Prop. 4. When ciphers are intermixed with the figures in the multiplier. Rule. Omit the ciphers, and let the first figure of each product be placed under its respective multiplier. Prop. 5. When there are ciphers at the end of the multiplicand or multiplier. Rule. Neglect the ciphers, and multiply as before, then to the right-hand of the product annex as many ciphers as were omitted. For the proof. Multiply the multiplier by the multiplicand, and if the product be the same with that of the multiplicand by the multiplier, the work is right. Note 1. If two numbers are to be multiplied together, they will make the same product, whichever number you make the multiplier. 2. If several numbers, as 5, 6, 7, &c. are to be multiplied together, it is the same thing whether 5 be multiplied by the product of 6 and 7, or it be multiplied first by 6 and then by 7, &c. And, if several given numbers are to be multiplied by any number, and the sum of the products taken; it will be the same thing, if you multiply the sum of those given numbers by that multiplier. 3. The product of any two numbers can have at most but as many places of figures as are in both the multiplier and multiplicand, and at least but one less. tion. 4. Multiplication may be proved by casting out the nines as in addiThus cast the nines out of the multiplier and multiplicand, and set down the remainders. Multiply the two remainders together; and, if the excess of nines in the product be equal to the excess of nines in the total product, the work is generally right. Examples to Proposition 1. (1.) Multiply 471347325 by 2 Product 942694650 (2.) Multiply 371493407 by 3. Examples to Prop. 2. (12.) Multiply 47134987 by 56 8 377079896 7 Product 2639559272 (13.) Mult. 47134784 by 21. Examples to Prop. 3. (23.) Multiply 471493475 Or, 471493475 Proof by multiplication. by 4395 2357467375 4243441275 1414480425 1885973900 Product 2072213822625 4395 1885973900 1414480425 4243441275 2357467375 2072213822625 Multiplicand 8 X 3 Multiplier. (24.) Mult. 430714934 by 743. 4395 471493475 21975 30765 17580 13185 39555 17580 4395 30765 17580 2072213822625 Examples to Prop. 4. (30.) Multiply 4713457 37707656 18853828.. 32994199. 23567285 26885596435656 31.) Mult. 371493407 by 700505. (37.) Multiply 47150000 37720 42435 14145 Product 187657000000000 (38.) Mult. 471000 by 40700. CLASS II. Exercising all the Propositions. (43.) Mult. 47149 by 7. (44.) Mult. 371594 by 12. (51.) Required the continued product of 56,750,54730, 64007, and 587504. (52.) Required the sum of 157 added 495 times to itself. (53.) Let 954 be added 435 times to itself, and shew what the last sum total exceeds or falls short of four hundred and fifteen thousand. (54.) Required the product of eleven thousand eleven hundred and eleven, by twelve thousand twelve hundred and twelve. (55.) What is the difference between thrice six and twenty and thrice twenty-six. (56.) There are two numbers; the greater is 19 times 508, and their difference is 15 times 112; required the sum and product of those numbers. SIMPLE DIVISION. Definition 1.-Simple Division is a rule by which we find how often one number is contained in another of the same denomination; being a short method of performing subtraction. 2. The number to be divided is called the dividend, the number you divide by is called the divisor; and hence will arise a third number, called the quotient, which shews how often the divisor is contained in the dividend. If the divisor does not exactly contain the dividend, a fourth number will occur, called the remainder, which must always be less than the divisor. Prop. 1. When the divisor does not exceed 12. Rule. Observe how often the divisor is contained in the first, or first and second figure of the dividend, and set the quotient figure under it, carry 10 for every unit remaining after subtraction to the next figure of the dividend; proceed thus, multiplying and subtracting mentally, till you have made use of all the figures in the dividend. C Prop. 2. When the divisor is a composite number. Rule. Divide the dividend by one of the component parts, and that quotient by the other, for the required quotient. If there be a remainder to each of the quotients, multiply the last remainder by the first divisor, and to that product add the first remainder for the true one. Prop. 3. When the divisor consists of several figures. Rule. Find how many times it may be had in as many figures of the dividend as are just necessary; multiply the divisor by the quotient figure, subtract the product from that part of the dividend which stands above it, and, to the right hand of the remainder, bring down the next figure in the dividend, which number divide as before; and so on till all the figures in the dividend are brought down. Prop. 4. When the dividend has ciphers on the right hand. Rule. Cut off the ciphers from the divisor by a dash of your pen, and also cut off as many ciphers, or figures, from the dividend. But, when the division is finished, the ciphers omitted must be restored to their proper places, and the figures cut off in the dividend must be placed to the right-hand of the remainder. Note 1. When the scholar is pretty ready at division, he may subtract each figure of the product as he produces it, and write down only the remainder. For the proof. Multiply the quotient by the divisor, to the product add the remainder, if any, and the sum will be equal to the dividend. 2. There are several methods of proving division. If you subtract the remainder from the dividend, and divide this number by the quotient, the quotient found by this division will be equal to the former divisor. 3. Or, add the remainder, and all the products of the several quotient-figures by the divisor, together, according to the order in which they stand in the work, and the sum will be equal to the dividend. Another method. Cast away the nines in the divisor and quo tient, take their product, and cast away the nines, to which add the excess of nines in the remainder after division: the excess of nines in this sum will be equal to the excess of nines in the dividend, when the work is right. |