ANALYSIS.-4.8 equals 48 tenths, and 1 of 48 tenths is 8 tenths, or .8.

9. Multiply 8 by 18 (.08 ×.09). Divide .0072 by .09. 10. The product of two factors is .096, one of which is .8; what is the other?

How many decimal places in the quotient when tenths are divided by units? Tenths by tenths? Hundredths by tenths? Thousandths by hundredths?

If there are two decimal figures in the divisor and three in the dividend, how many are there in the quotient? If three in the divisor and three in the dividend? If none in the divisor and three in the dividend? If two in the divisor and none in the dividend?

295. PRINCIPLES.-1. The dividend must contain at least as many decimal places as the divisor, before division is possible.

2. Since the dividend is the product of the divisor and quotient, it contains as many decimal places as both divisor and quotient. Hence,

3. The quotient must contain as many decimal places as the number of decimal places in the dividend exceeds those in the divisor.

WRITTEN EXERCISES.

296. 1. Divide .952 by .7.

OPERATION.

.7).952

1.36

ANALYSIS.-Divide as in fractions. (238.) Thus, .952÷.7=1985÷16=158% × 4=188=1.36. Or, Divide as in integers, and since the dividend contains three decimal places, and the divisor one, the

quotient must have two decimal places. (PRIN. 3.)

RULE.-Divide as in division of integers, and from the right of the quotient point off as many figures as the decimal places in the dividend exceed those in the divisor.

1. If the number of figures in the quotient be less than the excess of the decimal places in the dividend over those in the divisor, the deficiency must be supplied by prefixing ciphers.

2. If there be a remainder after dividing the dividend, annex ciphers, and continue the division: the ciphers annexed are decimals of the dividend.

3. In most business transactions, the division is considered sufficiently exact when the quotient is carried to 4 decimal places, unless great accuracy is required.

4. To divide by 10, 100, 1000, etc., remove the decimal point in the dividend as many places to the left as there are ciphers in the divisor. (266, 3.)

8. Divide 88.476 by 1.2; by 3.6; by .01; by 1.04. 9. Divide $56.05 by .59; $408.37 by 27.

10. Divide $6.45 by $.45; $52 by $.65; 293.75 by 454. 11. Divide .0026 by .003; 3 by .450; 75 by 1000.

12. What is the quotient of 75.15208 divided by 24? by .24? by .024? by .0024? by .00024 ?

13. Divide $3875 by 10; by 100; by 1000; by 10000.

27. If 64 tons of iron cost $4816, how many tons can

be bought for $1730.75 ?

28. How many coats can be made from 32.4 yards of cloth, allowing 2.7 yards for each coat?

29. At $287 each, how many horses can be bought for $4885.80?

30/ If 125 bushels of potatoes cost $824, how many barrels, each containing 24 bushels, can be bought for $224.40? 31. If 31 cords of wood cost $11.37, what will 201 cords cost?

32. How much sugar can be bought for $46.75, if of a hundred pounds cost $63?

33. Gave 10 cords of wood, worth $4 a cord, for 7.74 barrels of flour. What was the flour worth a barrel?

34. A man sold a horse for $125, and received in payment 12 yards of cloth at $34 a yard, and the balance in tea at $.621. How many pounds of tea did he receive? Find the second member in each of the following equations :

35. Of (1.00818+63÷4000 × 100)-4=?

36. Of 714-.714÷(.34-.034 x.25 of 6)=?

37. Of (.48÷800 × 10000+6.4÷.08)÷.125= ? 38. Of (34 x.193 +2.7 ×.41)÷(4.81—§ of 1.662) = ? 39. Of ($262.90÷$.56) x .0084+.027×100=? 40. Of ($1260 x 3.49)÷$10.47-$850÷$6.80=?

CIRCULATING DECIMALS.

ORAL

EXERCISES.

297. 1. What are the prime factors of 10? Of 100? 2. Change to the decimal form ; † ; † ; † ; &♬ (285.) What are the prime factors of each of the denominators of these fractions?

Are they the same as the prime factors of 10?

Can these fractions be reduced to perfect decimals?

3. Change to the decimal form, extending to four places, ; ; ; f·

Can these fractions be reduced to perfect decimals?

What are the prime factors of their denominators?

4. Change to the decimal form, extending to three places, ;; †

Can these fractions be reduced to perfect decimals?

What are the prime factors of their denominators?

How do the decimals produced by these fractions differ from the decimals produced by the fractions in examples 2 and 3?

What kind of decimals are all fractions equivalent to, that in their lowest terms have denominators containing the factors 2 or 5? 5. What figure is constantly repeated in reducing to a decimal ?? ? ?

C. If a decimal consists of 3 repeated indefinitely, what fraction is it equal to ?

7. Is there any difference between and ? and ? and ? and §§§§?

8. Is there any difference between and #? # ind 4848?

9. If the numerator is 4444, what must be its denominator so that the fraction may equal?

To change a repeating decimal number to an exact fraction, what figures must be used in the denominator?

DEFINITIONS AND PRINCIPLES.

298. A Finite Decimal is a perfect decimal, or one that terminates with the figures written; as, .25, .375. 299. A Circulating Decimal is a decimal in which a figure, or set of figures, is constantly repeated in the same order; as, .333+, .727272+.

300. A Repetend is the figure or set of figures, continually repeated.

The repetend is written but once, and when it consists of a single figure a point is placed over it; when it consists of more than one figure, points are placed over the first, and over the last figure. Thus, the circulating decimal .666+, and .297297+, are written .6, and 297.

301. A Pure Circulating Decimal is a decimal which commences with a repetend; as .7, or .279.

302. A Mixed Circulating Decimal is a decimal in which the repetend is preceded by one or more decimal places called the finite part of the decimal; as, .27, or .04648, in which .2 or .04 is called the finite part.

303. The law for the formation of repetends will be apparent from the following:

1..1111+

3.

=.i. 5. =.4444+ =.4.

=.01. 6.

=.2323+

=.23.

=.135.

2. .01010+ =.001001+ =.001. 7.384 =.135135+ 4.=.00010001+.000i 8.88.17281728+=.1728

304. PRINCIPLES.-1. Every fraction in its lowest terms, whose denominator contains no other prime factors than 2 or 5 is equivalent to a finite decimal.

2. Every fraction in its lowest terms, whose denominator contains other prime factors than 2 or 5 is equivalent to a circulating decimal.