59. Queftions to exercise Addition and Subtraction. Queft. 1. Two perfons A and B are of feveral ages, the age of the elder, being that of A is 70, the difference of their ages is 19, what is the age of B? Anfw. 51. Quest. 2. What number is that which being added to 168, makes the fum to be 205? Answ. 37. Queft. 3. The fum of two numbers is 517, the leffer is 40, what is the greater? Anfw. 477.. Queft. 4. A certain perfon born in the year of our Lord 1616, defired to know his age in the year 1676, what was his age? Anfw. 60. Quest. 5. The greater of two numbers is 130, their difference is 49, what is the leffer number? Anfw. 81. Queft. 6. What number of pounds, fhillings, and pence, added to 341. 16s. 9d. will make 1001? Anfw. 651. 35. 3d. Quest. 7. How many years fince the Spanish Invafion, it be ing in the year 1588, and the present year being 1750? Anfw. 162 years. 4 Quest. Queft. 8. From 100 pounds borrow'd, take 72 paid; 'Twasa Virgin that lent it, what's due to the Maid? Anfw. 281. Queft. 9. A Mifer hath three bags of money, containing in all 29841, 6s. of which the firft contains 324 10s. and the fecond 9134. os. 6 d. what doth the third contain? Answ. 17461. 15s. 6d. Quest. 10. A Merchant had 5 debtors, AB, C, D, and E, which, together, owe him 11567. now B, C, D, and E, together, owe him 737. what is A's debt? Anfw. 419l. Queft. 11. The three Towns of London, Huntingdon, and York, lie in the fame road; the diftance between the farthest of thefe Towns, viz. London and York, is 192 miles; now, if from London to Huntingdon be 57 miles, how far is it from Huntingdon to York? Anfw. 135 miles. 60. CHA P. V. MULTIPLICATION of Whole Numbers. MU ULTIPLICATION teaches how by two Numbers given to find a third, which fhall contain either of the Numbers given fo many times, as the other contains I or unity: Or Multiplication may be confidered as a manifold addition, or the repeating of a given Number as often as required. 61. Of the two Numbers given in Multiplication, one (which you will) is called the Multiplicand, and the other the Multiplier, or both are called Factors. 62. The Number fought, or arifing by the Multiplication of the two Numbers given, is called the Product, the Fact, or the Rectangle: So if 5 be given to be multiplied by 3, or 3 by 5, the Product is 15; that is 3 times 5, or 5 times 3 makes 15; and here 5 may be called the Multiplicand, and 3 the Multiplier, or 3 may be called the Multiplicand, and 5 the Multiplier; and as 3 (one of the two numbers given) contains I or unity thrice, fo 15 the Product contains 5 (the other given number) thrice; likewife as 5 (one of the given numbers) contains unity 5 times, fo 15 (the Product) contains 3 (the other given Number) 5 times: This fame Product may be found by Addition two ways, viz. either by writing down the number 5, three three times; or the number 3, five times; and adding them together, as below. 63. Multiplication is either fingle or compound. 64. Single Multiplication is, when the Multiplicand and Multiplier confift each of them of one figure only, as in the laft example. In like manner if you multiply 9 by 5, the product is 45; this is likewife fingle Multiplication: Now the feveral Varieties of fingle Multiplication are well expreffed in the following Table, ufually called Pythagoras's Table. The truth of which may be proved by Addition, as above. The TABLE of MULTIPLICATION. 2 3 4 5 6 7 8 9 2 4 6 8 10 12 14 16 18 15 18 21 2427 4 8121620|24|28|32| 5 10 15 20 25 30 35 40 121824 30 35 42 54 71421 28 35 42 49 5663 816 24 32 40 48 566472 918273645546372|81| The use of the Table is this: Having one figure given to be multiplied by another, to know the product of them, find the Multiplicand in the top of the Table, and the Multiplier in the first column thereof towards the left-hand; then the product will be found on the fame line with the latter, and under the former. So 9 being given to be multiplied by 5, I find 9 in the top of the Table, and 5 in the firft column towards the left-hand; then carrying my eye from 5 in a right line equidistant to the upper fide or top line of the Table, until I come to that square which is directly under 9, I find there 45, which is the product required. The particular varieties of this Table ought to be learned by heart, (that is, a man must be able to give the product of any fingle Multiplication, without the leaft paufe or ftay) before he can readily work compound Multiplication, as will appear hereafter. Note 1. The product of any number by an even number, will be an even number. 2. The product of an odd number by an odd number, will be an odd number. 3. The product of any two numbers can have, at moft, but as many places of figures as are in both Factors; and, at least, but one place fewer. Examp. 9+9=81, and 1+1=1. 65. Compound Multiplication, is when the Multiplier and Multiplicand, either one or both, confift of more figures than one. 134 2 66. In Compound Multiplication, when the numbers given end with fignificant figures, place them as in Addition and Subtraction. So 134 being given to be multiplied by 2, place them thus: Then proceeding to the Multiplication, fay thus, 2 times 4 is 8, which fet under the line in the rank of your multiplying figure; again fay 2 times 3 is 6, which likewife fet under the line in the next rank: Laftly, 2 times I is 2, which being likewife fet down under the line in the next rank, the product is discovered to be 268, and the work will ftand as in the margin. 268 67. The truth of this process may be made evident: Thus, Since 100 multiplied by 2 produces 30 multiplied by 2 produces And 4 multiplied by 2 produces 200 60 8 268 For fince the parts 100, 30, and 4, added together, make Therefore 134 multiplied by 2 produces 1232 23 3696 2464 28336 the whole 134; therefore the products of each of those parts, being added together, will be the product of the whole. 68. When the Multiplier confifts of more figures than one, as many figures as it has, fo many feveral products must be fet down under the line, which at Îaft being added into one fum, will give you the total product of all. So 1232 being given to be multiplied by 23, the operation will ftand thus; 1232 beingmultiplied by 3 (according to the last rule) the product is 3696. Again, 1232 being multiplied by 2, the product is 2464, which feveral products after they are placed in their due order, (that is, the first figure arifing in every product under its respective multiplying figure) and added together, produce 28336, the product required: In like manner 1321 being given to be multiplied by 123, the pro 1321 123 3963 2642 1321 162483 duc duct is 162483, and the operation will appear as in the margin. 69. The product of 1232 by 23, is equal to the product of 1232 by 20 and the product of 1232 by 3, added together. See Examp. 1. Art. 68. But 1232 multiplied by 20 produces And 1232 multiplied by 3 produces Therefore 1232 multiplied by 23 produces 24640 3696 28336 70. When the product of any of the particular figures exceeds ten, place the excess under the line, as before, and for every ten that it so exceeds, keep one in mind to be added to the next rank; as was taught in Addition. 3084 36 18504 9252 111024 Example, 3084 being given to be multiplied by 36, the work will ftand thus; 6 times 4 being 24, fet 4 under the line, and reserve 2 in mind for the two tens; then fay 6 times 8 is 48, to which add 2 kept in mind, the whole is 50; therefore fet down in the next rank under the line (o because there is no excess of 50 above 5 tens) and keep 5 in mind for the 5 tens; again, fay 6 times nothing is nothing, to which adding 5 that was kept in mind, the whole will be but 5, which fet down under the line in the next rank; again, 6 times 3 is 18, which (in regard 3 is the laft figure of the Multiplicand) fet wholly down; fo that the particular product arifing from the multiplying by the figure 6 is 18504: In like manner proceeding with the multiplying figure 3, the particular product arifing will be 9252. Laftly, thefe several products being placed in due order, and added together (after the manner of the laft Article) will give 111024, which is the total product arifing from the Multiplication of 3084 by 36, and the operation will stand as in the margin. After the fame manner, if 5073 be given to be multiplied by 256, the product will be found to be 1298688, and the operation will stand as you fee in the example. 5073 256 30438 25365 10146 1298688 71. When the two numbers given to be multiplied, do, one or both of them, end with a cypher or cyphers towards the right-hand, multiply the fignificant figures in both numbers, one by the other, neglecting fuch cyphers; and when the Multiplication of the fignificant figures is finished, annex, on the right-hand of the number produced by the Multiplication, the cypher or cyphers, with which one or both of the numbers firft given did end, fo will the whole give you the true product demanded. Example, 43100 being given to be multiplied by 15000, the product will be found 646500000; for, omitting the |