11

9

(5) Add together 1, 45, 7; 2, 33, 58; 84, 135,

2

Answers: 541, 10387, 49583, 2088.

(6) Find the sum of 3, 4, 5, 3; of 1, 3, 4, ; of 20,20,,; of, 1, .

7 6

5

14

Answers: 3,

(7) Find the sum of

659

115, 116, 2.

of 9, and of 8.

Answer: 4.

(8) Required the sum of 143 and , 4 and 3 of 2: of of of, of 3

of : of of, 9,

28

2

(9) Required the value of 13+

Answer: 5.

of

4.1 4
+

3 34 510

(10) Determine the quantity in its simplest form which shall be equivalent to the sum of the magnitudes, of of 11, 12 of 35 of, 25 of 43 of 12. Answer: 438.

(11) Find the respective sums of 14, 5, 1, 3: of 35, 28, 1, 7 and of 212, 31, 4, 510.

Answers: 568, 14,30 and 15%

(12) Add together, 35, 100, 110, 2500: also, 387,

of 3704 and 31, 7, 83, 41.

Answers: 1, 2548 and 233.

(13) Add together 3, 4,

(14) Find the sum of,, 5, 6, 12, 1, 1; and

(15) Shew that the simple fraction equivalent to the

87. RULE. Reduce the fractions to a common denominator; subtract the less numerator from the greater; under the remainder place the common denominator, and the result, properly reduced, will be' the required difference.

For, taking the quantities 5 and 24, and reducing them to fractional forms, we have the difference

=

16 18 112 54 58

=

=

= 21. 3 7 21 21 21

This operation may be performed in a more convenient form as follows:

the difference-5-21-5-21-21: where 14, being greater than 1, is subtracted from + 1 or

7

21

28

and 1

21

is carried to the whole number 2, as in Integers.

(2) What is the difference of 19 and 13; of 81% and 171; of 1000 and 3845; of 279% and 1688? Answers: 6, 9675, 61533, 1103.

(3) Required the difference of 13 of 35 and 27 of 16 of of and of of 25: of 24 of 44 and

7 of 10).

Answers: 4, 128, 7.

(5) Prove that the sum of 5

times their difference.

and 3, is equal to four

III. MULTIPLICATION OF FRACTIONS.

88. RULE. Multiply together the respective numerators and denominators, reduced to fractional forms if necessary; and the fraction thence arising will be the product, which may generally be simplified by means of the preceding Articles.

For, let the fractions be and ; then if & be mul tiplied by 7, the product will be 14 by Article (74): but 7 being 8 times as great as, the multiplier above used is 3 times too large, and the product will therefore be 8 times too large also: whence, the product required must be

89. If three or more fractions be proposed, as

3 1 6

1 2 3

6

1

2

3

2' 3

and their continued product is

× the product of

1 2 3

X X
2 3 4

6 1
4

2 × 3 × 4 24

=

24

and thus the rule may be proved to be general: also, in cases like this, the reduction is shortened by cancelling from the products of the numerators and denominators, any factor or factors common to both, and effecting the multiplications of what are left; thus,

Hence, the product of two or more fractions is the value of the compound fraction of which they are the constituent fractions: and thus, Multiplication is here extended so as to express the part or parts of any quantity: 5 7 5×7 5 7

for,

6 x 8

= X in the same way as
6 8

5

5

6

of 7,

5 of and 5 of 7 may be replaced by × 7, 5 x

of
6 8

7

6

and 35 respectively.

Examples for Practice.

(1) Required the product of and ; of 3 and §; of 23 and 73; of 8 and 10151; of 63 and 1417.

(2) Find the product of 3, and: of,

49 1339

of 17 and 38: of 48, 553 and 4 of 2%,

(3) Required the product of, 2, 3 and 4: of 3 4, and 11: of, 23, 1 and 1 of 63, 95, 12 and

11/9/ 41 31

(5) Multiply together of } and } of

of of 4 and 5 of 6 of 203.

57

(6) Find the continued product of the fractions, 381, 144, 443 and 2. 329

324

2116

Answer: 1.

IV. DIVISION OF FRACTIONS.

RULE. Multiply the dividend by the divisor inverted, and the result will be the quotient, which may be reduced, when possible: or, which is the same thing, invert the divisor, and then proceed by the rule for the Multiplication of Fractions.

3 28

For, let be to be divided by ; then ÷4 = is

3 7

5 times too small, because the divisor has been taken 5 times too great: whence, the quotient will be

and the operation may be expressed in the following form:

Hence, Division is here extended to express the finding of the fraction, the product of which and the divisor is the dividend: and the quotient shews what part or parts the dividend is of the divisor.

91. To denote the division of one integer by another, as of 4 by 5, we have the quotient

or, in words, a simple fraction may be considered as an adequate expression of the implied division of the numerator by the denominator: and therefore the fraction, in addition to its meaning as explained in Article (71), implies that if 4 were divided into five equal parts, one of these parts is expressed by it.