REVIEW QUESTIONS.

197. 1. Define a Fraction. A Fractional Unit. What two kinds of fractions are there? How distinguished? How is a common fraction expressed? Define the Denominator, and illustrate. The Numerator, and illustrate. The Terms of a fraction. 2. How are Common Fractions classified? Define a Proper Fraction. An Improper Fraction. A Mixed Number. fractions indicate? What is the Value of a Fraction? Principle. Name and illustrate the General Principles of Fractions, each in its order, I, II, III. The General Law.

What do

3. What is Reduction of fractions? When is a fraction reduced to higher terms? Principle. Rule. When is a fraction reduced to lower terms? When to its lowest terms? Principle. Rule.

4. How is an integer, or mixed number, reduced to an improper fraction? How is an improper fraction reduced to an integer or mixed number?

5. When have two or more fractions a common denominator? When the least common denominator? Principles 1, 2. Rules.

6. What kind of fractions only can be added or subtracted? Why? Rule for each. How may integers and mixed numbers, if small, be treated? What should always be observed as necessary to render complete the solution of an example in fractions?

7. How is a fraction multiplied by an integer? Rule. An inte ger by a fraction? Rule. A fraction by a fraction. Rule. Why use cancellation? How is a fraction multiplied by its denominator? What are fractions connected by the word of sometimes called? How are they to be treated?

8. How is a fraction divided by an integer? Rule. An integer by a fraction? Rule. A fraction by a fraction? Rule. How are integers and mixed numbers to be treated? What kind of expressions are sometimes called Complex Fractions? How are they to be treated?

198. 1. If a cubic block be divided into 10 equal parts, what part of the whole block is 1 part called? 2 parts? 3 parts? 4 parts? 5 parts?

2. If a unit be divided into 10 equal parts, what are the parts called?

3. What is the fractional unit?

4. If 1 tenth of a cubic block be divided into 10 other equal parts, what part of the whole block is 1 part? 2 parts? 3 parts? 4 parts? 6 parts?

5. What is of ? of? 3 of 16 ?

6. If a unit be divided into 100 equal parts, or each tenth into 10 equal parts, what are the parts called?

7. What is the fractional unit?

8. What part of 1 tenth is 1 hundredth? How many 1 hundredths is 1 tenth?

9. If 1 hundredth of a cubic block be divided into 10

equal parts, what part of the whole block is 1 part? 2 parts? 3 parts? 5 parts? 7 parts? 9 parts?

11. If a unit be divided into 1000 equal parts, or each 1 hundredth into 10 equal parts, what are the parts called?

12. What is the fractional unit?

13. What part of 1 hundredth is 1 thousandth? How many 1 thousandths is 1 hundredth ?

NOTATION AND NUMERATION.

199. A Decimal Fraction is one or more of the decimal divisions of a unit. Thus, 1, 0, 180, 1880, etc., are decimal fractions.

200. Decimal fractions are commonly called Decimals.

201. Decimal Fractions are unlike Common Fractions, because their denominators increase and decrease by the uniform scale of 10, the same as integers. The fractional units are, therefore, always tenths, hundredths, thousandths, etc.

202. The Decimal Sign (.), called the decimal point, is used to distinguish a decimal from an integer, and must always be placed at the left, or before the figure or figures that are to be regarded as decimals.

It also indicates the denominator, and determines the value of the decimal expression.

203. Decimal fractions may be numerically written in two ways, viz.: either as common fractions, the denominator being expressed, or in decimal notation, the denominator being omitted, but indicated from the place the numerator occupies in the scale. Thus,

fo, or .5, read 5 tenths.

204. The denominator of a decimal fraction, when expressed, is always 10, 100, 1000, etc., or 1, with as many ciphers annexed as there are figures in the given decimal. Thus, .3; .04; .005100, etc.

205. To express in decimal form a decimal fraction written in the form of a common fraction, omit the denominator, and if the numerator contain as many figures as there are ciphers in the denominator, place the point before it for the decimal required. Thus, .7; 38.36; 145.145, etc.

=

206. But if the number of figures in the numerator are not equal to the number of ciphers in the denominator, prefix ciphers to the numerator until the number of places are equal to the ciphers in the denominator, and place the point before the whole. Thus, .05; 108 = .012; 10%0.008, etc.

=

EXERCISES.

207. 1. Express in decimal form 18.

180 = .09

ANALYSIS.-The numerator is 9, the denominator 100; the decimal is expressed by a point,

naught, nine; thus, .09, and is read nine hundredths.

The teacher should exercise the class orally by rapidly dictating similar examples, thus: "The denominator is 100, the numerator 5: what will express the decimal?" The prompt response should be, "Point, naught, five, read five-hundredths" (.05).

Ques. "The denominator is 1000, the numerator 26 ?"

Ans. "Point, naught, two, six, read twenty-six thousandths" (.026).

208. 1. Express and read as a common fraction .09.

.09=180

ANALYSIS.-The numerator is 9, the denominator 100, and the fraction is T.

In like manner express and read,

2.

.6;

.02;

.28;

.45; .024; .010. 3. .009; .075; .150; .204; .040; .110.

Oral exercises similar to the following should be freely used. The teacher pronounces distinctly, "Point, naught, five." The pupil promptly responds, "The numerator is five, the denominator one hundred, and the fraction is five hundredths" (TOT).

Ques. "Point, naught, two, six?" Ans. "The numerator is twenty-six, the denominator one thousand, and the fraction is twentysix thousandths" (1880).