£. hence of a penny is equal to of1⁄2 of of a ££. The reason of the rule is therefore evident.

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

Hereofof}=} yard, the answer.

3. Reduce oz. avoirdupois to the denomination of tons.

Ans. 173,360 T.

4. Reduce of a pint to the fraction of a hogshead.

Ans. Thhd.

CASE II.

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. Hereof of y=24°d. Ans.

In this example of a £. is equal to 4 of 20 shillings. But 1 shilling is equal to 12 pence; hence + of a £=4 of 4o of 24°d. Hence the reason le is manifest.

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

3. Reduce

4. Reduce

of a £. to the fraction of a penny. Ans. d.

of a day to the fraction of a minute. Ans. 480m.

5. Reduce of an acre to the fraction of a pole. Ans. P.

CASE III.

136. To find the value of a fraction in integers of a less denomination. fa

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 denomi nate number.

Ex. 1. What is the value of 3 of a £.?

2

20.

3)40

13s... 1 Remainder

12

3)12

4d

Ans. 13s 4d.

2. What is the value of lb. troy?

Ans. 9oz. 12p

154

REDUCTION OF DENOMINATE FRACTIONS.

3. What is the value of of a cwt.?

Ans. 1gr. 7lb.

4. What is the value of § of an acre?

CASE IV.

Ans. 2R. 20P.

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 of 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. Thhd.

4. Reduce 1gr. 776. to the fraction of a hundred. Ans cut.

QUESTIONS.

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 same denomination?

133 What is reduction of denominate fractions?

How do you reduce a denominate fraction from a her 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 † 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 neither of a £. nor of a £.

of a £. make But of a £. and of a £. may be added: they make of a £. of a £. and of a £. make of a £.

So, I

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

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

tor.

2d. That they be reduced to a common denomina

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 frac tion to its lowest terms, or to its equivalent mixed number.

Ez. 1. Add †,†, and togther.

Here 1+3+6+3=13, the sum of the numerators › Hence is the sum of the fractions.

Ans. 61.

2. Add of a £., § of a £. and 4 of a £. together. Ans. of a £.=2}£.

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,, together.

6×3×5=90

4×2×5=40 Numerators.
2×3×2=12

2×3×5-30 common denominator. Hence,+1+1=f8+j8+jz=22+8=Y7

2. Add of a £., f of a £. and § of a £ together. Ans. £14 £1}=£1.

NOTE. 141. When there are mixed numbers, instead of reducing them to improper fractions we may add the whole numbers and the fractional parts separately, and then add their sums.

Ex, 3. Add 194, 63 and 44 together.

19+6+4=29 the sum of the whole numbers ++3+4=8+FzZ + Zy=fff=ly the sum of tional parts. Hence, 29+130.

Ans. 30.