thousandths, and nearly 3 ten thousandths.

Therefore, to divide one number by another, when there are decimals in both or either,

Proceed as in whole numbers, annexing as many Os to the dividend and to each remainder, as may be necessary to perform the division, and to carry the operation as far as desired. In the quotient, point off as many decimals as the number of decimals in the dividend, including the Os annexed to it, and to the remainders, exceeds the number of decimals in the divisor.

If the divisor have more decimals than the dividend, supply the deficiency by annexing Os to the dividend before dividing.

If there be not figures enough in the quotient for decimals, supply the deficiency by prefixing Os.

4. Divide .05 by .003.

5. 10 men are to receive equal shares in 27.25 dollars; what is each one's portion? Ans. 2.725 dollars. 6. A man bought 2.5 barrels of flour for 16 dollars; what price did he give a barrel ? Ans. 6.4 dollars.

7. If you give .75 of a dollar for .125 of a yard of cloth, what price do you pay a yard? Ans. 6 dollars. 8. 43 of a mile was divided into 6 equal parts; what portion of a mile was there in each part?

Ans. .07167 of a mile, nearly.

Explain how example 3, lesson 81, is performed.

How do we divide one number by another when there are decimals in both or either?

What if the divisor have more decimals than the dividend?

What if there be not figures enough in the quotient for decimals?

9. Divide .0004 by .2.

10. Divide 37 by .0403.

Ans. .002.

Ans. 918.114, about.

11. A man gave 9 cents for .75 of a pound of cheese; how much must he have given for a pound?

Ans. 12 cents 12. If 375 bushels of potatoes be obtained for 7.5 tons of hay, how many bushels should you receive for 1 ton of hay?

Ans. 50.

CONTRACTIONS IN THE USE OF DECIMALS.

LESSON 82.

When we have either of the decimal fractions in the following table to multiply or divide by, it will be much shorter and easier to employ the corresponding common fraction.

3. Multiply 3.68 by .0625.

Ans. 9.05 dollars.

Ans. 7.5

Ans. .23.

4. A man owned .5 of 432.54 acres of land; what quan

tity was that?

8. Divide .037 by .25.

Ans. 216.27 acres.

Ans. .04

Ans. 121.76.

Ans. 54.

Ans. .148.

Ans. 32 dollars.

9. .125 of a hogshead of molasses cost 4 dollars; what

did the whole hogshead cost?

Ans. 705.

11. What sum is .25 of 11.6 dollars? Ans. 2.9 dollars.

10. Multiply 5640 by .125.

12. Divide .43 by 333 &c.

Ans. 1.29.

PROMISCUOUS QUESTIONS IN FRACTIONS.

LESSON 83.

1. The head of a fish is of his whole length, his body is of his whole length, and his tail is 2 feet long; what is the length of the fish? Ans. 8 feet.

2. 4 boys found 7 large apples, which they agreed to share equally; what was each one's part in a whole number and a common fraction, and in a whole number and decimals? Ans. 12 and 1.75.

3. A man saved out of the wreck of his fortune 450 dollars, which was only of what he had possessed; what was his fortune before his loss? Ans. 9,000 dollars.

Ans. .025.

4. Subtract .875 from .9.
5. Express 323 in a whole number and decimals.

Ans. 32.667, nearly.

Ans. 18.

Ans. .

6. 12 is of what number? 7. is of what number? 8. Change 94 to an improper fraction, the denominator of which shall be 16?

Ans. 148.

9. Now change this fraction to its simplest form. Ans. 37. 10. A man had 1, §, 1, and of a dollar; how much money had he ? Ans. 2 dollars.

11. Change .15 to a common fraction, and change this fraction to its simplest form.

12. 9 is of what number?
13. 11 is 5 times what number?

LESSON 84.

Ans. 2.

Ans. 11.

Ans. 25.

1. If you have 9 dollars and spend 71 dollars, how much will you have left?

Ans. 11 dollar.

2. A gentleman made a will, giving of his estate to his wife, of it to his son, and the remainder, amounting to 2,000 dollars, to his daughter; how much was he worth? Ans. 12,000 dollars.

3. If you give of a dollar for of a bushel of wheat, what quantity can you get for 1 dollar? Ans. 2 bushels. 4. A man sold of of a ship for 3,000 dollars; what was the whole vessel worth at this rate?

5. What is .16667 of 2,472 ?

Ans. 12,000 dollars.
Ans. 412

6. A teacher stated that of his scholars learned to read and write, that of the remainder learned geography and grammar, and that the rest, amounting to 5, learned arithmetic; how many scholars did he have? Ans. 60.

7.

of a house is worth 516 dollars; what is the value of the whole house? Ans. 1,204 dollars. 8. If you give of a dollar for 1 bushel of corn, what must you give for 124 bushels ? Ans. 9 dollars. 9. .375 of a quantity of goods worth 4,000 dollars was destroyed during a fire; what sum will a man lose who owned .12 of the whole ? Ans. 180 dollars.

10. A man sold .875 of a firkin of butter for 7 dollars; what was the whole worth at this rate? Ans. 8 dollars.

11. If you give .625 of a dollar for 1 gallon of molasses, what must you give for .8 of a gallon? Ans. .5 of a dollar.

LESSON 85.

To be performed in the mind.

1. 15 ounces of honey were given to James and Henry, James receiving 3 ounces, and Henry 12 ounces; what part of Henry's share was James'? What part of James' share was Henry's?

Finding what part of 12 ounces 3 ounces are, we call finding the ratio of 3 to 12, and finding what part of 3 ounces 12 ounces are, we call finding the ratio of 12 to 3. Therefore, to find the ratio of one number to another, Find what part of the second number the first is.

2. What is the ratio of James' share of the honey to Henry's? Of Henry's share to James'?

3. What is the ratio of 1 to 2? Of 2 to 4? Of 9 to 15? Of 100 to 3? Of 8 to 32? Of 32 to 8? Of 60 to 12 ?

4. In example 1, what part or proportion of the whole 15 ounces did James get, and what proportion of the whole did Henry get?

Proportion is often used in the same sense as part, as in the preceding question. We also say that James and

Finding what part of 12 ounces 3 ounces are, we call what? Finding what part of 3 ounces 12 ounces are, we call what?

How do we find the ratio of one number to another?
In what sense is proportion often used?

Henry shared the honey in the proportion of 3 to 12, and that Henry and James shared it in the proportion of 12 to 3, meaning that James had or as much as Henry, and Henry 12 or 4 times as much as James.

5. A farmer mixed some rye and corn together in the proportion of 3 to 2; what part of the amount of rye was the amount of corn? What part of the amount of corn was the amount of rye ?

6. If the mixture had been composed of as much corn as rye, in what proportion would the corn have been to the rye? In what proportion would the rye have been to the corn?

7. A man in his will divided his estate between his son and daughter in the proportion of 7 to 5, giving the son 3,500 dollars; what sum did the daughter get?

For the Slate.

8. A man's estate was divided among his two sons in the proportion of 12 to 17; the share of the first being 1,800 dollars, what was the share of the second?

Ans. 2,550 dollars. Explanation. What part of the first one's share was that of the second?

9. Four men were paid a certain sum in the proportions of 2, 3, 5, and 7, the first receiving 8 dollars; what was given to each of the others?

Ans. the second had 12, the third 20, and the fourth 28 dollars.

Explanation. The shares of the first and second were in the proportion of 2 to 3, the shares of the second and third were in the proportion of 3 to 5, &c.

10. Four men, A, B, C, and D were weighed; A weighed 135 pounds, and B weighed 150; the weights of C and D were in the same proportion, but that of C was 180 pounds; what did D weigh? Ans. 200 pounds. 11. A is worth 4,500 dollars, and B 6,000 dollars; what part of B's property is that of A, the fraction being changed. to its simplest form? Ans. A's property is of B's.

In what proportion do we say that James and Henry shared the honey? In what proportion do we say that Henry and James shared it? What do we mean by this?