OPERATION. 4) 3.00 .75 Again, is equal to 3 divided by 4. Annexing ciphers as before, we find that 4 in 3 we cannot, but since 3 units are equal to 30 tenths, 4 in 30 tenths goes 7 tenths times and 7 tenths over, but 2 tenths are equal to 20 hundredths, and 4 in 20 hundredths goes 5 hundredths times and none over. 7 tenths and 5 hundredths, or 75 hundredths, is the value of the given fraction expressed decimally. We conclude, therefore, that to express the value of any vulgar fraction decimally, we place ciphers on the right of the numerator, regarding them as decimals, and then divide by the numerator, pointing off as many decimal places from the right of the quotient as there are ciphers annexed. When there are not as many figures in the quotient as there are ciphers used, ciphers must be prefixed to make up the deficiency. EXAMPLES. 1. Express the fraction 4 decimally. 2. Reduce and to decimals. Ans. .25. Ans. .333+ and .1666+, 3. Reduce to the form of a decimal. Ans. .9166+. Q. What does a fraction denote? How is its value found? How may a vulgar fraction be expressed decimally? How many tenths are there in 1 unit? How many hundredths? How many hundreds in 1 tenth? How many decimals are pointed off in the quotient? If the number of figures in the quotient is not equal to the number of decimals to be pointed off, how do you supply the deficiency? APPLICATIONS UPON DECIMAL FRACTIONS. 1. Find the value of the Flour shipped from the United States in each of the years from 1790 to 1838; the number of barrels shipped each year, and the average prices in Philadelphia being as follows: OF COMPOUND NUMBERS. 98. In all the operations which we have explained, we have considered the numbers in each example as expressing entire or fractional parts of the same kind of unit; that is, all dollars or parts of dollars; all yards or parts of yards, &c. To those numbers which are composed of the same kind of unit we have given the name of Simple Numbers. COMPOUND NUMBERS are those which are composed of units of different kinds. Thus, 8 dollars 5 cents and 3 mills is a compound number. 10 pounds 14 shillings and 4 pence is a compound number. Q. What kind of numbers are those upon which we have been operating? What name was given to these numbers? What are compound numbers? Give some examples of compound numbers. 99. Before explaining the operations used in compound numbers, we will first show the different kinds of units. In Art. 2 we have defined a unit to be a single quantity which is used to compare quantities of the same kind with each other.. Thus, were I to ask Mary how much more ribbon she had in her work-basket than Ann had, she might apply the middle finger of her right hand to her ribbon as often as it would contain it, and say, her ribbon was 4 fingers long; and by applying the same finger to Ann's ribbon in the same way, she might find that there were 2 fingers in Ann's ribbon. Mary has therefore 2 fingers more ribbon than Ann has. In this example the middle finger of Mary's right hand was the unit by which the quantity of ribbon in the two cases was compared. Again, were I to ask John how much longer his top-cord was than Henry's, he might take his pencil, and applying it to both cords, find that his cord contained the pencil 63 times, and Henry's contained it 5 times. John's cord would be 11⁄2 times the pencil longer than Henry's. Here the pencil is the unit of measure. Again, find how many more chestnuts James has in his bag than William has in his. You might take a tin cup, and on measuring find that James's filled the cup 20 times, and William's 15 times. James has 5 cups more of chestnuts than William, and the tin cup was the unit of measure. to What has been done in these three cases might be extended any kinds of measure whatever. Q. What has a unit been defined to be? How do you explain this by means of the ribbon? What is the unit in this case? In the case of the top-cords what is the unit? In the chestnuts? 100. But as the middle fingers of all little girls' hands are not of the same length, and as measures for pencils and cups also vary, it is found necessary to fix by law the magnitude of the different units used, in order that the measures may be uniform. Further, as convenience would require a small unit for small quantities measured, and a larger unit for larger quantities, different units are used, whose magnitude corresponds to the quantities measured. The following tables comprise the divisions of the different kinds of units used in the United States. They should be carefully committed to memory. Q. Why will not the middle finger answer as a unit? How are the units of measure made uniform? What arrangement is made to suit the quantity measured? In this table it will be seen that there are five different kinds of units used in the currency of the United States, viz: Mills, Cents, Dimes, Dollars, and Eagles; and that they increase in value in a tenfold proportion. Accounts, however, are kept only in dollars and cents. E Q. Repeat the United States currency table. How many different units are there in it? Write down the signs used to represent them. How many mills in 1 cent? How many cents in 1 dime? How many mills in 1 dime? How many dimes in 1 dollar? How many cents in 1 dollar? How many mills? How many dollars in 1 eagle? How many dimes? Cents? Mills? In which of these units are accounts kept? 102. ENGLISH CURRENCY, OR STERLING MONEY. 4 Farthings make 1 Penny. 12 Pence 20 Shillings 21 Shillings Sign D. 66 1 Shilling. Sign Sh. 66 1 Pound. Sign £ £ sh. d. far. 1=20-240=960 1=12=48 1= 4 This table is used in England, and was in common use in this country before the Revolution. It has 5 different units, viz: Farthings, Pence, Shillings, Pounds, and Guineas. Farthings are usually written in fractional parts of pence, thus: 1 far.=4d.; 3 far.=‡d. Q. Repeat the English currency table. What else is it called? What are the different kinds of units used? Write their signs? Where is this table used? Was it ever used in this country? How many farthings in 1 penny? How many pence in 1 shilling? How many farthings in 1 shilling? How many shillings in 1 pound? How many pence in 1 pound? How many farthings? How are farthings usually expressed? 3 farthings are equal to what? 1=20=80=2240-35840=573440 1= 4= 112= 1792= 28672 1= 16 By this table heavy and coarse articles are weighed, such as meat, hay, groceries, and all the metals except gold and silver. It has six different units, viz: tons, hundred weight, quarters, pounds, ounces and drachms. Although by this table 112 pounds constitute the hundred weight, and 2240 |