It is equally impossible to add fractions whose denominators are unlike. of a shilling added to of a shilling, makes neither of a shilling nor of a shilling. But of a shilling =2 of a shilling; and 2+ of a shilling. Fractions must therefore be reduced to a common denominator, before they can be united. (See Case 8th.) Hence we have the following rule.

RULE.-Reduce all the fractions to the same denomination, and also to a common denominator; then add their numerators and place their sum over the common denominator. If the fractions produced be improper, reduce them to a whole number or mixed quantity.

Note.-If any of the fractions are compound, they must be reduced to simple ones, before they can be reduced to a common denominator. (See Case 4th.)

and 1, and

Ex. 1. What is the sum of and? These fractions, reduced to a common denominator by Case 8th, become 2+1=24 or 11⁄2†, Ans.

63

48

2. What is the sum of 3 of 3, and of? By Case 4th, of=72, and of 5, and 1+3=10+108, (see Case 8th,) and 60+08=103 or 1708

48

3

3. What is the sum of and? Ans. or 13.

4. What is the amount of of and of? Ans. 28. 5. What is the amount of 18 and 16, and 1⁄2 of 4? Ans. 34.

Note 2d.-When whole numbers are combined in the same operation with fractions, add each separately and unite them, as in the above sum.

6. What is the amount of 21, 7. What is the amount of

7, 7, and of? Ans. 2941.
of a penny added to of a £. ?

Explanation. of a £. of a penny; and of a penny +80 of a penny-329 of a penny, and this equals 2 s. 3 d. 13 qr. Ans.

8. What is the amount of of a yard and of a nail? Ans. 3 qr. O na.

9. What is the sum of 5 of a pound added to of a shilling? Ans. 103 s. 17 s. 2 d.

6

=

83
70

70

10. What is the sum of of and 9 of 13. Ans. 3 or 148. 11. What is the sum ofofofofofof, and ? Ans. 1.

12. What is the sum of 5 of a ton added to of a cwt.? Ans. 113 cwt.

63

13. What is the sum of of a day added to 1⁄2 of an hour? Ans. 19

hours.

14. What is the sum of

of a pound, of a shilling, and

of a a penny? Ans. 3 s. 2 d. 2 qr.
15. What is the sum of
hour? Ans. 1 day, 22 hours, 15 m.

of a week, 4 of a day, and 4 of an

SUBTRACTION OF FRACTIONS.

As we can subtract a quantity only from another of the same kind, it is obvious that the same preparations are necessary to perform operations in this rule as in the preceding; therefore,

RULE.-Prepare the fraction as in addition, then subtract the less numerator from the greater, and place the remainder over the common denominator.

It will be obvious that the difference of the numerators is the difference sought.

Ex. 1. From take . Operation, —=—=},

Ans.

.168

96

2. From take 3-21=188—139-189, Ans. 3. From take. Ans. 4. In this last example it is evident, that, as the denominators are the same, the operation consists in subtracting the numerators only. The same is true of all similar examples, provided only that the fractions are of the same denomination.

4. From take Ans. 12 or §.

12'

5. From 1 take.

6. From take.

20

7. From 2 take 19.

Ans. 3

16'

Ans. 12 or 3.

Ans. 28% or 4.

of a pound

8. From of a pound take of a shilling.

=20 of a shilling: and 20-60 of a shilling. Therefore, 60 -=55 of a shilling, and 55 of a shilling 9 s. 2 d. 9. From of a league take of a mile.

1 league 3

miles; therefore, of a league of a mile; and of a mile = of the same; hence, 2-2 or 13 of a mile, which is the distance required.

10. From of a shilling take of a penny. Ans. 21 or 54

pence.

11. From of a day take of an hour. Ans. 144, or 204

hours.

12. From

Ans. 273

20

of a pound Avoirdupois take of an ounce. or 1313 ounces.

13. From take of 3 of 4. Ans. .of of 4=1, and 4-4-2 or 1.

14. From of an ell English take or 27 yards.

15. From 93 take 6. Ans. 34. 16. From 19 yards take 5 yards. 17. From 7 Ells English take 4 yards, or 33 Ells English.

of a yard. Ans. 1

Ans. 13ļ.

yards. Ans. =44

18. From of a pound sterling take of a penny. Ans. 3 of a penny, or 2 s. 1

d.

Ans. 31 or 10 feet.

19. From of a rod take of a foot.
20. From of an ounce take of a pwt.

1611 pwt.

Ans. 459 pwt. or

MULTIPLICATION OF FRACTIONS.

A fraction may be multiplied by a whole number, either by multiplying the numerator or dividing the denominator by that number. This has been fully illustrated in Section 4th, of Fractions.

by 3.

Ans. 29, or 33.

8. Multiply

A fraction is multiplied into a whole number equal to its denominator, by rejecting that denominator.

by 21. By dividing by 21, I take part be repeated 21 times, it is

9. Multiply part of 15; if, then, this evident that the value of 10. Multiply 72 by 82.

all the parts will equal 15.

Ans. 72.

Ans. 7.

by 73.

Ans. 41.

13. Multiply

2

by 43.

Ans. 21.

TO MULTIPLY FRACTIONS BY FRACTIONS.

RULE.-Multiply the numerators of the given fractions together for the required numerator, and the denominators, for the required denominator; then by Case 1st reduce the terms as far as practicable.

Note.-Mixed numbers are to be reduced to improper fractions before multiplying; or we may first multiply the integris and then the fractions, and add their products.

RULE FOR CANCELING.- -Place the numerators of the given fractions above a horizontal line, and their denominators below the same. Cancel, multiply, &c. as before.

One important advantage of the above rule will be found in the fact, that it always gives the answer in its lowest possible terms.

therefore, 11=numerator, and 2×6×9×12=1296, denominator, and 6, Ans.

11

3

6. Multiply by 18'

Statement,

Ans. 35, or 2.
Ans. 203.

7. Multiply by 2.

8. Multiply 16 by 12.

9. Multiply 134 by 93. Ans. 12472.

10. Multiply of 5 of 3 of 3 of 3 of of of by of of of 18. Ans. 384

In solving sums by canceling, like the above, the necessity of reducing compound fractions to simple ones is avoided.

11. Multiply of 19 by 3.

Statement,

-1. 19. 3
2. 6'

Ans. 19, or 44.

If any whole numbers are given as parts of the dividend, it is only necessary to write them above the line as in the last example; or, if they are given as parts of the divisor, they only require to be placed below the line; the operation then proceeds as before.

17. What will 2 barrels of sugar cost at 18 dollars per barrel? Ans. 427 dollars.

18. What will 8 pounds of tea cost at 14 dollars per pound? Ans. 1018 dollars.

19. What will 44 cords of wood cost at 34 dollars per cord? Ans. 1713 dollars.

20. What will 9 yards of cloth cost at Ans. 7 dollars.

dollar

per yard?

21. What will 34 gallons of wine cost at 1

dollars per

gallon? Ans. 46 dollars.

22. What will 12 barrels of sugar cost at 15

dollars per

barrel? Ans. 190 dollars.

23. What will 22 pounds of lard cost at

dollar per

pound? Ans. 213.

12*