yard? 2. If } of a dollar will pay for 102lb. of nails, how much is the price per pound? Ans. $ 3. If 4 of a yard of cloth cost $3, what is the price per Ans. $51 4. If $211 will buy 713 barrels of apples, how much are they per barrel ? Ans. $2-94 125: 5. If 41 gallons of molasses cost $25, how much is it per quart? Ans. cts. 6. If ignhd. of wine cost $250], how much is the wine per quart? Ans. $1581=88 3101 cts. 7. If eight pounds of tea cost 7 of a dollar, how much is it per pound? Ans. 95 cts. 8. In 8.1 weeks a family consumes 165, pounds of butter: how much do they consume a week? Ans. 1963 1b. 9. If a piece of cloth containing 1763 yards costs $3759 what does it cost per yard? Ans. $ DECIMAL FRACTIONS. § 119. If the unit 1 be divided into 10 equal parts the parts are called tenths, because each part is one-tenth of unity. If the unit 1 be divided into one hundred equal parts, the parts are called hundredths, because each part is one hundredth of unity. If the unit 1 be divided into one thousand equal parts, the parts are called thousandths, because each part is one thousandth of unity: and we have similar expressions for the parts, when the unit is divided into ten thousand, one hundred thousand, &c., equal parts. The division of the unit into tenths, hundredths, thousandths, &c., forms a system of numbers called Decimal Fractions. From which we see, that in each case the denominator gives denomination or name to the fraction; that is, determines whether the parts are tenths, hundredths, thousandths, &c. Q. When the unit 1 is divided into 10 equal parts, what is each part called? What is each part called when it is divided into 100 equal parts? When into 1000? Into 10,000, &c.? How are decimal fractions formed? What gives denomination to the fraction? $ 120. The denominators of decimal fractions are seldom set down. The fractions are usually expressed by means of a point, or comma, placed at the left of the numerator. Thus, four-tenths, forty-five hundredths, ,1047 The denominator of every decimal fraction, however, is always understood. It is a unit 1 with as many ciphers annexed as there are places of figures in the numerator. The place next to the decimal point is called tenths; the next place to the right, the place of hundredths; the next, the place of thousandths; and so on for places further to the right, according to the following Table. DECIMAL NUMERATION TABLE. ,4 ,125 Hundredths. ,6 4 is read 4 tenths. 64 hundredths. Decimal fractions are numerated from the left hand to the right, beginning with the tenths, hundredths, &c., as in the table. 94. 4 10 Q. Are the denominators of decimal fractions generally set down? How are the fractions expressed ? Is the denominator understood ? What is it? What is the place next the decimal point called? The next? The third, &c. ? Which way are decimals numerated ? $ 121. Let us now write and numerate the following decimals. Four-tenths, ,0 4. Four thousandths, ,00 4. Four ten thousandths, ,00 0 4. Four hundred thousandths, ,0 0 0 0 4. Four millionths, ,0 0 0 0 0 4. Four ten millionths, ,00 0 0 0 0 4. Here we see, that the same figure expresses different values, according to the place which it occupies. But so of ਹ is equal to To=,04. Tooo=,004. 10000=,0004. 100000=,00004. 1000000=,000004. to of 1000000 4 10000000=,0000004. Therefore, the value of the places diminish in a tenfold proportion from the left hand to the right. Hence, ten of the parts in any one of the places, are equal to one of the parts in the place next to the left; that is, ten thousandths make one hundredth, ten hundredths make one-tenth, and ten-tenths a unit 1. This law of increase from the right hand towards the left, is the same as in whole numbers. Therefore, whole numbers and decimal fractions may be united by placing the decimal point between them. Thus, 36,95 thirty-six, and 95 hundredths. 127,4 127 and four-tenths. 163,03 163 and three hundredths. 627,0047 627 and 47 ten thousandths. A number composed partly of a whole number and partly of a decimal, is called a mixed number. Q. Does the value of a figure depend upon the place which it occupies? How does the value change from the left towards the right? What do ten parts of any one place make? How do they increase from the right towards the left? How may whole numbers be joined with decimals? What is a number called when composed partly of whole numbers and partly of decimals ? Write the following numbers in figures, and numerate them. 1. Forty-one, and three-tenths. 41,3. 2. Sixteen, and three millionths. 16,000003. 3. Five, and nine hundredths. 5,09. 4. Sixty-five, and fifteen thousandths. 65,015. 5. Eighty, and three millionths. 80,000003. 6. Two, and three thousand millionths. 2,000000003. 7. Four hundred and ninety-two thousandths. 0,492. 8. Three thousand, and twenty-one ten thousandths. 9. Forty-seven, and twenty-one ten thousandths. 10. Fifteen hundred and three millionths. 11. Thirty-nine, and six hundred and forty thousandths. 12. Three thousand, eight hundred and forty millionths. 13. Six hundred and fifty thousandths. 14. Fifty thousand, and four hundredths. 15. Six hundred, and eighteen ten thousandths. 16. Three millionths. 17. Thirty-nine hundred thousandths. $ 122. The denominations of Federal Money will correspond to the decimal division, if we regard 1 dollar as the unit. For, the dimes are tenths of the dollar, the cents are hundredths of the dollar, and the mills, being tenths of the cent, are thousandths of the dollar. EXAMPLES. 1. Express $16, 3 dimes 8 cents and 9 mills decimally. Ans. $16,389. 2. Express $95, 8 dimes 9 cents 5 mills decimally. Ans. 3. Express $107, 9 dimes 6 cents 8 mills decimally. Ans. $107,968. 4. Express $47 and 25 cents decimally. Ans. $47,25. 5. Express $39,39 cents and 7 mills decimally. Ans. $39,397. 6. Express $12 and 3 mills decimally. Ans. 7. Express $147 and 4 cents decimally. Ans. $147,04. 8. Express $148, 4 mills decimally. Ans. $148,004. 9. Express four dollars, six mills decimally. Ans. Q. If the denominations of Federal Money be expressed decimally what is the unit ? What part of a dollar is i dime? What part of a dime is a cent? What part of a cent is a mill? What part of a dollar is 1 cent? 1 mill ? § 123. A cipher is annexed to a number, when it is placed on the right of it. If ciphers be annexed to the numerator of a decimal fraction, the same number of ciphers must also be annexed to the denominator; for there must be as many ciphers in the denominator as there are places of figures in the numerator (see § 120.) The numerator and denominator will therefore be multiplied by the same number, and consequently the value of the fraction will not be changed (see $ 84.) Hence, Annexing ciphers to a decimal fraction does not alter its value. We may take as an example, 3=ió. If now we annex a cipher to the numerator, we must, at the same time, annex one to the denominator, which gives ,30 3.0 by annexing one cipher, 3000 by annexing two ciphers, 3000 all of which are equal to .=,3. Also, ,5= *=,50=500=,500=300%. Also, ,8 ,80=,800=,8000,=,80000. Q. When is a cipher annexed to a number? Does the annexing of ciphers to a decimal alter its value? Why not? What does threetenths become by annexing a cipher? What by annexing two ciphers ? Three ciphers? What does ,8 become by annexing a cipher? By annexing two ciphers? By annexing three ciphers ? $ 124. Prefixing a cipher is placing it on the left of a number. If ciphers be prefixed to the numerator of a decimal fraction, that is, placed at the left hand of the ,300 = ,3000=10000 |