26. A man who had spent of his money and $3 more, found that he had $37 left. How much money had he at first? 27. A and B together can do a piece of work in 10 days. Assuming that A can do but as much as B, in what time can each do the work?

28. A man who owned

of a shoe-factory sold of his share for $12000. What would the factory be worth if sold at that rate?

29. Find the sum and the difference of 20 and 21; divide the sum by the difference; the difference by the sum; and find the product of the quotients.

30. A lad lost that he had but

had he at first?

of his marbles. After buying 6, he found as many as he had at first. How many

QUESTIONS FOR REVIEW.

187. What kind of fractions only can be added? What must be done to dissimilar fractions before they can be added? How should mixed numbers be added? What kind of fractions only can be subtracted? What must be done to dissimilar fractions before they can be subtracted? How should mixed numbers be subtracted?

How may a fraction be multiplied? What is the rule for multiplying a fraction by a fraction? Show that this rule includes the previous cases in multiplication of fractions. What is a compound fraction? How may a fraction be divided? Give the rule for dividing a fraction by a fraction. Show that this rule includes the previous cases in division of fractions. What is a complex fraction?

What is a fraction? What is the numerator? What is the denominator? What is reduction of fractions? How are fractions reduced to their smallest terms? How are mixed numbers reduced to improper fractions? How are improper fractions changed to integers or mixed numbers? How are dissimilar fractions made similar? What is meant by the least common denominator of fractions?

DECIMAL FRACTIONS

188. 1. If a line is divided into 10 equal parts, what is each part called?

2. If of a line is divided into 10 equal parts, what part of the whole line is each part? How much is

of? 3. If of a line is divided into 10 equal parts, what part of the line is each part? How much is of To? 4. How many hundredths are equal to ?

5. How many thousandths are equal to

?

6. The divisions of any thing into tenths, hundredths, thousandths, etc., are called Decimal divisions.

189. A Decimal Fraction is one or more of the decimal divisions of a unit.

The word decimal is derived from the Latin word decem, which means ten.

Decimal fractions are usually called decimals.

190. Since decimals have the same law of increase and decrease as integers, the denominator may be indicated by the position of the figures. Hence:

191. The figures in the first place at the right of units represent tenths; in the second, hundredths; in the third, thousandths; in the fourth, ten-thousandths; etc.

192. The Decimal Point is a period placed before the decimal.

Thus .6 represents ; .24 represents

The decimal point is also called the Separatrix, inasmuch as it is used to separate integers from decimals.

.6 represents. .08 represents 18. .005 represents 1000

1. .04

66

100. .006

66

1000

66

9

10. .09

16. .07

The orders below millionths in their order are: ten-millionths, hundred-millionths, billionths, ten-billionths, hundredbillionths, etc.

EXERCISES IN NUMERATION.

194. 1. Read 42.356.

ANALYSIS. The decimal part of the number expresses 3 tenths, 5 hundredths, 6 thousandths, or 356 thousandths. The whole expression is therefore read: 42 and 356 thousandths.

RULE.-Read the decimal as integer and give it the denomination of the right-hand figure.

In reading an integer and decimal, use the word and only between the integral and decimal parts of the number.

195. 1. Express decimally 29 thousandths.

ANALYSIS. Since 29 thousandths is equal to 2 hundredths and 9 thousandths, 9 is written in thousandths' place, 2 in hundredths' place, and since there are no tenths, O in tenths' place. We then place a decimal point before the tenths.

Hence 29 thousandths: -.029.

RULE. Write the numerator of the decimal, prefix ciphers, if necessary, to indicate the denominator, and place the decimal point before tenths.

The number of places in a decimal will be equal to the number of ciphers in the denominator.

Express decimally:

2. Nine tenths. Eight tenths. Seven tenths.

3. Fourteen hundredths. Twenty-one hundredths.

4. Sixteen thousandths. Two hundred two thousandths.
5. Nine ten-thousandths. Eighty-four millionths.

6. Nine and two hundred sixteen hundred-thousandths.
7. Eleven and six hundred twenty-two thousandths.
8. Forty and ninety-six ten-thousandths.

9. Thirty-seven and forty-four hundred-thousandths. 10. Seventy-three and eight millionths.

11. 426 hundred-thousandths. 36 millionths.

12. 729 ten-thousandths. 324 thousandths.

13. 406 millionths. 370 ten-thousandths.

14. 35 hundred-thousandths. 310 thousandths. 15. 215 ten-millionths. 14 billionths.

REDUCTION OF DECIMALS.

196. To reduce dissimilar decimals to similar decimals.

1. Read the decimals .8; .80; .800.

1000

2. How does compare in value with? With ? 3. Read the decimals .43; .430; .4300.

4. How does 43 compare in value with 430? With 4800?

100

1000

10000

5. How does annexing a cipher to a decimal affect its value?

197. PRINCIPLE.-Annexing a cipher to a decimal does not alter its value.