PROBLEM V.-To reduce an improper fraction to a whole or mixed number.

105. RULE.-Divide the numerator by the denominator; the quotient will be the whole number required; and if there be any remainder, write it over the given denominator for the fractional part of the required result.*

Ex, 1. Reduce and go to whole or mixed numbers,

56
7)56

8, Ans.

200

9)200

22, Ans.

Exercises.-Reduce the following fractions to whole or

PROBLEM VI.-To reduce a whole or mixed number to

an improper fraction.

106. RULE.-Multiply the whole number by the denominator of the fractional part, and to the product add the numerator; the sum will be the required numerator; below which write the given denominator.

A whole number may be expressed in a fractional form by writing a unit below it; or by multiplying it by any whole number, and writing that number below the product as denominator. Thus 9== 36, &c.

*The reason of this rule and of that for the next problem, is evident from the nature of fractions.

Ex. 9. Reduce 51 to an improper fraction.

PROBLEM VII.-To express any given quantity as a fraction of another given quantity of the same kind, considered as an integer.

107. RULE.-Make the integer the denominator of the required fraction, and the other given quantity its numerator, both being reduced to the same denomination, if they be not of the same already.

Ex. 10. Reduce 13s. 9d. to the fraction of £1.

In 13s. 9d. there are 165 pence, and in £1, 240 pence; the fraction therefore is £13, which, by reduction to its lowest terms, becomes £. The reason of this is evident, since 1 penny is of a pound, and in 13s. 9d, there are 165 pence, or of a pound.

Ex. 11. Express a pound troy weight as a fraction of a pound avoirdupois.

A pound avoirdupois weight contains 7000 grains, and a pound troy weight 5760 grains. (see page 54.) Hence the fraction is $368, or in its lowest terms 14.*

Exercise 1. Reduce 12s. 6d. to the fraction of £1.

Ex. 2. Reduce £32 10s. to the fraction of £100.

Ex. 3. Reduce 1qr. 20lb. to the fraction of 1cwt.

Ans. §.

Ans. 18.

Ans. .

*This rule enables us to find the ratio of two quantities. Thus, the ratio of 13s. 9d. to 17. is that of 1 to 1, or 11 to 16; and a pound avoirdupois is to a pound troy, as 1 to 14, or as 175 to 144.

Ex. 4. Reduce 2 hours to the fraction of 23h. 56m. 4s.

Ans.

Ex. 5. Reduce 2s. and 1d. to the fraction of 6s. 8d.
Ans..

Ex. 6. The height of Ben Nevis, the highest mountain in Britain, is 4350 feet, and that of Mount Ararat 9500 feet: express the former as a fraction of the latter in its simplest form. Ans. fo Ex. 7. Express 96 pages as a fraction of a book containing 432 pages. Ans. . Ex. 8. If one farm contains 27 acres, 2 roods, 14 poles, and another 160 acres, 28 poles; what is the simplest form of the fraction expressing their comparative magnitudes ?

434

Aos. 97 Ex. 9. The height of the highest peak of the Himmaleh mountains is supposed to be 27600 feet, and that of Chimborazo, the highest peak of the Andes, in South America, 21440 feet above the level of the sea: express the latter as a fraction of the former in its simplest form.

Ans. 268

345

PROBLEM VIII.-To find the value of a fraction in the denominations contained in the integer.

108. RULE. Consider the numerator as expressing the integer, taken as often as there are units in the nu merator, and divide it by the denominator.*

The rea

£

s. d.

74

00

0 11 54, Ans.

Example 12. Required the value of £4. Here the integer is £1; and the numerator being regarded as 4 times. that integer, or £4, is divided by the denominator: the quotient 11s. 54d. is the value of the fraction. son of the rule is manifest from the nature of fractions. Ex. 13. The height of the Antesana hamlet, near Quito, South America, the highest inhabited spot on the surface of the globe, is about 13400 feet: if a person have ascended through of this space, to what height has he ascended?

This problem is, in strictness, a particular case of the next. On account of its frequent use, however, it is better in a separate form.

N

Here, the height of the hamlet is the integer; and the numerator expressing three times the quantity, we multiply by 3 and divide by 8, and we find for the answer 5025 feet.

13400 feet. 3

8) 40200

Ans.

5025 feet:

Exercise 1. Required the value of £5. Ans. 11s. ląd. Ex. 2. Required the value of of a foot.

13

Ans. 5 in. 6 lines. Ex. 3. Required the of 756 dollars, 25 cents.

Ans. $453.75.

Ex. 4. Required the value of & of 175 tons.

Ans. 145 tons, 16cwt. 2qr. 184lb. Ex. 5. The Earth revoles on its axis in 23 hours, 56 mizutes, 4 seconds; in what times does it perform & of a rotation ? Ans. 10 hours, 38 min. 151 sec. Ex. 6. By the articles of the union of Great Britain and Ireland, which took place in 1801, during the first 20 years, Britain was to contribute †, and Ireland, of the amount of the public expenditure. How much did each country contribute in making a million sterling?

Ans. £882352 18s. 91 d. and £117647 18. 2d.

PROBLEM IX. To reduce a given fraction to another of a lower denomination.

109. RULE.-As in common reduction, reduce the given numerator, considered as an integer, to the denomination to which the fraction is to be reduced, and write below it the given denominator.

Example 14. Reduce T dollar to the fraction of a

cent.

Here by reducing 5 dollars to cents, and writing 1600 below it, we have for the required fraction cent, which, by reduction to its lowest terms, becomes cent.*

Exercise 1. Reduce £ to the fraction of a shilling. Ans. + shilling.

*The reason of this is manifest, since.by the nature of fractions, the given fraction expresses a sixteen hundredth part of 5 dollars, and must therefore express a sixteen hundredth part of the cents in 5 dollars.

Ans. $

Ex. 4. What part of a dollar is & cent? The converse of this problem, or the reduction of a fraction to a higher denomination is seldom of use. It is performed by operating on the denominator in the way mentioned in the rule. Thus, if it be required to reduce pence to the fraction of a pound, let the denominator be multipled by 20 and 12, and the numerator be retained the result £, or, in its simplest form, £100 is the result required.*

Addition of Fractions.

110. RULE.-Reduce the given fractions to others having a common denominator, if they be not such already. When they are in this state add all the numerators together, and below their sum write the common denominator: the result will be the sum of the fractions; which, if it be an improper fraction, must be reduced to a whole or mixed number.†.

If some of the given quantities be mixed numbers, the fractions are to be added by the preceding part of the rule;

* The method of reducing compound to simple fractions will be given in Multiplication of Fractions, and that of reducing complex fractions to simple ones, in Division of Fractions. These are the natural and proper positions of these problems; but should any teacher conceive their introduction necessary here, with a view to prepare fractional quantities for Addition, Subtraction, &c. it will be easy for the learner to turn over for them to Multiplication and Division.

When the fractions whose sum we want to find have the same denominator, the method of performing the operation is as obvious as the addition of whole numbers. For it is as plain that the sum of two ninths and five ninths is seven ninths, (that is, 5-7,) as that the sum of 2 dollars and five dollars is equal to seven dollars. Ninths in the former case, and dollars in the latter, are but the denomination, of the numbers which we add: and in place of the fractional notation, the columns in which the numbers 2 and 5 stand might be headed with the denomination ninths, as it is commonly with the denomination dollars. And if the fractions which we are required to add have different denominators, they can be brought by reduction of fractions, to equivalent fractions of the same denominator; and the same reasoning will apply as before.