repeated; it must déninish it to nothing. How is multiplication proved cipher to the multiplicand, we 'repeat it 10 times; and then subtract , tiplier is 99 we join two ciphers which increases our multiplicand 100 times , then subtracting the multiplicand once, leaves it 9 times re peated; and we observe the same rule, in multiplying by 3 nines, or any number of nines. How do you proceed to multiply by a composite number? A. First multiply the multiplicand by one of the component parts, and then that product by the other; and the last product will be the total product. When is a number said to be a composite number? A. When it is produced by multiplying together any two figures in the multiplication table. Is thirty-five a composite number? A. Yes. What are its component parts ? A. 7 and 5. Why? A. Because 7 times 5 are 35. In multiplying by the component parts of 35, why should it produce the same product, as multiplying directly by 35 ? A. Because when 7 times; and when we multiply this product by 5, we repeat a num we multiply by 7, we repeat our multiplicand ber 5 times, which is 7 times greater than our first multiplicand; and 5 times 7 times must make 35 times repeated.- What is multiplication of decimal, or federal money? A. It is repeating the parts of an integer, or integers and parts of integers, a given number of times.What are integers in federal money ? A. Dollars. What are the de. cimal parts of an integer? A. Dimes, cents and mills, or cents and mills, as they are commonly expressed.' What do you understand by a decimal ? A. I understand, that it is something less than a unit; in calling tenths, I understand that it takes 10 for a unit; and 'hundredths, that it takes 100 for a unit; and thousandths, that it takes 1000 to equal a unit; so that the name of each figure, shows the ex. act number required to equal a unit. How do you multiply decimal money? A. The same as whole numbers. How do you distinguish decimals from whole numbers ? A. By a separatrix placed between them. Why are decimals multiplied the same as Because they increase in a tenfold proportion, ame as whole num produet? A. As far bers. How do you point off decimals in yonle sa ole numbers? A. from the right, as the separatrix is in the multiplicand. When the price of one yard, one pound, or the price of a unit of any kind, is given; how do you obtain the price of a quantity ? A. By multiplying the price of one, by the number expressing the quantity. Why should that give the price of the quantity?' A. Because it is repeating the price of one as many times, as the quantity expresses, or contains units. If one yard cost $4,50 cts. ; how would you find the price of 4 yards ? A. I would multiply $4,50 cts., the price of one yard, by the units expressed in the number of yards, that is, by 4. Why should that give the price of 4 yards ? A. Because, it is plain, that 4 yards must cost 4 times as much, as one yard; and multiplying the price of one yard by 4, is repeating the price of one yard, 4 times. concise manner how often one number is contained in another, or how often,one number may be subtracted from another. The two given numbers in division are called divisor and dividend. The number arising from the operation of the work, is called the quotient. And there is sometimes an uncertain number called the remainder, The dividend is the number to be divided. The divisor is the number by which the dividend is to be divided. The quotient shows how many times the dividend contains the divisor, or how many times the divisor can be subtracted from the dividend. The remainder is something left of the dividend after the divisor can no longer be subtracted, and is always less than the divisor. In multiplication we have the price of a unịt given to obtain the price of a quantity. In division we have the price of a quantity given to obtain the price of a unit; so it will be perceived, that division is exactly the reverse of multiplication, as well as a short way of performing subtraction; by it, we change lower denominations into higher; and by it we divide any number or quantity into an equal number of parts or shares, containing equal quantities. RULE.Place the divisor at the left hand of the dividend, separated from it by a vertical line, (1).. Then, assume as many of the left hand figures of the dividend, as shall be sufficient to contain the divisor something less than ten times, and inquire how often the di-, visor is contained in the assumed number, and place the result for a quotient, at the right of the dividend, separated therefrom by a small vertical line. Multiply the divisor by this quotient figure, and place the product under the assumed part of the dividend. Then subtract it therefrom, and to the remainder, bring down the next figure of the dividend, which must be divided as before, placing the result in the quotient; and so proceed, till all the figures of the dividend, have been brouglat down." After multiplying by your quotient figure, if the product be greater than the assumed part of your dividend, you must lese. sen your quotient figure; but if the remainder (after snbtracting) be greater than the divisor, you must increase your quotient figure.And when you bring down (to the right of your remainder) a figure from the dividend, and it does not then contain the divisor, you must write a cipher in the quotient, to signify, that the divisor is not contained, and then bring down another figure from the dividend; and so con!inue to tlo, till your divisor is contained, or all the figures brought down from the dividend. METHODS OF PROOF.-- The best method of proving division, is by multiplication. 1. Multiply the quotient by the divisor, and if the product be equal to the dividend, the work is right. But should you have a remainder, it must be added to the product to make it equal to the diyidend. 2. Division may be proved by addition. Add the divisor as many times, as the quotient expresses a unit: and to the amount add the remainder, (if you have any,) and if the sum equal the dividend, the work is right. But the better way to prove division by addition, is to add the remainder to the several products produced from the multiplication of the divisor in the order in which they stand; and if the amount equal the dividend, the work is right. 3. Division may be proved by subtraction. Subtract the divisor from the dividend, as often as the quotient contains a unit; and if it diminish it to nothing, the work is right; but when you have a remainder in dividing, your remainder after subtracting, must equal your remainder, when the work was performed by division. 4. Division may be made to prove itself. Divide the dividend by the quotient, and if your quotient equal your divisor in the first operation, the work is right; and if you had a remainder in the first work, the remainder in the last operation must equal the remainder in the first. Division is denoted by the following character +; thus, 48+6 signifies, - that 48 is to be divided by 6. It is sometimes denoted by a horizontal line, (-) drawn under the 48 sig. ! dividend, and the divisor placed below the line; thus, 6 nifies, that 48 is to be divided by 6. It is likewise denoted by two vertical lines, | | or inverted parenthesis, )(; thus, 6)48( signifies, that 48 is to be divided by 6. NOTE.-Under this, as well as the other simple rules, the methods of proof introduced, are in themselves sufficient to illustrate the principles of the rules, and the relation which they bear to one another; and for the sake of becoming ac quainted with these principles, the student should as carefully examine every method of proof offered, as the rules themselves. Every difficulty Jvill vanish before the student, every principle will appear easy and simple, and his progress will be rapid and interesting, if he will only attentively examine the illustrations and demonstrations given with the examples, DIVISION TABLE. 11 4 • 4 11 5 + 5 = 1 4 21 6 3 28 4 2)10 6 319 3 3/12 4 3 15 8 4112 3 4 16 4 420 5 15 3 5 20 4 5125 5 6 728 4 7 9 27 3 9 36 4 9145 20 . . . 10/30 10 22 11133 3 11 44 4 1155 5 11 24 2 12136 3 12148 4 12160 5 12 .. - 28.- 36 9.3 8|56 . 981 o 60 12184 : . . 522 6 33 12 DIVISION TABLE -- CONTINUED. 6.46 = 11 7 = 11 8 8 1499 z 1 12 6 2 14 7 2/16 8 2/18 :: 9 2 18..6 3 21..7 3/24 8 3/27 : 39 3 24 .. 6 4 7 4/32 8 4 36 :-9... 4 30 5/35 5.7 * 540 8 5 6 642 7 6 48 654 6 42 6 7/49.7 756 8 7163 9 7 48 6 7 8164 8 8172 4.9 54 6 9/63 .. 7 872 8 9 9 6 10170 7 10/80 8 - 10/90 9 10 66.5 6 1177 7 .1188 8 11 72 6 7 12/96 8 12 108 9 12 10 10 = 1/100. 10. 10177 11 7148 12 4 20 10 2 110.10.-1188 11 8160 -: 12 5 30 3 120 10..1299 11 9172 - 12 6 40.- 10 4 11 + 11 = 1 110. 11. 10/84 12 7 50 10 11.- 2.121 11.. 11/96 .- 12.. 8 60 10 1:1... 31132. li 121108. 9 70 10 744 11 - 4/12 + 12 1|120. 12.-10 80 : 10 II 12 2132 - 12 ..11 90 10 - 9166 11.636 12 3 144. 12.-12 Note. The student will notice in the table, that the dividend is placed first, the divisor next, and the quotient or answer to each, at the right of the parallel lines. The student can make use of language similar to this, in repeating the table, 2 in 2 once, 2 in 4 twice, 2 in 6 three times; that is, 2 is contained, or can be subtracted from 2, once; and 2 is contained, or can be subtracted from 4, twice; &c. 8+2+how many? 32+ 4=how many ? 16+4=how many ? 28; 4=how many? 36+6=how many? 81+9=how many 2 45+5=how many? 64: 8=how many? 72+9=how many? 72-12=how many ? 96+8=how many? 48+ 6-how many? Questions to be mentally answered by the Student. 1. If 10 oranges cost 20 cents; what will one cost ? 2. If 3 apples cost 3 cents; what will i cost? 3. IT 3. bushels of oats cost 6 shillings; what was that a bushel? 8155 - - 5 24 ? |