7. What part of one ounce is one dram? What part of 1 pound is one dram? 2 drs.? 3 drs.? 1 oz. 1 dr.? 1 oz. 2 drs.? 2 oz. 4 drs.? 3 oz. 6 drs.? 8 oz. 3 drs.? 9 oz. 11 drs.? 8. What part of a pound is of an oz.? of an oz.? What part of a pound is an oz.? 2 oz.? of 31 oz.? 41 oz.? 9. What part of a pound Troy is 1 dwt.? 5 dwt.? 6 dwt.? 9 dwt.? 11 dwt.? 10 dwt.? 1 oz. 1 dwt.? 3 oz. 4 dwt.? What part of oz. Troy is 1 dwt.? 3 dwt. 1 gr.? 4 dwt. 6 gr.? 7 dwt. 3 grs.? 8 dwt. 9 grs.? 10 dwt.? 12 dwt.? 16 dwt.? 10. What part of an ell English is 1 qr. of a yard? 2 qrs.? 3 qrs. ? What part of a qr. is 1 nail? 3 nails? 11. What part of a yd. is 1 qr. 1 nail? 2 qrs. 3 n.? 3 qrs. 2 n.? What part of an ell English is 3 nails? 1 qr. 3 n.? 4 qrs. 1 n. ? 12. What part of a yd. is 1 inch? 4 inches? 7 inches? 9 inches? What part of a yard is 1 qr. 2 in.? 2 qrs. 3 in.? 1 in.? 3 qrs. 13. From a vessel containing 3 gallons of wine, 3 gills leaked out; what part of a gallon leaked out? What part of a gallon remained? 14. From a barrel full of wine 7 quarts were drawn; how many quarts remained? What part of the barrel had been drawn out? What part of the barrel had remained? 15. If of a barrel of beer be divided into 4 equal parts, what part of a barrel will each of the parts be? How many gallons will each part be? 16. If one quart be taken from a barrel full of beer, what part of a barrel will remain? If 3 pints be taken out, what part will remain? If 7 gallons be taken out, what part of a barrel is taken out? What part of a barrel remains? 17. A man distributed 74 gallons of milk what part of a gallon did he give to each? among 5 persons; 18. If you have 3 gallons of milk, and distribute it to some poor persons, giving of a gallon to each, how many persons will you give it to? How much will remain? 19. What part of 1 foot is 1 barley-corn? 2 bar.? 5 bar.? 1 inch 1 bar.? 3 in. 2 bar.? 5 in. 1 bar. ? 20. What part of a yard is 2 inches? 34 inches? 14 in.? 5 in.? 6 in.? 17 in.? 24 in. 2 bar.? 21. What part of a rod is a foot? 1 feet? 2 feet? 4 feet, 3 in.? 6 feet, 7 in.? 10 feet, 5 in.? 22. What part of 3 rods is a foot? 1 foot? 3 feet? What part of a furlong are 24 rods? 5+ rods? 23. What fraction of a foot is What fraction of a foot is a rod? of a yard? of a yd.? of of a rod? of a rod? 24. A man measured the length of his barn with a stick half a yard long, and found the barn 314 times the length of his stick; how long was it? 25. A carpenter is cutting up a board 17 feet in length, into pieces 24 feet long; how many pieces will there be, and how long will be the piece that remains? 26. A man measures a piece of fence with a pole 94 feet long; the fence is 15 times the length of the pole; how many rods is it in length? 27. What part of a peck is of a bushel? of a peck? of a peck? of a bushel? of a bush.? of a bush.? 28. What part of a peck is of a bush.? of a bush.? of a bush.? 29. Two men bought a lot of standing wood in company, for 11 dollars; one cut off 2 cords, the other 1 cord; what ought each to pay? 30. Two boys bought the chesnuts on a tree for 50 cents; one had 11 quarts, the other 6 quarts and 1 pint; what ought each to pay? 31. Three men bought a piece of cloth for 24 dollars? the first took 24 yds., the second the same quantity, and on measuring the remainder it was found to be 3 yards; what ought each to pay? 32. Two men hire a horse for a month for 12 dollars; one travels 200 miles with the horse, the other 150; how much should each pay? SECTION XII. DECIMAL FRACTIONS. [See Numeration, Part II.] In the calculations in common fractions, a great inconvenience arises from their irregularity. There is no law regulating the magnitude of either of the terms. The denomina From tor may be in any ratio whatever to the numerator. seeing one you can make no inference at all respecting the magnitude of the other. In calculations of addition, it is often more than half the work to bring the fractions into a common denomination. Now it is evident that if fractions could be written in the same manner as whole numbers, that is, increasing in a tenfold rate as you advance to the left, and decreasing in a ten fold rate as you advance to the right, an immense gain would be made in the convenience of calculating them. Operations in fractions would then be just as easy as operations in whole numbers. Now this advantage is gained in decimal fractions. They are brought under the same law as whole numbers. Let us observe the manner in which whole numbers are written. Take the number 222; the right hand figure signifies two units, the next two tens, the next two hundreds; just as if it were written in this manner, 2×100+ 2x10+2: two multiplied by 100 plus two multiplied by 10, plus two; making two hundred and twenty-two. But thi cumbersome method of writing is unnecessary, because the law of notation determines what number the figures in each place shall be multiplied by. It must not be forgotten that the figure 2 in the above example in no case signifies of itself more than two. It is the place it occupies that gives it the higher value of tens or hundreds. Now it would evidently be a great convenience if we could reduce fractions to the same law, so that they would, like whole numbers, decrease in a decimal ratio, in advancing from the left to the right. To show this regularity to the eye we will write the following numbers: two multiplied by 1000, two multiplied by 100, two multiplied by 10, two units, two divided by 10, two divided by 100, and two divided by 1000. Written in full they would stand thus: 2×1000+ 2×100+2×10+2+fotiŝotro‰0. But we have seen that we may write the whole numbers without the multipliers, thus, 2222, because we know from the place each figure occupies what its multiplier must be. Just so we can write fractions without the denominators, provided we know, from the place of the numerator, what the denominator must be. Thus the whole of the above series may be written as follows; 2222.222. A decimal, therefore, is the numerator of a fraction, whose denominator is never written, but is always understood to be 1, with as many ciphers as there are places in the decimal. In You observe that, in writing the series given above, there is a period placed at the right hand of the whole numbers, separating the unit figure from that of tenths. The period must never be omitted when there are fractions, for it enables you to determine the value of each figure in the sum. stead of reading .22 two tenths and 2 hundredths, we may call it 22 hundredths, which is more convenient and amounts to the same; for 2 tenths is equal to 20 hundredths; so .222 is two hundred and twenty-two thousandths. So, in all cases, read the decimal numbers as whole numbers, and for their denominator take 1 with as many ciphers as there are places in the written decimals. In all your study of decimals, be careful not to confound the words which express fractions with the similar words which express whole numbers; as tenths with tens, hundredths with hundreds. The following questions will aid you in fixing this distinction clearly in mind. 1. How many tenths are equal to ten whole ones? 2. How many tenths are equal to two and a half whole ones? 3. How many hundredths are equal to three and a quarter whole ones? 4. How many hundredths are equal to one hundred whole ones ? 5. How many thousands are equal to ten whole ones? 6. In fifteen whole ones how many tenths? How many hundredths? 7. In seventy-five hundredths how many tenths? 8. In three tenths how many hundredths? 9. In six tenths how many thousandths? Thus, you observe, fractions have been brought under the same law that regulates the writing of whole numbers. They may now be added, subtracted, multiplied, and divided, like whole numbers. But in doing this it is important to determine the place of the period that separates the whole numbers from the fractional part of the sum. Where must the period be placed in the answer? ADDITION AND SUBTRACTION OF DECIMALS. Let us first observe how important it is that the rule in this case be entirely correct. If I have this number, 32.5, to write, and by any mistake I should write it 3.25, it would denote a quantity only one tenth as great as it should be; or, if I should write 325. it would denote a quantity ten times greater than it should be. Moving the period one place to the right, makes the number ten times as great as it was before, for tens become hundreds, and hundreds, thousands; and each figure ten times as great as before. So, by moving the period one place to the left, the number becomes just one tenth what it was before. Removing the period two places from its true place, makes the number 100 times larger or smaller than it should be, according as you remove it to the right or the left. Hence you may see that in order to multiply a number that has decimals, by 10, you have only to remove the period one place to the right; to multiply by 100, remove it two places, and so on. To divide by 10, remove the period one place to the left; to divide by 100, remove it two places, and so on. From the above you may see the importance of being perfectly accurate in fixing the place of the decimal in the answer to any question. We will begin with addition. Add 4.46 to 3.21. Here you observe the two whole numbers make 7, and 46 hundredths added to 21 hundredths make 67 hundredths: the answer, then, must be 7. 67, having two decimal places. Add 6.8 to 5. 23. The 3 hundredths must evidently stand alone, since there is nothing like it to add to it; 2 tenths added to 8 tenths make 10 tenths, or one whole one; this we carry to the 5, which gives us for the answer, 12. 03. This will serve to suggest the rule for placing the period in the answer to questions in addition. The number of decimal places in the answer must be as great as can be found in any one of the numbers to be added. |