GEOMETRY AND MENSURATION.

PART FIRST.

INTRODUCTION.

CHAPTER I.

FRACTIONS.

SECTION I.

Definitions.

Fractions signify parts of a thing broken, or of a number divided.

When a thing or number is divided into two equal parts, the parts are called halves. If a thing or number be divided into three equal parts, the parts are called thirds. If divided into four equal parts, the parts are called fourths. If divided into five equal parts, the parts are called fifths, &c. always receiving a name from the number of parts into which the thing or number has been divided.

When fractions are expressed in figures, two numbers are used, with a short line between them, as, one half; , one third;, two thirds;, five sevenths; 1, eleven twelfths; &c.

The number below the line, which gives the NAME to the fraction, is called the DENOMINATOR.

The number above the line, which expresses the NUMBER of parts used, is called the numerator.

In the fraction, the denominator, 6, expresses that some thing or number has been divided into six equal parts; and the numerator, 5, expresses that five of those parts are taken. The fraction, then, is five sixths.

Write the following fractions in figures. Three fourths ; seven eighths; ten twelfths; fifteen sixteenths; nineteen twenty-fifths; twenty-five thirty-fifths; thirty-nine fortyeighths; fifty-seven seventy-fifths; eighty-nine ninetieths; one hundred five one hundred twenty-fifths; two hundred thirty-five three hundred forty-fifths; five hundred fiftyfive seven hundred seventy-sevenths; eight hundred eighty nine nine hundred eighty-eighths.

SECTION II:

Proper and Improper Fractions.

If the value of a fraction be less than a unit, that is, if the numerator be less than the denominator, it is a PROPER FRACTION; as 3, §, 7, 12, &c.

If the value of a fraction be equal to, or greater than a unit, that is, if the numerator be equal to or greater than the denominator, it is an IMPROPER FRACTION; as 3, 4, 8, 12, 25, &c.

5 9

Write the following fractions in figures; and separate the proper from the improper fractions.

Two thirds; three halves; four thirds; three fifths; six fourths; eight sevenths; nine elevenths; seven sixths

five eighths; five fifths; fifteen sixteenths; seventeen eighths; five twenty-fifths; twenty-five fifths; twenty-five twenty-fifths; ninety-eight eighty-ninths; eighty-nine ninety-eighths.

Although fractions are properly parts of a unit, or one single thing, they may be used to express a part or parts of any number of things: thus we may express two thirds of ten dollars, as well as two thirds of one dollar; or three fourths of one hundred miles, as well as three fourths of one mile.

SECTION III.

Mixed Numbers.

A whole number with a fraction is called a MIXED NUMBER; as, one and five eighths, 1ğ; two and three fourths, 22.

Express the following mixed numbers in figures.

Four and two thirds; six and four fifths; five and one half; seven and four sixths; nine and three sevenths; thirteen and seven eighths; twenty and eleven twelfths; thirty-seven and fifteen sixteenths.

Any number may be written in the form of a fraction : thus, 5; 13. These numbers may be expressed with any denominator required, by multiplying the whole number by the denominator, and placing the product as a numerator over the given denominator,

Express 5 as a fraction, having 2 for its denominator.

Write 10 in the form of a fraction, having 5 for its denominator. Change 15 to a fraction, with 15 for a denominator.

ANS. 50.

ANS. 225

Change 25 to a fraction, having 25 for its denominator. ANS. 625.

SECTION IV.

Change of Fractions.

It is often necessary to change the form of a fraction or mixed number. This may be done without changing its value.

To change an improper fraction to a whole or mixed

number.

Divide the numerator by the denominator: the quotient will be a whole number.

1. Change 12 to a whole number.

12 divided by 4=3.

ANS. 3.

If there be a remainder, write it over the denominator, and annex it to the quotient.

2. Change to a mixed number.

16 divided by 5=3 and 1 fifth remaining. ANs. 31. 3. Change to a mixed number.

ANS. 5.