Multiply the divisor by the quotient figures, as before, and subtract the product of each figure in the divisor, from the dividend, as you produce it; set down the remainder, and carry as many to the pro duct of the next figure, as there were tens borrowed before; and so op, till all the dividend figures have been employed.

I

204 re mainder.

In the first example, I find the first quotient figure to be 9, then, say 9 times 6 are 54, which I take from 62, and 8 remains; next, I say 9 times 5 are 45 and 6 that I carry (for 6 tens which I borrowed) make 51, which I take from 51, and nothing remains; again, I say 9 times 7 are 63 and 5 that I carry (for 5 tens which I borrowed) make 68, which I take from 70, and 2 remains; so, the whole remainder is 208, to which I annex 6, with a dot, over it, and the sum is 2086. The second quotient figure is 2, consequently I say twice 6 are 12, which I take from 16, and 4 remains; next, I say twice 5 are 10 and 1 that I carry (for 10 which I borrowed) make 11, which I take from 18, and 7 remains; again, I say twice 7 are 14 and 1 I carry (for 10 I borrowed) make 15, which I take from 20, and 5 remains; wherefore, the second remainder is 574, to which I annex 5, with a dot over it, and the sum is 5745, in which the divisor goes 7 times; of course, the third quotient figure is 7, consequently, I say 7 times 6 are 42, which I take from 45, and 3 remains, which I set down under the 5, and proceed on in the same manner, through the whole dividend.

APPLICATION.

1. One hundred cents make a dollar; therefore, how many dol lars are in 5000 cents?

2. How many dollars are in 8400 cents?

Ans. 50 dollars.

Ans. 84 dollars.

3. Twenty shillings make one pound sterling; therefore, how many pounds sterling are in 1500 shillings?

Ans. 75.

4. Twelve pence make one shilling; therefore, how many shillings do 1152 pence make?

Ans. 96.

5. Four farthings made a penny; therefore, how many pence do 48 farthings make? Ans. 12 pence. 6. If a man go a journey of 432 miles in 12 days, how many miles did he travel each day?

Ans. 36 miles.

7. If a man travel 36 miles in a day, in how many days will he gó a journey of 432 miles?

Ans. 12

8. The expense of building a certain court-house is 5022 dollars, which is to be defrayed equally by 186 men. How much must each man pay? Ans. 27 dollars.

9. I am desirous to plant out 2812 apple trees in 19 rows. How many trees must I put in each row?

Ans. 148. 10. A man planted 2812 peach trees, and put 148 trees in each How many rows were there in his orchard?

row.

each bale?

Ans. 19.

11. How many pieces and bales are contained in 478800 yards of linen, allowing 25 yards to be in cach piece, and 56 pieces in Ans. 19152 pieces and 342 bales. 12. Several boys went out to gather chestnuts and collected 9900, which were so divided among them that each boy had 825. How many boys were there in company?

Ans. 12.

13. How many times will a wheel, which is 208 inches in circumference, turn round between Richmond and Staunton, which is 7603232 inches? Ans. 36554 times!

ARITHMETICAL PROBLEMS.

PROBLEM I.-Having the least of two numbers, and the difference between them, given, to find the greater number.

RULE.-Add them together, and their sum will be the greatest

number.

1. The least of two numbers is 127 and their difference is 198.

What is the greatest number?

Ans. 325.

2. The least of two numbers is 9709, and the difference between them is 1192. What is the greatest number? Ans. 10901.

PROBLEM II. Having the sum of two numbers, and one of them given, to find the other one.

RULE.-Subtract the given number from the given sum, and the remainder will be the other number.

1. If 325 be the sum of two numbers, and one of them 198, what is the other number?

Ans. 127. 2. If 10901 be the sum of two numbers, and one of them 9709, what is the other number? Ans. 1192. PROBLEM III. Having the greater of two numbers, and the difference between them given, to find the less.

RULE.-Subtract the difference from the greater number, and the remainder will be the less number.

1. If the greater of two numbers be 1001, and the difference between them 825, what is the less? Ans. 176. 2. If the difference between two numbers be 334, and the greater one 2000, what is the less? Ans. 1666. PROBLEM IV. Having the divisor and quotient given, to find the dividend.

RULE. Multiply them together, and the product will be the dividend.

1. The divisor of an operation in division is 12, and the quotient 144. What is the dividend?

Ans. 1728. 2. What dividend will produce 9276 for the quotient, when the divisor is 756? Ans. 7012656.

PROBLEM V. Having the product of two numbers, and one of them given, to find the other one.

RULE. Divide the product by the given number, and the quotient will be the number required.

1. If the product of two numbers, be 1728, and one of them 144, what is the other number?

Ans. 12. 2. The product arising from the multiplication of two numbers is 7012656, and one of them is 9276. Required the other number.

Ans. 756.

PROBLEM VI.-Having the dividend and quotient given, to find the divisor.

RULE.-Divide the dividend by the quotient, and the quotient thence arising will be the divisor.

1. A certain dividend is 288 and the quotient 32. divisor?

What is the

Ans. 9.

2. The quotient of an operation in division is 365 and the dividend 18980. What is the divisor?

Ans. 52.

PROBLEM VII.-Having the sum and difference of two numbers given, to find those numbers.

RULE.-1. To half the sum add half the difference, and that sum will be the greatest number.

2. From half the sum take half the difference, and the remainder will be the least number.

1. What are those two numbers whose sum is 144 and difference 32.

The sum 144÷2-72 the half sum.

The difference 32÷2-16 the half difference.

Therefore, 72+16=88 the greatest number, and 72-16-56 the least number.

Proof. 88+56=144 the given sum.

And 88-56-32 the given difference.

2. What are those two numbers whose sum is 2000 and difference 1200? Ans. 1600 and 400. PROBLEM VIII.-Having the sum of two numbers, and the difference of their squares given, to find those numbers.

RULE.-Divide the difference of their squares by the sum of the two numbers, and the quotient will be their difference. You will then have their sum and difference to find the numbers by problem the seventh.

1. What are those two numbers whose sum is 32, and the difference of their squares 256?

32)256(8 the difference of the two given numbers. Now, by the seventh problem,

To 16 half of the given sum,

Add 4 the half difference.

20 the greatest number. And from 16 the half sum, Subtract

4 the half difference.

12 the least number.

2. What are those two numbers whose sum is 2000, and the difference of their squares 240000 ? Ans. 1600 and 400.

PROBLEM IX. Having the difference of two numbers, and the difference of their squares given, to find those numbers.

RULE.-Divide the difference of their squares by the difference of the numbers, and the quotient will be their sum. You will then have their sum and difference to find the numbers by the seventh problem.

1. What are those two numbers whose difference is 20, and the difference of whose squares is 2000?

20)2000(100=the sum of the two given numbers. Now, by the seventh problem, To 50 the half sum,

Add 10 the half difference.

60 the greatest number.

50-10-40 the least number.

2. What are those two numbers whose difference is 50, and the difference of whose squares is 5000?

Ans. 75 and 25.

APPLICATION OF THE PROBLEMS.

1. I once borrowed a number of dollars from my friend, and have since paid him 275; I still owe him 125. How many did I bor

row at first.

Ans. 400 dollars. 2. A boy put 750 hazlenuts in two bags, one of which held 380. How many were in the other bag?

Ans. 370.

3. If I sell goods to the amount of 1000 dollars, and receive 825 in payment, what sum remains due to me?

Ans. 175. 4. A gentleman by his will left his whole estate to be equally divided among his 12 children, each one of whom received 1275 dollars. What was the whole amount of the estate? Ans. 15300 dollars.

5. The sum of 4600 dollars is to be distributed among a regiment of soldiers, so that each one may have 25 dollars. Please to inform me how many soldiers there were in the said regiment? Ans. 184.

6. A gentleman who happened to die intestate, left a tract of land containing 520 acres, which was so divided among his children that each one inherited 65 acres. How many heirs were there in his family? Ans. 8.

7. Two men, namely, A and B, together deposited 1000 dollars in the Bank of Virginia, but A put in 200 dollars more than B. Please to inform me the amount of each man's deposite?

Ans. A deposited 600 dollars and B 400.-By problem 7. 8. Two boys, namely, Alexander and Benjamin, had 14 marbles apiece when they commenced playing, but, after several games, Benjamin refused playing any longer, because he had lost some of his marbles, at which time it was found that the difference of the squares of the numbers, which each of them then had, was 336. How many marbles had Benjamin left, and what number did he lose? Ans. He had 8 left, and consequently he lost 6.-By problem 8. 9. Said Henry to his friend Charles, my father gave me 12 apples more than he gave to my brother James, and the difference of the squares of our separate parcels was 288. Now, if you are arithmetician enough to tell how many he gave to each of us, you shall have half of mine.

Ans. He gave Henry 18 and James 6.-By problem 9.

DECIMAL FRACTIONS.

A decimal fraction is a part, or parts of a unit, denoted by a dot placed before it-thus, .5 .25 and .125 are decimal fractions. The first figure after the dot is so many tenths of a unit, the second is so many hundredths, the third, so many thousandths, &c .They are commonly read .5 tenths, .25 hundredths, .125 thousandths, &e. But, more convenient and equally accurate-thus, .5 decimals, .25