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SUBTRACTION of Troy-WEIGHT.,

Ib. oz. p.w. gr. oz. p.w. gr. Bought 352 · 10 · 13 · 19

19205

13 19 Soli

019
II 16

18 118 16

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• 20

Rejt

332 · 10 · 10 · 71

86 . 16

23

Proof 352 · 10 · 13 · 19 | 205 ..13 · 19

SUBTRACTION of AvoiRDUPOISE-WEIGHT.

C.

9.
lb. I 16. oz.

dr.
Bought 256 2325 · 13 · 12
Sold
079. 3

26 00 14 · 13

2

.

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Proof 256 . 2 . 231 25 · 13 · 12 SUBTRACTION of SUPERFICIAL MEASURES of LAND.

Acres, Roods, Per. A. R. P.
Bought 780 2. 35 2040 • I. 20
Sold

090. 3.36 919. 3. 30

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Reft

689 · 2 · 39

II20.

I. 30

I

20

Proof 780. 2 : 35 2040 . 59. Questions to exercise Addition and Subtraction.

Queft. 1. Two persons A and B are of several 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? Answ. 51.

Quest. 2. What number is that which being added to 168, makes the sum to be 205? Answ. 37.

Queft. 3. The sum of two numbers is 517, the lesser is 40, what is the greater ? Answ. 477..

Queft. 4. A certain person born in the year of our Lord 1616, defired to know his age in the year 1676, what was his age ? Answ. 60.

Queft. 5. The greater of two numbers is 130, their differences is 49, what is the leffer number? Answ. 81.

Queft. 6. What number of pounds, shillings, and pence, added to 34. l. 16 s. 9d. will make 1001? Answ. 65l. 35. 3d.

Queft. 7. How many years since the Spanish Invasion, it be. ing in the year 1988, and the present year being 1750? Answ.

4

Queft.

162 years.

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Quest. 8. From 100 pounds borrow'd, take 72 paid;

'Twasa Virgin that lent it, what's due to the Maid? Answ. 281.

Queft. 9. A Miser hath three bags of money, containing in all 29841: 6 s. of which the firft contains 324. 10 s. and the fecond 9131. os. 6 d. what doth the third contain ? Answ. 1746 l. 155. 6 d.

Quest. 10. A Merchant had 5 debtors, 4 B, C, D, and E, which, together, owe him 11561. now B, C, D, and E, toa gether, owe him 7371. what is A's debt? Answ. 4191.

Quest. 11. The three Towns of London, Huntingdon, and York, lie in the same road ; the distance between the farthest of these 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? Answ. 135 miles.

CH A P. V.
MULTIPLICATION of Whole Numbers.

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60. CULTIPLICATION teaches how. by two Numbers

given to find a third, which shall contrin either of the Numbers given so many times, as the other contains I or unity: Or Multiplication may be considered 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 arising 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 50 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, lo 15 the Product contains 5 (the other given number)

thrice ; likewise 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.

5
5
5

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15 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 likewise single Multiplication : Now the several Varieties of single Multiplication are well expressed in the fallowing Table, usually called Pythagoras's Table. The truth of which may be proved by Addition, as above.

The TABLE of MULTIPLICATION,

11_2_3_4_51_6_2819
2 4 6 8 10 12 14 1618
3] 6 9 12 15 18 21 24
41 8 12 16 20 24 28 32 36
5 10 15 20 25 30 35 40 45
6 12 18 24 30 35 42 4854
7114212835424956163
8 16 24 32 40 48 5664172

9|18|27|36|45|5416317281 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 same 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 first column towards the left-hand; then carrying my eye from 5 in a right line equidistant to the upper side 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 single Multiplication, without the least pause

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or stay) 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 most, but as

many places of figures as are in both Factors; and, at least;

but one place fewer. Examp. 9+9=81, and iti=I. 65. Compound Multiplication, is when the Multiplier and Multiplicand, either one or both, consist of more figures than

one.

200

60

: 66. In Compound Multiplication, when the numbers given
end with fignificant figures, place them as in Addition and Subs
traction. So 134 being given to be multiplied by 2,
place them thus : . Then proceeding to the Multiplica-

134
tion, say 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 likewise set under the line in the next 268
rank: Lastly, 2 times I is 2, which being likewise set
down under the line in the next rank, the product is discovered
to be 268, and the work will stand as in the margin.
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

8 Therefore 134 multiplied by 2 produces

268 For since the parts 100, 30, and 4, added together, make 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 consists of more figures 1232 than one, as many figures as it has, so many several

23 products muft be set down under the line, which at

3696 fast being added into one fum, will give you the total

2464 product of all. So 1232 being given to be multiplied by 23, the operation will stand thus ; 1232 being

28336 multiplied by 3 (according to the last rule) the product is 3696. Again, 1232 being multiplied by 2,

1321

123 the product is 2464, which several products after they are placed in their due order, (that is, the first 3963 figure arifing in every product under its respective 2642 multiplying figure) and added together, produce 1321 28336, the product required : In like manner 1321 162483 being given to be multiplied by 123, the pro

dud

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

24640 And 1232 multiplied by 3 produces

3696 Therefore 1232 multiplied by 23 produces

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.

Example, 3084 being given to be multiplied by 3084 36, the work will stand thus ; 6 times 4. being 24,

36 fet 4 under the line, and reserve 2 in mind for the two tens; then fay 6 times 8 is 48, to which add 18504 2 kept in mind, the whole is 50; therefore set down

9252 o in the next rank under the line (o because there is no excess of 50 above 5 tens) and keep 5 in mind I11024 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 set down under the line in the next rank; again, 6 times 3 is 18, which (in regard 3 is the last figure of the Multiplicand) set 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 arising will be 9252. Lastly, these several products being placed in due order, and added to

5073 gether (after the manner of the last Article will give 111024, which is the total product arising from

30438 the Multiplication of 3084 by 36, and the opera

25365 tion will stand as in the margin. After the same 10146 manner, if 5073 be given to be multiplied by 256, the product will be found to be 1298688, and

1298688 the operation will stand as you see in the example.

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 significant figures in both numbers, one by the other, neglecting such 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 first 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

256

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