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3 128. Reduce TE of a pound sterling to its proper quantity. Ans. 12 shillings. 2 129. Reduce 7 of a pound to its proper sum. Answer 5s. 8d.: #.

130. Reduce # of a crown to its proper sum. Ans. 2s. 11d.

131. Reduce { of a cwt. to its proper quantity. Answer 3gr. 14lb. 25 132. Reduce ; of a quarter of corn to its proper quantity.

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14 133. Reduce 15 of an acre to its proper quantity. An

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19 134. Reduce 20 of a week to its proper time. Answer

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ADDITION OF VULGAR FRACTIONS. 187. When simple fractions are to be added together.

RULE I. Reduce the given fractions to a common denominator, by Art. 180.

II. Add the new numerators together, under their sum place the common denominator, and reduce the fraction to its lowest terms for the answer, by Art. 170.

III. If the resulting fraction be an improper one, reduce it to its equivalent whole or mixed number, by Art. 173*.

g As whole numbers of different denominations cannot be added or subtracted, so dissimilar fractions (namely, such as are not like parts of the same

whole) cannot; so, in Example 1, # cannot be added to + until they are

- - - - - - - 2 reduced to the same denomination: this reduction being performed, + becomes ExAMPLEs,

16 , and 3. becomes 21, now these two fractions, namely 16 and 31 , being 56 56 56 56

2 < 3 1. Add 7 8. together.

OPERATIox. 2 X S = 16 t Erplanation. 3 x 7 = 21 j "* 7214 merg tors. I first reduce the given fractions to a - - common denominator; I then add the 37 their sum. new numerators 16 and 21 together,

and under their sum 37 place 56, the

8 = 56 common denom. - 7 x common denominator, which gives the

- 37 answer. Answer ---. 56 2 3 3 2. Add --, -, and -- together. 5, I., and is tog OPERATION. 2 × 4 × 8 = 64 Earplanation. ; C : – : Having reduced the fractions 3 x 3 x 8 = }} 72610 7274merators. to a common denominator, and 3 × 3 × 4 = 36J added the new numerators toge172 sum. ther as before, I place the sum 172 over the common denominator 3 × 4 × 8 = 96 common denom. 96, which being an improper 172 76 - fraction, I reduce it to a mixed Ans. = 1 – sum required. number for the answer. 96 96 q 3. Add +and+ together sun * - a - ether. on — . 5 and I tog 12

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16+ 21 37

added together, the sum is evidently To , or ; as in the example; and the same of others.

Nothing can be plainer than the grounds of this process; for since the denominator only indicates what parts the fraction consists of, therefore, when several fractions having the same denominator are proposed, the comparative value of each will be expressed by its numerator. Thus, let the fractions

+. +, and + be proposed; it is plain that + is double of +. and + triple of it, and that the sum of all three can be nothing but sevenths, that is, it will consist of as many sevenths as there are units in all the numerators

I 2 3 taken together; wherefore 7, 7–, and + added together will amount to

6 7, which is the rule.

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188. When there are whole or mired numbers to be added.

RULE I. Add the fractions together by the preceding rule.

II. Add the whole numbers together, and prefix their sum to the sum of the fractions (found by the preceding rule) for the anSWer".

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2O1 67 -
375 T35 sum of the fractions, Art. 187; also 3 + 4

+5 = 12 sum of the whole numbers. Wherefore 18;

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189. When compound or compler fractions are to be added.

Rule. Reduce the compound and complex fractions to simple ones, reduce their equivalent simple fractions to a common denominator, and proceed as before".

* This rule is evident; for if the sum of the whole numbers be prefixed to the sum of the fractions, it is plain that the result will be the sum of all the given numbers, both whole and fractional.

* Fractions cannot be added together until they are first reduced to simple

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First, I reduce the compound fraction + of 5. to 10. Secondly and

7 21 thirdly, I reduce the two complex fractions to ** and +. their respective 35

simple ones. Fourthly, I reduce these three fractions, # # and + to a o

common denominator. Lastly, I reduce the answer # to its lowest terms by dividing successively by 7, 3, and 3, and then I reduce the improper frac

. 54 tion 35 to a mixed number.

fractions of the same whole, and then to similar parts of the same whole, namely, to a common denominator; when all this is effected, the sum of the whole is evidently found by adding all the numerators of the reduced fractions together, and placing under the sum the common denominator, as in Art. 187; wherefore the rule is manifest.

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190. When the fractions to be added are of different denominations in money, weights, or measures.

Rule. Reduce the fractions to their proper quantities, by Art. 186, then add the proper quantities together by the rules of Compound Addition”.

y It is plain that the fractions treated of in this rule being dissimilar cannot be added together until they are reduced as the rule directs; when this reduction is performed, the remainder of the operation (depending on the rules of Compound Addition) is sufficiently obvious.

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