243. Complex fractions, after being reduced to simple ones, may be added, subtracted, multiplied, and divided, according to the respective rules for simple fractions.

4. What is the continued product of the following numbers:

Ans. 174.

1. At 2 bushels to an acre, how many bushels of wheat will be required to sow 7 acres? 2. Bought 8 bushels of apples for $4.684; what did they cost per bushel? Ans. $0.574.

3. Bought a bale of cloth for $963; I dispose of it for § of the cost, and by so doing I lose $2 on a yard; required the number of yards in the bale. Ans. 18 yd.

4. If a dividend be 183 times and a quotient 6 times 3, what was the divisor?

5. By what number must 12 be multiplied, that the product 1 be multiplied, that the product Ans. $.

shall just equal 1?

6. Bought a horse and chaise for $250, and paid for the harness of what I paid for the horse. The chaise cost What was the price of each? Ans. Horse, $13018; chaise, $11913; harness, $837

the value of the horse.

7. S. Walker has engaged to work at yearly wages of $200 and a suit of clothes. At the end of 9 months, falling sick, and being unable to labor longer, he receives the suit of clothes and $144, as the amount justly due. What was the cost of the clothes? Ans. $24.

8. What will be the result if of 3 of 3 be multiplied by of itself, and the product divided by ?

9. Bought 137 acres of land at $25 per acre, and paid for it in wheat at $23 per bushel. How many bushels did it require ? Ans. 13717 bushels. 10. How long will it take a man to travel 553 miles, provided he travels 3 miles per hour, and 97 hours per day?

11. If $% per cord is paid E. Holmes for sawing into three pieces wood that is 4 feet long, how much more should he receive per cord for sawing into pieces of the same length wood that is 8 feet long? Ans. $0.224.

12. A steamboat leaves New Orleans, January 1st, bound up the river to a place distant 2317 miles. Her forward motion is at the rate of 94 miles per hour for 162 hours each day, and she lies at anchor in the night for fear of running upon a snag. But having lost her anchor on the fifth day, she each succeeding night drifts backward, at the rate of 2 miles per hour. On what day of January will she reach her point of destination? Ans. 15th day.

A PROPOSED NUMERATOR, OR DENOMINATOR. 244. To reduce one fraction to another of equal value, having a proposed numerator, or denominator.

Ex. 1. Reduce to an equivalent fraction having 4 for a

numerator.

OPERATION.

5 X 7

48415

4

53

Ans.

The proposed numerator, 4, is such a part of the given numerator as 4 divided by 5, or . Now, as the numerator proposed is only as large as the given numerator, in order that the value of the two fractions be the same, the denominator of the proposed fraction should be only as large as the denominator of the given fraction. Taking of the given denominator, 7, we obtain

5g, which, written under the proposed numerator, gives

fraction required.

2. Reduce & to a fraction of equal value having 12 for a denominator.

RULE.

103
12

Ans.

Since the proposed denominator, 12, is 12 of the given denominator, 9, we find 12 of the given numerator, 8, for numerator of the proposed fraction; 12 of 8 103, which, written over 103 the proposed denominator, gives 12

as the fraction required.

Take of both terms of the given fraction such a fractional part as the proposed numerator, or denominator, is of the given numerator, or denominator, and the result will be the required fraction.

EXAMPLES.

3. Change to a fraction whose numerator shall be 34. Ans..

4. Change 39 to a fraction whose numerator shall be 9.

Ans.

5. Reduce 4 to a fraction whose numerator shall be 5.

9

24

6. Reduce 1 to a fraction having 12 for its denominator.

10. J. Holton owns of a wood-lot, and his brother of the same lot; what fraction whose denominator shall be 12 will express the part each owns?

Ans.

A COMMON NUMERATOR.

245.° A COMMON numerator of two or more fractions is a common multiple of their numerators.

246. To reduce fractions to a common numerator.

Ex. 1. Change 2, 4, §, and to other fractions of the same value, having a common numerator. 36 36 Ans. 28, 29, 38, 28.

OPERATION.

36, least common multiple of the numerators, = common nu

3,6 of 4 36 of 5 36 of 7 36 of 10

=

=

=

48, new denominator.

45, new denominator.

42, new denominator.

40, new denominator.

We find the least common multiple of all the numerators, which is 36, for the common numerator; and to obtain the several new denominators we take such a part of the given denominators, respectively, as the common numerator, 36, is of each given numerator. Thus, both terms of each fraction being proportionably increased, its value is not changed.

RULE. Find the least common multiple of the given numerators for a common numerator. Take, for the new denominator of each fraction, respectively, such a part of its given denominator as the common numerator is of its given numerator.

NOTE. Compound fractions, or whole and mixed numbers, must be reduced to simple fractions, and all to their lowest terms, before finding the common numerator.

EXAMPLES.

2. Reduce §, 4, 4, and to other fractions of equal value having a common numerator. Ans. 34, 3, 31, 21. 3. Change, 22, and 13 to fractions having a common nu

merator.

4. A can travel round a certain island, which is 50 miles in circumference, in 45 days, B in 63 days, and C in 63 days. If they all set out from the same point, and travel round the island the same way, in how many days will they all meet at the point from which they started, and how many times will each have gone round the island?

Ans. They will meet in 320 days; A will have gone round the island 75 times; B, 50 times; and C, 48 times.

GREATEST COMMON DIVISOR OF FRACTIONS.

247. THE greatest common divisor of two or more fractions is the greatest number that will divide each of them, and give a whole number for the quotient.

248. To find the greatest common divisor of two or more fractions.

Ex. 1. What is the greatest common divisor of, 2,

and 5.

=

OPERATION.

240.

15, 28, 51-15, 20, 16 = 12 100 240 Greatest common divisor of the numerators Least common denominator of the fractions

= 4

45

Ans. 5.

Greatest com

mon divisor required.

Having reduced the fractions to equivalent fractions having the least common denominator, we find the greatest common divisor of the numerators 12, 100, and 240 to be 4. Now, since the 12, 100, and 240 represent forty-fifths, their greatest common divisor is not 4, a whole number, but 4 forty-fifths; therefore we write the 4 over the least common denominator, 45, and have as the answer. RULE. - Reduce the fractions, if necessary, to their least common denominator. The greatest common divisor of the numerators, written over the least common denominator, will give the greatest common divisor required.

18?

EXAMPLES.

3. What is the greatest common divisor of 123, 4. What is the greatest common divisor of ≥,

3

2. What is the greatest common divisor of 4, 4, 8, and Ans. 15 92, and 81? , and ? Ans. 51, and 21? bushels of rye, He wishes to put this grain, without mixing, into the largest bags, each of which shall contain the same quantity. Required the number of bags and the quantity each will contain.

5. What is the greatest common divisor of 34, 6. A farmer has 33 bushels of corn, 67 70 bushels of wheat.

Ans. The capacity of each bag, 33 bushels; and the number of bags, 51.

7. I have three fields; the first contains 737 acres, the second 88 acres, the third 13910 acres. Required the largest-sized house-lots of the same extent into which the three fields can be divided, and also the number of lots.

Ans. Size of each lot, 7+ acres; number of lots, 41.

LEAST COMMON MULTIPLE OF FRACTIONS.

249. THE least common multiple of two or more fractions is the least number that can be divided by each of them, and give a whole number for the quotient.