and the remainder to Mr. L at a profit of $1250. What Ans. $5975.

was his whole gain?

9. A huckster bought 125 bushels of apples at $1 per bushel; 35 bushels at $3 per bushel; and 25 bushels at $ per bushel. At what price per bushel must he sell them to gain $30.614?

Ans. $.87. 10. A hunter in pursuit of a fox that was 3 miles ahead of him, ran 14 hours, at the rate of 5 miles an hour. The fox advanced at the rate of 3 miles per hour, for 14 hours. How far in advance of the hunter is the fox? Ans. 18 miles. 11. Henry, William, and Jacob can remove a pile of wood in 4 hours; and Henry and William together can remove it in 9 hours. In how many hours can Jacob alone remove it? Ans. 619 hours.

12. Two men 115 miles apart start toward each other, and travel, one at the rate of 21 miles per hour, and the other at the rate of 3 miles per hour. In how many hours will they meet ? Ans. 1834 hours.

13. L, M, and N bought a drove of cattle: L paid for of the drove, M, for of it, and N, for the remainder. It was found that L paid $640 more than N. What did each pay, and what was the cost of the drove ?

Ans. L, $3840; M and N, each $3200; $10240. 14. How many barrels of apples at $9 per barrel will pay for 95 cords of wood at $51 per cord? Ans. 55.

15. A gentleman purchased a farm for $9000; he paid $95 for fencing and other improvements, and then sold it for $381 less than he had expended. What did he receive for it? Ans. $8713.75. 16. If a family of 5 persons consume of a barrel of flour in one month, in how many months will they consume 9.1

barrels ?

Ans. 1313.

17. If a man earns $112 per month and spends $77 in the same time, how long will it take him to save $941 ? Ans. 263 months. 18. There are two outlets to a cistern; by one 15 gallons, and by the other 20 gallons, can be emptied in one minute. How many gallons may be emptied by both in 171⁄2 minutes?

19. The product of three numbers is 79. The first number is 7, the second is 94. What is the third? Ans. 1

185. Converse Reductions.

We have already learned that Fractions with reference to the mode of expressing them are either called "Decimals” or “Fractions.”

In the decimal form, the denomination, or fractional unit, is indicated by the position of the decimal point (119); but, in the fractional form, the denomination, or fractional unit, is expressed by the denominator (152).

We shall now proceed to learn how to change Decimals to Fractions and Fractions to Decimals under the heading Converse Reduction.

EXAMPLE 1.-Reduce .875 to a common fraction.

EXPLANATION.-.875 is read 875 thousandths. Therefore our numerator is 875, and our denominator, 1000, and the fraction, $75, which reduced to its lowest terms by RULE (178) =},

1000

EXPLANATION. We write the number without the decimal point, and express its fractional

unit by the known denominator 100, which gives us the complex frac

31

tion which reduced to a simple fraction by Rule for Division of

100'

Hence, to reduce a decimal to a fraction, we have the following

RULE. Write the given number of decimal units. Omit the decimal point, and express the fractional unit by a denominator. Reduce the resulting fraction to its simplest form.

8

EXAMPLE 3.-Reduce to a decimal.

SOLUTION.

=

1ST EXPLANATION.—Annexing three ciphers to 7, 8)7.000 reduces it to thousandths, and of 7000 thousandths

.875

2D EXPLANATION.-Since expresses the quotient of 7 divided by 8, we annex decimal ciphers to 7, and divide by 8, as in Division of Decimals.

Hence, to reduce a fraction to a decimal, we have the following

RULE.-Annex a decimal cipher or ciphers to the numerator and divide by the denominator.

PROBLEMS.

1. Reduce .625 to a fraction. 3. Reduce .75 to a fraction. 5. What fraction = .4375? 7. What fraction = .008 ? 9. Reduce.28125 to a fraction. 11. Reduce .0375 to a fraction. 13. What fraction = .56? 15. What fraction = .096 ? 17. Reduce to a decimal.

2. Reduce to a decimal. 4. Reduce to a decimal. 6. What decimal = ? 8. What decimal=? 10. Reduce to a decimal. 12. Reduce to a decimal. 14. What decimal=? 16. What decimal=? Ans. .33or.33 or .333 &c.

Sometimes the decimal is interminable, and in such cases a fraction may be written after the decimal figures, the quotient being carried to any desired number of decimal places; or the sign + may be written after the decimal figures, to show that the divisor is not exact.

Decimal figures which continually repeat, are called a Repetend.

The value of a repetend is expressed by a fraction whose numerator is the repeating figures and whose denominator is as many nines as there are figures in the repetend. Thus,

333 = .333 &c. =

999

=

= .66 &c. = §§= }; &c., &c.

That this is correct is shown from the fact that produces .1111 &c.; therefore, & will produce .222 &c.;, .33 &c.; §, .555 &c. In like manner will produce .0101 &c.; therefore, will produce .02 &c., &c., &c.

99

18. Reduce the repetend .384615 to a fraction. 19. Reduce to a decimal.

20. Reduce the repetend .238095 to a fraction. 21. Reduce to a decimal.

22. Reduce the repetend .142857 to a fraction. 23. Reduce to a decimal.

24. Reduce the repetend .428571 to a fraction. 25. Reduce to a decimal.

186. Multiplication and Division by Aliquot Parts.

In common oral transactions, resort is often had to short methods of calculations, which in many, though not by any means in all cases, facilitate calculations.

We shall learn these short processes under the heading Multiplication and Division by Aliquot Parts.

An Aliquot Part of a number is any exact divisor of it. Thus, 6 is of 12; 3 is of 12 ; &c.

The Unit of an Aliquot Part is that number which is divided to obtain the part.

The Aliquot Parts of $1 are as follows:

MULTIPLICATION.

EXAMPLE 1.-What cost 144 pencils, at $.061 a piece?

SOLUTION.-Since 6 cents = of a dollar, the cost of the pencils is as many dollars as there are pencils. of 144 is 9. Therefore,

144 pencils cost $9.

PROBLEMS.

1. What cost 32 yards of muslin, at $.124 per yard? 2. What cost 36.6 yards sheeting, at 163 cents per yard? 3. What cost 24 yards of alpaca, at 334 cents per yard? 4. What cost 28 pounds of butter, at 20 cents per pound? 5. What cost 64 wooden pails, at 25 cents per pail? 6. What cost 19 pounds of tobacco, at 50 cents a pound? 7. What cost 120 oranges, at 10 cents each?

8. What cost 24 slates, at 84 cents a piece?

9. What cost 60 gum erasers, at 5 cents a piece?

10. What cost 128 bottles of ink, at 64 cents a bottle?

In like manner we may multiply when the unit of the aliquot part is 100, 1000, &c.

EXAMPLE 2.-What cost 61 shares of mining stock, at $144 per share?

SOLUTION. —At $144 per share 100 shares cost $14400, but since 61 is of 100, 61 shares cost of $14400, or $900.

11. What cost 12 12. What cost 16

pound?

yards of alpaca, at 32 cents per yard? pounds of Java coffee, at 36.6 cents per

13. What cost 334 pounds of coffee, at 24 cents per pound? 14. What cost 20 pounds of butter, at 28 cents per pound? 15. What cost 25 wooden pails, at 64 cents apiece? 16. What cost 50 papers of tobacco, at 19 cents per paper?