8. To multiply by 21, 31, 41, &c. to 91, in one line :--

RULE.

9

=

5 4X1 0 9

First, bring down the unit figure of the multiplicand, which will always be the unit figure of the product; then multiply every figure of the multiplicand by the tens figure of the multiplier, and to each pro duct add the figure which stands next on the left hand of that which you multiplied, and you have the product required.

EXPLANATION OF THE FOURTH EXAMPLE.

I first begin and bring down the unit figure, (6) of the multiplicand, for the unit figure of the product, and then say 5 times 6 are 30, and 9, the left hand figure, make 39; I set down 9 and carry 3; then 5 times 9 are 45, and 3 1 carry make 48, and 7, the left hand figure,

make 55'; I set down 5 and carry 5, saying 5 times 7 are 35, and 5 I carry make 40, and 3, the left hand figure, make 43; I set down 3 and carry 4; lastly, 5 times 3 are 15, and 4 1 carry make 19, which I set down, and the work is completed.

QUESTIONS FOR EXCERCISE.

1. A gentleman bestwowed 120 apples to each scholar in a neighbouring school, which consisted of 25 scholars; how many apples did ke give, in all ? Ans. 3000.

2. How many dollars must be given to 28 persons, so that each may have 76 ?

Ans. 2128.

3. A stationer sold quills to 3070 school boys, in one year: the number he sold each, by estimation, was 324; I demand how many quills he sold them. Ans. 994680.

4. A general, on besieging a garrison, divided among his army, (as a bounty for their conduct, &c.) 20 dollars a man; the number of men was 19374; what was the sum given by him? Ans. $387480. 5. A farmer employed 128 men to complete a piece of fencing; how much money did he pay them, allowing that he gave each man 17 dollars, at an average ? Ans. $2176.

6. A vintner sold a cargo of wine, just imported; the invoice bill was estimated, from the captain's return, at 117 tierces; each tierce contained 120 gallons, and each gallon 5 bottles; how many bottles did he sell, at that rate? Ans. 70200.

7. If a man earn 3 dollars per week, how much will his earnings amount to in 3 years and 7 weeks, each year being computed at 52 weeks? Ans. $489.

8. Suppose a man chop 4 cords of wood in one week, how many cords will 48 men chop in 21 weeks? Ans 4032 cords.

9. How much wheat will 36 persons consume in 2 years, if they consume 12 bushels in one week; each year being 52 weeks?

10. What is the amount of 144 multiplied by 37 ?

Ans. 1248.
Ans. 5328.

DIVISION

Shews how many times one number is contained in another, or, to find how many times one number of a given denomination, is contained in another number of the same denomination; or from any two numbers given, to find a third, which shall consist of as many units, as the one of those given numbers is comprehended in the other; and is a concise way of performing several subtractions.

It consists of 4 principal parts, viz :

1. The dividend, or number given to be divided.

2. The divisor, or number given to divide by.

3. The quotient, or answer to the question, which shews how often the divisor is contained in the dividend.

4. The remainder (which is always less than the divisor, and of the same name with the dividend) is very uncertain, as there is sometimes a remainder, and sometimes none.

Division is both simple and compound; simple, when the divisor consisteth of one figure only, and the dividend of two at least. Any thing of this kind is answered by the multiplication table; as if 54 were to be divided by 6, the answer would be 9; tor 6 is contained 9 times in 54; here 54 is the dividend, 6 the divisor, and 9 is the quotient or answer.

Compound Divison is, when the dividend consists of many places of figures, and the divisor of one or more figures. As if 365, the days in a year, were to be divided by 7, the days in a week; 365 is the dividend, 7 the divisor, 52 the quotient, and 1 the remainder.

Divisor. 7)365(52 Quotient. Divisor. 6)6390(1065 Quotient.<

Division is thought to be the most difficult lesson of arithmetick; but I shall by plain rules and familiar examples, render it easy to the meanest capacity; and as in multiplication, so in this rule, I shall eadeavour to make some new improvements; discovering that many things in this rule may be abridged, and shew its excellent use in answering many questions, which seem to require a more progressive knowledge in arithmetick.

EXAMPLES.

1. Let it be required to divide 7420 by 5, in order for which, I place my dividend (or sum to be divided into 5 equal parts or shares,) thus, 7420; then I place my divisor 5 at the left band of the dividend, with a parenthesis between them, and a straight line drawn underneath, thus:

5 ) 7 4 2 0

then I proceed according to rule; and first, I seek, saying, how many times 5, (the divisor) can I have in 7, the first figure of the dividend, and the answer is one, which I place under the line, just under the 7; then, according to the rule, 1 multiply, saying, once 5 (the divisor) is 5; then, as the rule directs, I subtract, saying, 5 from 7, and there remain 2, which are two tens, which I suppose stand before the next figure in the dividend, viz. 4, and so make it 24, for a new dividend. Then again I seek, saying, how many times 5, (the divisor,). can I have in 24, (the dividend) and the answer is 4 times, which E place under the 4, the second figure in the dividend; then I multiply, saying, 4 times 5 (the divisor) are 20, which I subtract from 24, and there remain 4, which being 4 tens, make the next figure 2, in the dividend, 42. Then again I seek how many times 5 I can have in 42, and the answer is 8 times, which I put under the line, just under 2, the third figure in the dividend; then I multiply, saying, 8 times 5 are 40, which I subtract from 42, and there remain 2, which 2 tens make, with the cypher in the dividend, 20; then saying how many times 5, (the divisor) in 20, and the answer is 4 times; then 4 times 5 are 20, from 20 and there remains nothing. And so the work is done; as appears by the following example:

So I find by the above work, that 5 is contained in 7420 just 1484 times; or, if the dividend had been so many shillings or pounds to be divided between 5 persons, each person must have had so many shillings or pounds for his share.

2. Let it be required to divide 123156 by 36.

Observe the following instructions.

Having placed your numbers as in the work, make a point under 3, the third figure of your dividend, you cannot have 36 in 12, the two first figures thereof.

Then ask how many times 36 in 123? or, which is better, how many 3's in twelve? suppose 4; but I cannot have 4 times 6 in 3, for you must take the figure of your divisor no oftner in the first figure, or first and second figures of your dividend, than you can have the 2d figure of your divisor in the remainder of the first, or first and second joined to the second or third. Suppose therefore 3 times; but can I have 3 times 6 in 23? That I can; wherefore place 3 in your quotient, and multiplying your divisor thereby, the product (100) place under 123, and to the remainder, (15) point and bring down the next figure 1, of your dividend; then how often 36 in 151, or how often 3 in 15? Ans. 4 times; place 4 in your quotient, and multiplying the divisor thereby, your product, to wit, 144, subtracted from 151, leaves 7, to which point and bring down the next figure 5; then how often 36 in 75, or, 3 in 7? Ans. 2 times; place 2 in your quotient, multiplying your divisor thereby, your product (72) brought from 75, to the remainder, (3) point and bring down your next and last figure of your dividend, to wit, 6, asking how often 36 in 36, or 3 in 3? Ans. 1 time; place 1 in your quotient, saying 1 time 36 is 36, which subtracted from 36, leaves , and your work is finished; and the quotient is found to be 3421. See the work.

Dividend.

Divisor. 36) 1 2 3 1 5 6 (3 4 2 1 Quotient.

Division may be performed without charge to the memory, by making a tablet of your divisor multiplied into the 9 digits, and may prove of good use to the learner; not only in great numbers, but by practicing a few, this way. he will attain to a good knowledge of division and be enabled to work easily without such a table.