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£.: hence of a penny is equal to of t'a of z's of a £=££. The reason of the rule is therefore evident.

2. Reduce of an inch to the denomination of yards.

Here of it of į=ita yard, the answer. 3. Reduce foz. avoirdupois to the denomination of tons.

Ans. 1735 T. 4. Reduce f of a pint to the fraction of a hogshead.

Ans. Thhd.

CASE II. Ø 135. To reduce a denominate fraction from a higher to a lower denomination.

RULE. I. Consider how many units of the next lower denomination make one unit of the given denomination, and place 1 under that number forming a second fraction.

II. Then consider how many units of the denomination still lower make one unit of the second denomination and place 1 under that number forming a third fraction, and so on, to the denomination to which you would reduce.

III. Connect all the fractions together, forming a compound fraction. Then reduce the compound fraction to a simple one by Case V. Ex. 1. Reduce £ to the denomination of pence.

Here 4 of < of Y=24od. Ans. In this example of a £. is equal to 4 of 20 shil. lings. But 1 shilling is equal to 12 pence; hence of a £=4 of 7 of y=40d. Hence the reason of the rule is manifest.

2. Reduce fcwt. to the fraction of a pound.

Ans. Pelb. 3. Reduce it of a £. to the fraction of a penny.

Ans. yd. 4. Reduce 1 of a day to the fraction of a minute.

Ans. 480m. 6. Reduce ft of an acre to the fraction of a pole.

Ans. 412P.

CASE III. $136. To find the value of a fraction in integers of a less denomination.

RULE. Multiply the numerator by that number which makes one of the next lower denomination, and divide the product by the denominator. If there be a remainder, multiply it by that number which makes one of the denomination still less, and divide again by the denominator. Proceed in the same way to the lowest denomination. The several quotients being connected together, will form the equivalent denominate number

Ex. 1. What is the value off of a £.?

2

20
3)40
13s ..
...1 Remainder

12
3)12
4d

Ans. 13s 4d. 2. What is the value of fib. troy?

Ans. 9oz. 127

154

REDUCTION OF DEXOMINATE FRACTIONS.

3. What is the value of Le of a cut. ?

Ans. lqr. 72. 4. What is the value of of an acre ?

Ans. 2R. 20P.

CASE IV. 137. To reduce a denominate number to a fraction of a given denomination.

RULE.

Reduce the number to the lowest denomination mentioned in it: then if the reduction is to be made to a denomination still less, reduce as in Case II. ; but if to a higher denomination reduce as in Case I. Ex. 1. Reduce 4s 7d to the fraction of a £.

4 12 55 of zo of t= of a £. Ans.

55d by adding in the 7d. 2. Reduce 2 feet 2 inches to the fraction of a yard.

Ans. Hyd. 3. Reduce 3 gallons 2 quarts to the fraction of a hogshead.

Ans. hhd. 4. Reduce 1gr. 716. to the fraction of a hundred.

Ans to cut.

QUESTIONS.

g 132. What is a denominate number? What is a de. nominate fraction ? What is the unit of a fraction? When are fractions of different denominations ? When of the samo denomination ?

$ 133. What is reduction of denominate fractions ?

Š 134. How do you reduce a denominate fraction from a lower to a higher denomination ?

& 135. How do you reduce a denominate fraction from a higher to a lower denomination ?

$ 136. How do you find the value of a fraction in integers of a less denomination ?

$ 137. How do you reduce a denominate number to a fraction of a given denomination ?

ADDITION OF VULGAR FRACTIONS.

$ 138. Addition of integer numbers teaches how to express all the units of several numbers by a single number. Addition of fractions teaches how to express

the values of several fractions by a single fraction.

It is plain, that we cannot add fractions so long as they have different units : for, of a £. and f of a shilling make neither 1£ nor 1 shilling.

Neither can we add parts of the same unit unless they be like parts ; for } of a £. and of a £. make neither of a £. nor of a £. But of a £. and I of a £. may be added: they make of a £. So, į of a £. and of a £. make of a £.

Hence before fractions can be added, two things are necessary.

1st. That the fractions be reduced to the same denomination.

2d. That they be reduced to a common denominator,

CASE I. $ 139: When the fractions to be added are of the same denomination and have a common denominator.

RULE, Add the numerators together and place their- sum over the common denominator : then reduce the fraction to its Lowest terms, or to its equivalent mi: number.

Ex. 1. Add , , , and if togther.
Here 1.+3+6+3=13, the sum of the numerators 5
Hence y is the sum of the fractions.

Ans. Y=64 2. Add of a £., $ of a £. and of a £. together.

Ans. Y of a £.=27£.

CASE II. $ 140. When the fractions are of the same denomination but have different denominators.

RULE. Reduce compound fractions to simple ones, mixed numbers to improper fractions, and all the fractions to a common denominator. Then add them as in Case I. Ex. 1. Add f, }, } together.

6*3*5=90
4x2x5=40 Numerators.
2x3x2=12)

2x3x5=30 common denominator.
Hence, f+*+1=1$+$8+}}=2#ffŁW =
=47=411

Ans. 471. 2. Add of a £., f of a £. and f of a £ together.

Ans. £H4=£1M=£114. NOTE. $ 141. When there are mixed numbers, instead of reducing them to improper fractions we may add the whole numbers and the fractional parts sepa. rately, and then add their sums.

Ex. 3. Add 194,63 and 44 together.
19+6+4+29 the sum of the whole numbers

1=1+34 +=+1=1' the sum of
ional parts. Hence, 29+14 30.04.

Ans. 30 M.

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