero. REM.—In the multiplication, each figure may be regarded as the unit of its order, PROBLEMS. 1. 10 x 10 = 100. 11 x 10 = 110 121 3. 12 x 12 = 144 = 12 x (10 + 2) = 12 x 2 = 24 12 x 10 = 120 144 4. Multiply 432 by 4 = (400 + 30 + 2) * 4. 2 x 4 = 8 and 432 4 1728 1728 5. Multiply 432 by 14 = 432 X (10 + 4). .. 432 x 4 = 1728 432 432 x 10 = 4320 or 14 6048 REM.—The problems should be carefully impressed on the mind before proceeding. COR. 1.—When the multiplicand has several figures and the multiplier one that is only units, the first product of units by units will be units, or units and tens; the units must be placed in the right-hand or units place; if there be tens, it must be reserved and placed in or added to the column of tens; in the next product of tens by units, the right-hand figure will be tens, and must be united with the tens reserved, and placed in the column of tens; the left-hand figure, if there be one, must be treated as the previous one, reserved until the next product is obtained, and united with the right-hand figure; the process is the same in every successive order. COR. 2.—When the multiplier also has several figures, the process of each successive multiplier is the same, except that the right-hand figure of each product must be placed in the order of its multiplier. (Cor. 3, Prob. 2, page 22.) REM.-A multiplicand may be either an abstract or a concrete number, but a multiplier cannot be concrete, as it cannot refer to things, but merely indicates how many times the multiplicand is to be taken ; but the product will be of the same name as the mul. tiplicand; for twice $5 are $10; 3 times 20 yards of cloth are 60 yards of cloth; twice 4 are 8; 3 times 4 are 12, etc. In computation, it is best to regard all numbers as abstract. (7.) (9.) 26432 3004 105728 79296 2748928 79401728 11801700 (11.) (12.) 123 234 492 369 234 246 1321705728 28782 28782 REM.—The product is not changed by alternating the multipli. cand and multiplier. EXAMPLES. 1. Multiply 54326 by 346. 6. Multiply 9876325 by 356. 2. Multiply 23748 by 543. 4. Multiply 879654 by 2175. 3. Multiply 46874 by 697. 8. Multiply 986432 by 8704. 4. Multiply 36975 by 476. 9. Multiply 326875 by 3005. 5. Multiply 236874 by 2134. 10. Multiply 468753 by 2100. Examples may be added, or the same repeated, as the student will more readily comprehend by repetition than by different examples. REM. 1.-In multiplication, two factors are given to find their product. REM. 2.-In division, two numbers also are given to find the third; the one called the dividend corresponds to the product in multiplication, the other given number is called the divisor, and the required number is called the quotient; the two latter correspond to the factors in multiplication. DIVISION. PROBLEMS. When the product of two numbers is 4, and one of the numbers is 2, the other number is also 2; for 2x2 =4, and 4 divided by 2, or 4 divided into 2 equal parts, each part is 2, that is, the quotient is 2. 1. 9 = 3 = 3. 4. 16 ; 4= 4. 2. 12:2= 6. 5. 15:3= 5. 3. 12 : 3 6. 15 : 5 = 3. = 4. COR. 1.-The product of the divisor and quotient equals the dividend. COR. 2.-The divisor and quotient may be alternated. 24 18 12 COR. 3.-Division is the reverse of multiplication and addition, and is similar to subtraction ; for, it is separating a number into equal parts, which is the same as subtracting the same number from a larger one; that is, subtracting the divisor from the dividend and then from the remainder, repeating this process until there is no remainder, or until the remainder is less than the divisor. 6 is subtracted 4 times, hence it is contained four times. 24:6= 4. 6 6 11 (1.) (3.) 12 24 24 (4.) (5.) 11 ) 121 ( 10 + 1 12 ) 144 ( 10 + 2 110 120 11 = 8. 6. 48 ; 12 = 4. 12. 120 • 10 = 12. 7. 64 ; 8 = 8. 13. 130 ; 10 = = 13. 8. 96 ; 12 14. 140 ; 10 = 14. 9. 12 x 4 = 48. 15.. 10 x 12 = 120. 10. 8 x 8 = 64. 16. 10 X 13 = 130. 11. 12 x 8 = 96. 17. 10 x 14 = 140. CoR. 1.–Adding a zero to the right of a number multiplies the number by 10; taking a zero away from the right of a number divides the number by 10. Divide 60536 by 4; thus, 4) 60536 ( 10000 or 4 ) 60536 15134 The divisor 4 is contained once 536 100 in the unit of the highest order of 400 the dividend, which is one ten thousand; into the remainder 136 30 120 5000 times, then 100, 30 and lastly 4 16 ( 4 16 15134 |