CHAPTER V.

ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION OF

FRACTIONS.

ADDITION OF FRACTIONS.

§ 109. ADDITION OF FRACTIONS consists in finding the sum of two or more fractions.

The sum of two or more fractions is found by means of a common denominator.

Thus the sum of 4 and 3, is §; just as the sum of 2 cents and 3 cents is 5 cents.

How many units or whole ones are there in the sum of 3 and 2? In the sum of ,, and 3? In the sum of 5, and 2? In the sum ɔf 3, 5, and 7? In the sum of,, and?

§ 110. Two or more fractions may often be reduced, mentally, to a common denominator, (§ 99), and then added together.

Thus to add together and, we say is equal to and is equal to ; then+2=1=1/2

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What is the sum of and? Of and ? Of 3 and 5? Of, , and ? Of, %, and 1? Of 3, 4, and

RULE XX.

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§ 111. To add two or more fractions together.

1. If the fractions have not a common denominator, reduce them to a common denominator.

2. Add all the numerators together, and place the sum, as a numerator, over their common denominator.

3. Mixed numbers may be added under the form of improper fractions; or, the fractions contained in them may be added, separately, and their sum added to that of the integers.

EXAMPLES.

1. To add together, 3, 4, and §.

Reducing these fractions to a common denominator,

Adding the new numerators together, we have

168160+ 175 = 503.

Placing the sum 503 over the common denominator 280, which is equal to 13.

we have ++}

=

593
2809

2. To add together 56, 153, 84, and 7,5.

223 280

we have, 188
129 129

105
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Then 158 +7,5=188 +1 +12=3= 313 ;

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and the integer 56 added to 311⁄2 makes the entire sum 87%. Otherwise. Adding together the fractions contained in the given mixed numbers, we shall find +3+√51⁄2=112 = 1}. Adding together all the integers, 56 +15+8+7=86; then the entire sum is 86+1=87%, as before.

Note. In all subsequent exercises, improper fractions in the answers are to be reduced to integers or mixed numbers; and proper fractions, to their lowest terms.

6. Find the sum of 31, 5, 10, and 35. 74, 9, 25§, and 19,1⁄2.

7. Find the sum of 8. Find the sum of 9. Find the sum of 10. Find the sum of

11. What sum should be paid for a vest at 43 dollars, and a hat at 5 dollars? Ans. 10 dollars. 12. What sum should be paid for a cord of wood at 31⁄2 dollars, a barrel of flour at 5 dollars, and a shote at 21 dollars?

Ans. 11 dollars. dollars, a ton of hay 192 dollars. What did Ans. 48 dollars.

13. Bought a quantity of corn for 15 for 13 dollars, and a lot of pork for the whole amount to?

14. Sold wheat for 275 dollars, oats for 373 dollars, and rye for 27 dollars. What did the whole amount to?

Ans. 33938 dollars.

15. A merchant's bill was as follows: for calico of a dollar, linen 3 dollars, silk 134 dollars, and for groceries 21 dollars. What was the amount of the bill? Ans. 39 dollars.

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16. A farmer paid three laborers for a month's work as follows: to the first, 15 bushels of corn; to the second, 194 bushels; to the third, 237 bushels. How much corn did he pay them all? Ans. 58 bushels. of cloth. yards, the other two

17. A manufacturer sold four pieces piece contained 393 yards, the second 41 each 934 yards. How many yards did he

sell?

The first

Ans. 267 yards.

18. A person on a journey traveled the first day 31 miles; the second and third, each 293 miles; the fourth and fifth, each 27 miles. How far did he go in the five days?

Ans. 145 miles. 19. A stage coach ran for two hours at the rate of 8,7 miles per hour, and for two hours more at the rate of 7 miles per hour; how far was that in the whole time? Ans. 312 miles.

20. Sold to A, 25 barrels of apples, for 564 dollars; to B, 30 barrels, for 75 dollars; and to C, 10 barrels for 21 dollars. Required the quantity sold, and the sum received.

Ans. 66 barrels: 153 dollars. 21. Bought of a grocer a sack of coffee, for 13 dollars; a barrel of sugar, for 18,4 dollars; and a keg of rice, for 5 dollars. What sum should be paid for the whole?

Ans. 37% dollars. 22. Laid out for goods, at one time, of a dollar; at another time, 3 dollars; at another, 21 dollars; and at another, 93 dollars. What was the whole sum disbursed?

Ans. 353 dollars.

23. A merchant sold to one person, 4 yards of cloth for 24 dollars; to another, 93 yards for 43, dollars; and to another, 13 yards for 40 dollars. Required the quantity of cloth sold, and the sum received. Ans. 26 yards: 107 dollars.

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24. Bought in market a pound of butter for 18 cents, a

dozen eggs for 12 cents, a quarter of veal for 56 cents, and a quart of peas for 64 cents. What did the whole amount to? Ans. 93 cents. 25. Bought of a merchant a bunch of tape for 64 cents, a yard of cotton for 15 cents, a paper of pins for 314 cents, a yard of cambric for 25 cents, and a pair of gloves for 43 cents. What was the amount of the bill? Ans. 1214 cents.

26. Bought of a farmer a quarter of beef for 8 dollars, a cord of wood for 2 dollars, a ton of hay for 13 dollars, a quantity of corn for 184 dollars, and a lot of bacon for 15% dollars. What did the whole amount to? Ans. 58 dollars.

27. On a journey I traveled the first day 41 miles, the second 40 miles, the third and fourth each 45 miles, the fifth and sixth each 39 miles. What distance did I accomplish in the six days? Ans. 250 miles.

28. Going out to collect money, I received from A 37 dollars, from B 20 dollars more than from A, from C 52 dollars more than from B, and from D as much as from the other three together. What was the whole sum collected?

Ans. 316 dollars.

29. A farmer bought at one time 974 acres of land, for 1000 dollars; at another, 127 acres, for 1375 dollars; at another, 500 acres, for 6831 dollars; and at another, 333 acres, for 4013 dollars. What was the whole quantity of land that he purchased, and the sum that he paid for it?

Ans. 1058% acres; 13219 dollars.

30. A contributed toward a charitable purpose, 23 dollars, B contributed twice as much as A, C as much as B, D as much as A and B together, and E as much as all the rest. What was the whole contribution? Ans. 376 dollars.

31. An agriculturist sold to A, 135 bushels of corn, for 65 dollars; and 20 bushels of oats, for 5 dollars. He also sold to B, 17 bushels of oats, for 4 dollars; and 79 bushels of corn for 39 dollars. What quantity of each did he sell, and what sum did he receive for the whole ?

Ans. 214 bushels of corn, and 37% of oats: sum received 114 dollars.

SUBTRACTION OF FRACTIONS.

§ 112. SUBTRACTION OF FRACTIONS consists in finding the difference between two fractions; that is, the remainder when the less is taken from the greater.

The difference between two fractions is found by means of a common denominator.

Thus the difference between and is 3; just as the difference between 4 dollars and 7 dollars is 3 dollars.

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What is the difference between 3 and? Between 5 and 11? Between and? Between 1 and? Between 3 and 13?

§ 113. Two fractions may often be reduced, mentally, to a common denominator, (§ 99), and then subtracted, the one from the other.

Thus to subtract from 4, we say is equal to 1, and is equal to ; then 3-•

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What is the difference between and 2? Between

and ? Between and 7? Between and ? Between and? Between and? Between and I? Between 9 and 9?

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§ 114. A proper fraction may be subtracted from an integer, by first subtracting the fraction from a unit, and then subtracting a unit from the integer.

Thus to subtract from 7, we say from 1 or leaves, and 1 from 7 leaves 6; then 7-3=63.

Subtract

from 13. from 17. from 20. from 18. from 20.

RULE XXI.

$ 115. To subtract one fraction from another.

1. If the fractions have not a common denominator, reduce them to a common denominator.

2. Subtract the less numerator from the greater, and place the remainder, as a numerator, over the common denominator.

3. Mixed numbers may be used in subtraction under the form of improper fractions: or, the fractions in them may be taken first in subtracting, and then the integers.