EXAMPLES. 1. From 4738 take 2123. OPERATION, 4738 Diff. 2615 Proof 4738 Explanation. Having placed the less number under the greater, I take 3 from 8, and 5 remain to put down; then 2 from 3, and 1 remains; 1 from 7, and 6 remain; 2 from 4, and 2 remain; the whole remainder then is 2615. I then add the second line 2123 and this remainder together, which gives the proof. 2. From 135167 take 19103, and from 720981 take 10029. OPERATIONS. 135167 Diff. 116064 Proof. 135167 720981 710952 720981 Explanation. In the first of these operations the work is easy, till we come to the 9, where we say 9 from 5 I cannot, but borrowing 10, and adding it to the 5, the sum is 15; therefore 9 from 15, and 6 remain; put down 6, and carry 1 to the next figure 1, which makes 2, then 2 from 3, and 1 remains to put down. There being no figure under the left hand figure 1, I say 0 from 1, and 1 remains to put down. In the second operation, I say 9 from 1 I cannot, borrow 10, then 9 from 11, and 2 remain to put down; then carry 1 to the 2 makes 3, 3 from 8, and 5 remain to put down: the rest as before. 7. If from one thousand two hundred and thirty-four, seven hundred and eighty-nine be taken, what will be left? Ans. 445. 8. If three hundred and sixty-five be taken from five hundred and sixty-three, what is the remainder? Ans. 198. 9. If one thousand two hundred and thirty-four wheat corns be taken out of a bin containing one million, how many will be left? Ans. 998766. 10. A man was born in the year 1767, and died in 1799; what was his age? Ans. 32 years. 11. One was born in 1773, and another in 1801; required the difference of their ages? Ans. 28 years. 12. The art of printing was discovered in 1449, how many years is it since, this being 1812? Ans. 363 years. 13. A courier travelled three thousand miles in one year, and only one thousand nine hundred and nine in the next; how much does the former distance exceed the latter? Ans. 1091 miles. 14. Out of a thousand pounds, a person paid away eight hun dred and fifty-two; how many pounds had he remaining? Ans. 148 pounds. 15. Borrowed one thousand two hundred and thirty-four guineas, and paid in part nine hundred and eighty-seven; what sum is there still remaining due? Ans. 247 guineas. 16. There is a person, who if he lives until the year 1820 will be seventy-five years old; in what year was he born? Ans. in 1745. 17. King George the Second came to the throne in 1727, and died in 1760; how long did he reign? Ans. 33 years. 18. The western empire was destroyed by Odoacer, king of the Heruli, in the year 476, and the eastern empire submitted to Mahomet the Second, emperor of the Turks in 1453; how many years did the latter exist after the former? Ans. 977 years. MULTIPLICATION. 28 Multiplication' is a short method of addition; it teaches how to find the sum that arises from repeating one number, called the multiplicand, as often as there are units in another number, called the multiplier. The number sought, or that which arises from the operation, is called the product. The multiplicand and multiplier are frequently called Terms or Factors, and the product is sometimes called the Factum. The mark denoting Multiplication is x; it is named into, and shews that the number standing before the sign is to be multiplied into (or by) the number which follows it; thus 3×4 denotes that 3 is to be multiplied by 4, or taken 4 times o. To perform the operations in this rule with ease, a table of the products of every two numbers, each not exceeding 12, has been contrived; it is called the Multiplication Table, and must be well understood, learned by heart, and remembered. f The name Multiplication comes from the Latin multus many, and plico to fold; product from produco to produce; factor a maker or doer, and factum a thing done or made. When two or more numbers are to be multiplied by any number, the vinculum is placed over all the former, and a point is frequently interposed between the factors instead of the sign x; thus 3+4.5 denotes that the sum of 3 and 4 (or 7) is to be multiplied by 5; also 3+7+2.6-4 shews that the sum of 3, 7, and 2, (viz. 12,) is to be multiplied by the difference of 6 and 4, (viz. 2 ;) and the like in other cases. To find the product of two numbers, look for one of them in the top line of the Table, and for the other in the left hand column; then the number which stands directly under the first, and level with the second, is the product required: thus to find 3 times 5, look for 3 at top, and 5 on the left, then under 3 and level with 5 stands 15, which is the product of 3 and 5. To find 10 times 9, under 10 and level with 9 stands 90, the product. To find 6 times 11, under 6 and level with 11 stands 66, the product, &c. To shew that Multiplication is derived immediately from Notation, let it be required to find 3 times 4; put down the units in 4 count the whole, (Art. 12.) thus, 1, 1, 1, 1 .... 1, three times, and then 1, 1, 1 1, 1, 1, 1; these counted amount to 12, therefore 3 times 4 are 12; and the like may be shewn in all cases. But the rule is commonly derived from Addition thus; to find the product of 3 times 4, put 4 down three times, and find the sum. So if I want to find 9 times 8, I must put nine eights under each other, add them together, and the sum will be 72, or the product of 9 times 8; by this method the Multiplication Table was first formed. 4 4 4 12 Simple Multiplication, or Multiplication of whole numbers, is performed by the following rules. 29. When the multiplier does not exceed 12. RULE I. Under the right hand figure of the multiplicand write the multiplier. II. Multiply every figure in the multiplicand by the multiplier, and set the product, if it be less than 10, under the figure multiplied. III. But if the product be 10, or more, set down the units only, and carry 1 for every 10 to the next; multiply the next figure, and after you have multiplied it, (not before,) carry the ones to the product. IV. Proceed in this manner till all the figures are multiplied; at the last (or left hand) figure, the whole product must be set downi. i The sum of numbers is evidently of the same denomination with the numbers added; it cannot be of any other; consequently the product of any number multiplied by another must be of the same denomination with the number multiplied. Hence if units be multiplied by any whole number, the product is units; if tens be multiplied, the product will be tens; if hundreds be multiplied, the product will be hundreds, &c. Let it be required to multiply 574 by 4; now the 4 units multiplied by 4 produces 16 units; the 7 tens multiplied by 4 produces 28 tens, or 280; and the 5 hundreds multiplied by 4 produces 20 hundreds, or 2000. Wherefore these several products when added together will give the product of the two given numbers. -- By the Rule 574 4 2296 From this example, the truth and reason of the Rule will be manifest. With respect to the method of carrying, it is plain from Notation that 10 units are I ten, 20 units 2 tens, &c. 10 tens are 1 bundred, 20 tens 2 hundreds, &c. And in general, any number of tens of an inferior denomination will be so many units of the next superior; and therefore 1 is always carried for every ten, from the inferior to the next superior denomination, as directed in the Rule. 1. Multiply 375294 by 2. OPERATION. Multiplicand 375294 Multiplier 2 Product 750588 EXAMPLES. Explanation. Having written the multiplier 2 under the right hand figure of the multiplicand, namely, under the 4, I begin by multiplying that figure; thus I say twice 4 are 8, and put it down; next I say twice 9 are 18, put down 8 and carry 1; then twice 2 are 4 and 1 I carried 5, put down 5; then twice 5 are 10, put down o and carry 1; then twice 7 are 14 and 1 carried 15, put down 5 and carry 1 ; lastly, twice 3 are 6 and 1 carried 7, I put it down, and the work is finished. 2. Multiply 968754 by 3. OPERATION. Multiplicand 968754 Multiplier 3 Product 2906262 Explanation. Here I say 3 times 4 are 12, put down 2 and carry 1; 3 times 5 are 15 and 1 carried make 16, put down 6 and carry 1; 3 times 7 are 21 and 1 carried 22, put down 2 and carry 2; 3 times 8 are 24 and 2 carried 26, put down 6 and carry 2; 3 times 6 are 18 and 2 carried 20, put down 0 and carry 2; lastly, 3 times 9 are 27 and 2 carried make 29, I put down the whole 29, and the work is finished. |