76. A boat whose rate of sailing in still water is 14 miles an hour, was accelerated 3 miles per hour in going down stream, and retarded the same length of time in coming up. It was 5 hours longer in coming up a certain distance than in going down. What was the distance?

77. Eight men cut 97 which they received $48. pay of each man ?

cords of wood in 47 days, for

What was the average daily

78. A farmer engaged a man to work for him a year for $216 and a suit of clothes. The man left at the end of 10 months, and received $175 and the suit of clothes.

was the value of the suit?

What

79. A man has of his property invested in real estate, & of the remainder in bonds, of what still remains in bank stock, and the rest, which is $3500, he has invested in business. What is the value of his entire property?

80. The sum of two numbers equals 53, and one of

71

them is the difference between 491 and 22. What is the

other number?

11

9

81. A and B together can do a piece of work in 12 days. If A can do only as much as B, how long will it take each of them to do the work?

82. A man bought a house and paid of the price in cash at the time of the purchase. A year afterward he paid

of what then remained unpaid, and the two payments amounted to $5260. How much did he pay for the house?

83. A cistern which holds 280 gallons is empty. It has a supply pipe which will fill it in 10 hours, and a discharge pipe which will empty it in 7 hours. If the supply pipe has been running into it for 4 hours, and then both pipes are opened, in what time will it be emptied?

DECIMAL FRACTIONS.

164. 1. What is 1 of the ten equal parts of anything? 2. What is 1 of the ten equal parts of? 3 parts? 3. What is 1 of the ten equal parts of ?

10

35 parts?

6. What part of is? Of 150 is 1000? Of 1000 is Toboo?

165. The divisions of anything into tenths, hundredths, thousandths, etc., are called Decimal Divisions.

166. One or more of the decimal divisions of a unit are called a Decimal Fraction.

The word decimal is derived from the Latin word decem, ten.
Decimal fractions are commonly called decimals.

167. Since tenths are equal to ten times as many hundredths, hundredths equal to ten times as many thousandths, etc., decimals have the same law of increase and decrease as integers, and the denominator may be indicated by the position of the figures.

In the decimal system of notation, the representative value of a figure is decreased tenfold by each removal one place to the right; hence:

The figure at the right of units expresses tenths.

The figure at the right of tenths expresses hundredths.

* The figure at the right of hundredths expresses thousandths. The figure at the right of thousandths expresses tenthousandths.

The figure at the right of ten-thousandths expresses hundred-thousandths, etc.

168. A period, called the Decimal Point, is placed before the decimal.

Thus, .5 represents; .58 represents.

The number is read, 3 million, 415 thousand, 673 and 413 thousand 207 millionths.

The orders of decimals below millionths are ten-millionths, hundredmillionths, billionths, ten-billionths, hundred-billionths, trillionths, etc.

EXERCISES IN NUMERATION.

169. 1. Read the expression 5.239.

The whole expression is read, 5 and 239 thousandths.

RULE.

Read the decimal as an integral number, and give

it the denomination of the right-hand figure.

170. 1. Express decimally seventy-nine thousandths. EXPLANATION. Since thousandths occupy the third place, three figures are required to express the decimal. Hence, the number seventy-nine is written, and a cipher prefixed to cause the figures to occupy their proper position. Hence, the decimal is written .079.

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

Express decimally:

2. Eight tenths.

Three tenths. Five tenths. Four hundredths. Eight hundredths. Six hundredths.

3. Five thousandths. Three hundred four thousandths. Seven ten-thousandths. Eight hundred sixty-five tenthousandths. Sixty-eight thousandths.

4. Fifteen hundredths. Twelve hundred-thousandths. Forty-eight thousandths. Four hundred-millionths.

5. Ninety-five ten-thousandths. Ninety billionths. Sixtytwo hundred-thousandths. Fifty-five thousandths.

6. 210 millionths. 403 thousandths. 15 hundredths. 4256 hundred-thousandths. 1268 hundred-millionths.

7. 56345 millionths. 389 ten-thousandths. 3854 hundred-thousandths. 518 ten-millionths.

8. Sixty-seven, and three hundred forty-nine ten-thousandths. Fifty, and fifty-five millionths.

9. Eighty-eight, and five thousand five hundred-thousandths. Nine, and nine ten-thousandths.

22. How do the fractions in 16, 17, and 18 compare in value? How do the decimals compare?

effect of annexing ciphers to decimals?

What is the

23. How do the fractions in 19, 20, and 21 compare in value? How do the decimals compare? What is the effect of prefixing decimal ciphers to a decimal ?

24. How does the number of places in a decimal compare with the number of ciphers in the denominator.

171. PRINCIPLES. -1. Annexing ciphers to a decimal does not alter its value.

2. Each decimal cipher, prefixed to a decimal, diminishes the value of the decimal tenfold.

3. The denominator of a decimal, when expressed, is 1 with as many ciphers annexed as there are figures in the decimal.

172. In expressions of the currency of the United States, the cents, mills, etc., may be read as decimals of a dollar : Thus, $5.375 may be read 5 dollars and 375 thousandths, or 5 dollars 37 cents.

Read the following as dollars and decimals of a dollar :

1. $6.495. 2. $5.083.

3. $7.394.

4. $5.865.

5. $4.004.

6. $8.056.