30. Reduce,,,13, to equivalent fractions having Ans. 3003, 355, 3003, 5008. a common denominator. 3185 4235 Ans. 11. Ans. 143228. 32. What is the sum of 4, 4, 5 ? 33. From a piece of cloth and of the whole was cut 108 off. What part of the whole was thus taken away? 37. A tree 150 feet high had broken off in a storm. What was the length broken off? Ans. 30 feet. 38. A and B together possess 1477 sheep, of which A owns and B. How many belong to each man? 39. A owns Ans. {A's, 844. B's, 633. of a ship, valued at $15422; he sells to B of his share. What is the value of what A has left; also, what is the value of B's part? Ans. {{ A's remaining part is $1402. 40. A cotton mill is sold for $30000, of which A owns of the whole, B and C each ownofof the whole. many dollars does each one claim? How A claims $6000). Ans. B claims $5000, 41. A and B have a melon, of which A owns, and B ;C offers them one shilling, to partake equally with them of the melon, which was agreed to. How must the shilling be divided between A and B ? Ans. A must have of it. 42. A farmer had of his sheep in one field, in a 。 second field, and the residue, which was 779, in a third field How many sheep had he in all? Ans. 1230 sheep. 43. If I divide 616 dollars between A, B, C, and D, by giving Aof the whole, B of the remainder, C & of what then remained, and D the balance, how much will each receive? A had 154 dollars 49. A Decimal Fraction is that particular form of a Fraction, whose denominator consists of a unit, followed by one or more ciphers. Thus,, 100, 100, 180, 1000, 1000, &c., are Decimal Fractions. In practice, the denominators of Decimal Fractions are not written, but always understood. The above Decimal Fractions are usually written as follows: 0.1, 0.3, 0.04, 0.37, 0.08, 0.003, 0.0047, &c. The period, or decimal point, serves to separate the decimals from the whole numbers. The first figure on the right of the decimal point, is in the place of tenths; the second figure is in the place of hur.dredths; the third figure in the place of thousandths, and so on; the value of the units of the successive figures decreasing from the left towards the right, in a tenfold ratio, as in whole numbers. The following table will exhibit this. This table is in accordance with the French method of numeration (ART. 6,) where each period of three figures changes its name and value. Since decimals, like whole numbers, decrease from the left towards the right in a ten-fold ratio, they may be connected together by means of the decimal point, and then operated upon by precisely the same rules as for whole numbers, provided we are careful to keep the decimal point always in the right place. Annexing a cipher to a decimal does not change its value, because it is the same as multiplying its numerator and denominator by 10. Thus: 0·3=0·30=0·300=&c. But prefixing a cipher is the same as removing the decimal igures one place farther to the right, and therefore each cipher, thus prefixed, reduces the value in a ten-fold ratio. Thus 0-3 is ten times 0·03, or a hundred times 0·003. 0.2 is read two tenths. 0.25 0.365 0.105 0.03 twenty-five hundredths. three hundred and sixty-five thou sandths. one hundred and five thousandths. 66 three hundredths. 0.1234 is read one thousand two hundred and thirty four ten thousandths. four and three tenths. thirty-seven and three tenths. three hundred and sixty-five and thres A number composed of a whole number and a decimal part, is called a mixed number. What is a decimal fraction? Of what form is the denominator? Give examples of decimal fractions. In practice, which part is not written, but understood? What purpose does the decimal point serve? What place is the first figure on the right of the decimal point said to occupy? What place does the second figure occupy! What place does the third figure occupy? In what ratio do the values decrease in passing to the right? Is the above table in accordance with the French or English method of notation? Does annexing a cipher to a decimal alter its value? What effect is produced by prefixing a cipher? A number which is composed of a whole number and decimal is called, what? Let the pupil be exercised in decompounding decimals, as we have done in the following EXAMPLES. The expression, 7468, implies that 7468 is to be divided by 1000. Performing the division by the method of CASE IV, ART. 30, we obtain 7 for a quotient and 468 for a remainder. So that 68-748-7.468-7 units, 4 tenths, 6 hundredths, 8 thousandths, or, which is the same thing, it equals 7++180+1000. In a similar manner we find that 364587-36458-7-36458+. 364587-3645 87-3645++180. 100 364587-364-587=364++180+1000· 50. SINCE decimals, like whole numbers, increase from the right towards the left, they may be treated by the same rules as for whole numbers, provided we are careful to keep the decimal point in the right place, so that like orders may stand under each other. Hence we have this RULE Place the numbers so that the decimal points shall be directly under each other; add as in whole numbers. In the amount place the poin. under the points in the numbers added. How do you place the numbers to be added? How, the point in the amount" |