Example 2. Required the sum of the fractional expressions 3,

In this example one of the addenda, 34, is a mixed number: it is brought to an improper fraction. This preparation made, the reduction of the given fractions to a common denominator, the addition of the numerators, and the reduction of the result to a mixed number, are all effected precisely as in the preceding example.

Example 3. Required the sum of,, 4,

4, and 2=

and 2? (Art. 157 & 144).

The fractional expressions whose sum is required are therefore,

Calculation of l. c. m. of

the denominators.

2)8. 10. 5. 16. 1

5 3 21 9

82 10 52 18 20

Reduction to the same denominator and addition of the given expressions.

80+8=10

= 50

80÷10= 8

80+5=16

80+16=5

80+1=80

The sum of the numerators
The common denominator

615

is therefore equal to 80
is the sum of the given numbers.

171. From Articles 169 and 170 it follows,

That the addition of fractions having the same denominator is made by adding together the numerators of the fractions, and placing the sum over the common denominator;

That the addition of fractions having different denominators is made by reducing them to equivalent fractions having a common denominator, adding together the numerators and placing their sum over the common denominator;

And that the addition of quantities consisting of fractions having different denominators, mixed numbers, and whole numbers is made by reducing all the quantities to the fractional form, reducing these fractional expressions to a common denominator, adding together the numerators, and placing their sum over the common denominator.

Whence the general rule for the addition of any given quantities consisting of fractions, fractions and mixed numbers, or fractions, mixed numbers, and whole numbers, is,

Prepare the given numbers for addition by reducing mixed numbers to improper fractions, whole numbers to the form of fractions, and all the fractions to a common denominator. Then, add the numerators together, and place their sum over

the common denominator. The result is the sum of the given quantities.

If this result is expressed by an improper fraction, it is reduced to a mixed number by Article 156, and the fractional part of the mixed number to the lowest terms by Article 153.

172. Exercises in the addition of fractions:

173. The sum of any numbers consisting of integers and fractions, integers and mixed numbers, mixed numbers and fractions, or of all the three forms of expression, is found by the rule of Article 171. But when the integers, or the integral parts of the mixed numbers, are expressed by several figures the calculation by the general rule is very laborious. Examples 19 and 20 may suffice to prove this assertion.

Hence it is of importance to discover a more concise process for the addition of such numbers.

The sum of an integer and a fraction can always be indicated by interposing the symbol + between the integer and fraction. Thus the sum of the integer 8 and the fraction is indicated by the expression 8+.

The whole number being composed of eight repetitions of one simple unit, and the fraction of three fifths of one simple unit, the expression 8+ is equal to eight and three fifth repetitions of one simple unit. This is written 83 (Art. 155), and, generally, the sum of any integer and fraction, which are both related to the same unit, is expressed by writing the fraction to the right of that figure which expresses the simple units of the integer.

The addition, however, of a fraction to a whole number of the same kind may be considered a matter of notation merely, and the employment of the

rule unnecessary.

174. Next, let it be required to add another fraction, for example, to the mixed number 83, without reducing the latter to an improper fraction. 83+7=8+3+};

Or, since and, reduced to the same denominator, are 34, 35, 83+7= 8+3+38

But +8=14°=1+1% ..83+18+1+13=9+13=918.

This result is obtained by, first, adding together the two fractions, and, second, adding the sum of the fractions to the integral part of the mixed number. The process is independent of the values of the addenda 83 and , or of any particular numbers; whence to find the sum of a mixed number and a fraction,

Rule. Add the fractions together, and add their sum to the integral part of the mixed number.

175. Lastly, let it be required to find the sum of two mixed numbers (for example, 27 and 3513) without reducing the mixed numbers to improper fractions. 275 27+ and 351=35+17. ..27+3513=27++35+1},

or 27+3513-27+35++1} (Art. 14).

Now 27+35=62=sum of the integral parts,

and+13=28+32=32=141, or 1+1=sum of the fractional parts; ..27+3514-62+1+11, and 62+1+11=63+11=6311.

Whence, in conclusion, 275+3511=6311.

In this case also the process of addition is independent of the particular values of the mixed numbers which are added together.

The sum of any two mixed numbers is, consequently, obtained by adding the integral parts of the given mixed numbers into one sum, the fractional parts into another, and adding together the two results.

176. In like manner it is made evident if the mixed numbers, whose sum is required, are three, four, . . . . in number, or if some of the addenda are integers, others mixed numbers, and others fractions; that in any case the integral numbers or parts may be added into one sum, the fractions or fractional parts into another, and the two results into one, which is the sum of the given numbers.

Whence, to find the sum of any given whole numbers, mixed numbers, and fractions,

Rule. Combine the integers or integral parts of the given quantities into one number, the fractions or fractional parts into another, and add together the two results.

177. Exercises in the addition of mixed numbers and fractions.

1st. Required the sum of 163, 105, and 197 ?

Calculation. 163=16+2, 10=10+2, 197=19+2,

.. 16+10+193=16+3+10++19+7 (Art. 173),

or 16+10+193=16+10+19+++

The sum of the integral parts is 16+10+19=45.

(Art. 14).

The sum of the fractional parts is found by Article 171, thus:

Least common multiple

of 4. 12. 8.

Reduction to least common denominator, and addition of the fractions.

The sum of the integral parts is 45.

The sum of the fractional parts is 2+

.. 16+10+193=45+2+4=47+24=4724

17th. 17%+35}+241%+47818+3811 ?..................... Ans. 593231.

18th. 26++ } ? ......

19th. 54+26+9+12813?..

20th. 819+2043+46137?.

400

Ans. 34. Ans. 219,36

Ans. 33244

177'. The sum of any given algebraic fractional expressions is found in the same manner as the sum of fractions which are expressed by numbers; for example, the fractional expressions are equal, the first to n repetitions

1

n n'

of the apart of unity; and the second, to n' repetitions of the

unity; their sum is, consequently, equal to n+n' repetitions of the

cated by the interposition of the symbol + between them; thus, +; but to obtain the expression of the sum as in the preceding example, it becomes necessary to reduce the fractions to the same denominator (Art. 160).

is required, the addition may be indin'

n

n n' ď ď

The given fractions reduced to the same denominators are equal,

n ď nd

a to dx ď

n'

n' d n'd ď ď dor dď

and to

or

dd

nd' +n'd

dd

178. In subtraction of fractions, two fractions or two fractional expressions are given, and it is required to find the excess of the greater over the less.

If the given fractions have the same denominator, that is the greater which has the greater numerator (Art. 166). Of the two fractions and, for example, its excess over:

The fraction is equal to 11 repetitions of
And the fraction is equal to 5 repetitions of
Therefore and

the same quantity.

is the greater. To find

of unity;

of unity :

are equal, respectively, to 11 and to 5 repetitions of

But if from 11 repetitions of a quantity 5 are taken away, 6 repetitions of that quantity are left. Whence, from 11 repetitions of the quantity, taking away 5, the remainder is 6 repetitions of or (Art. 168 a), or §; that is, 5=4=+

This result is obtained by subtracting 5, the numerator of the subtrahend, from 11, the numerator of the minuend, and placing the remainder 6 over the common denominator.

The reasoning employed in this particular case being applicable to any fractions having a common denominator, it follows that the difference of any two fractions having a common denominator is obtained by subtracting the less numerator from the greater, and writing the remainder over the common denominator.

179. If the given fractions have different denominators, they can be reduced to equivalent fractions having a common denominator by the rule of Article 168. Their difference is then found in the manner already explained.

Detail: the least common multiple of the denominators is 56.

The fractions, after reduction to the least common denominator, are

The difference of 52 and 35, the numerators, is 17.

Therefore is the difference of the fractions and §.

180. The minuend and subtrahend may be, the one a whole or mixed number, and the other a fraction, or both may be mixed numbers.

The whole or mixed numbers in such cases are reducible to fractional expressions by Articles 157 and 144, and the fractional expressions to a common denominator by Article 168. After these reductions the subtraction is made as in Article 178.