8. State the five problems in reduction of fractions, and illustrate each by the use of lines or objects. 9. Show that multiplying the denominator of a fraction by any number divides the fraction by that number (269). 10. Show by the use of lines or objects the truth of the following: (1.) of 2 equals of 1. (3.) (2.) of 1 equals of 3. (4.) of 5 equals of 1. of 14 silver dollars, how 11. To give to another person many of the dollar-pieces must you change, and what is the largest denomination of change you can use? 12. Show by the use of objects that the quotient of 1 divided by a fraction is the given fraction inverted. 13. Why is it impossible to perform the operation in † + §, or in, without reducing the fractions to a common denominator? 14. Why do we invert the divisor when dividing by a fraction? Illustrate your answer by an example. 5 15. What objection to calling a fraction (226)? 16. State, and illustrate with lines or objects, each of the three classes of so-called Complex Fractions. 17. Which is the greater fraction, & or 27, and how much? 18. To compare the value of two or more fractions, what must be done with them, and why? 31 and 4; 24 7 3 5% 23 and show in each case which is the greater fraction, and how much? 20. State the rule for working each of the following examples: (1.) 3 + 4 + 88. (2.) (7+51)—(8—21). (3.) 5 x or of 27. (4.) 8 × 5. (5.). Explain by objects. (6.) 3. Explain by objects. 21. Illustrate by an example the application of Cancellation in multiplication and division of fractions. DECIMAL FRACTIONS DEFINITIONS. 309. A unit is separated into decimal parts when it is divided into tenths; thus, 310. A Decimal Fractional Unit is one of the decimal parts of anything. 311. By making a whole or unit into decimal parts, and ́ one of these parts into decimal parts, and so on, we obtain a series of distinct orders of decimal fractional units, each of the preceding, having as denominators, respectively, 10, 100, 1000, and so on. = Thus, separating a whole into decimal parts, we have, according to (246), 1= 18; making into decimal parts, we have, according to (252), = 1; in the same manner, 1080, 1000 = 100%, and so on. Hence, in the series of fractional units, o, Too, doo, and so on, each unit is one-tenth of the preceding unit. 312. A Decimal Fraction is a fraction whose denominator is 10, 100, 1000, etc., or 1 with any number of ciphers annexed. Thus, f‰, 130, 13, are decimal fractions. 43 313. The Decimal Sign (.), called the decimal point, is used to express a decimal fraction without writing the denominator, and to distinguish it from an integer. NOTATION AND NUMERATION. 314. PROP. I.-A decimal fraction is expressed without writing the denominator by using the decimal point, and placing the numerator at the right of the period. Thus, is expressed .7; is expressed .35. Observe that the number of figures at the right of the period is always the same as the number of ciphers in the denominator; hence, the denominator is indicated, although not written. Thus, in .54 there are two figures at the right of the period; hence we know that the denominator contains two ciphers and that .54. Express the following decimal fractions without writing the denominators: 315. A decimal fraction expressed without writing the denominator is called simply a Decimal. Thus, we speak of .79 as the decimal seventy-nine, yet we mean the decimal fraction seventy-nine hundredths. 316. PROP. II.-Ciphers at the left of significant figures do not increase or diminish the number expressed by these figures. Thus, 0034 is thirty-four, the same as if written 34 without the two ciphers. From this it will be seen that the number of figures in the numerator of decimal fractions can, without changing the fraction, be made equal to the number of ciphers in the denominator by writing ciphers at its left; thus, 1. 007 1000 Hence, 1 is expressed by using the decimal point and two ciphers thus, .007. Observe, that while the number of parts in the numerator is not changed by prefixing the two ciphers, yet the 7 is moved to the third place on the right of the decimal point. Hence, according to (314), the denominator is indicated. 317. PROP. III.-When the fraction in a mixed number is expressed decimally, it is written after the integer, with the decimal point between them. Thus, 57 and .09 are written 57.09; 8 and .0034 are written 8.0034. Express as one number each of the following: From these illustrations we obtain the following rule for writing decimals: 318. RULE.-Write the numerator of the given decimal fraction. Make the number of figures written equal to the number of ciphers in the denominator by prefixing ciphers. Place at the left the decimal point. EXERCISE FOR PRACTICE. 319. Express the following decimal fractions without writing the denominator: 1. 180. 19. Three hundred seven hundred-thousandths. 20. Nine thousand thirty-four millionths. 21. Seventy-five ten-millionths. 22. Eight thousand sixty-three hundred-millionths. Express the following by writing the denominator; thus, 320. PROP. IV.-Every figure in the numerator of a decimal fraction represents a distinct order of decimal units. Thus, is equal 50%+880 +1000. But, according 537 1000 500 1000 1000 = to (255), %, and 1880 180. Hence, 5, 3, and 7 each represent a distinct order of decimal fractional units, and 13, or .537 may be read 5 tenths 3 hundredths and thousandths. 537 1000 Analyze the following; thus, .0709 = 170 + 10800. 321. PROP. V.-A decimal is read correctly by reading it as if it were an integer and giving the name of the right-hand order. Thus, .975=80% + 1880 + 1000. Hence is read, nine hundred seventy-five thousandths. 1. Observe that when there are ciphers at the left of the decimal, according to (316), they are not regarded in reading the number; thus, .062 is read sixty-two thousandths. 2. The name of the lowest order is found, according to (314), by prefixing 1 to as many ciphers as there are figures in the decimal. For example, in .00209 there are five figures; hence the denominator is 1 with five ciphers; thus, 100000, read hundred-thousandths. From these illustrations we obtain the following 322. RULE.-Read the decimal as a whole number; then pronounce the name of the lowest or right-hand order. |