a remainder of 2; for 7 is contained in 65, 9 times, and 2 over. Now this remainder 2 is g, (§ XXVII.) and added to makes 7. 3÷7=1. The answer then is 9 1. Hence, when a remainder is left from dividing the whole number, it must, with the given Fraction, be reduced to an improper Fraction, and then divided. 57 4 17. Divide 53 by 23. 53-23. 23÷23=1. 18. Divide 25 17. 1841 by 7. 19. Divide 88 593. 175 by 23. 5,650 by 215. 420 by 20. A. 118. 114 by 280. A. A. 26,5 7,64611 by 24. A. 318347. by 17. A. 5. 4764 by 8. A. A. 75. 253 by 17. A. 14334. A. 261881. 96 MENTAL EXERCISES. LIV. 1. What part of a hogshead of wine is I gallon? is 57 gallons? 2. What part of a gallon is 1 quart? 3 quarts? 3. What part of an hour is 1 minute? 5 minutes? 4. What part of a week is 1 day? 2 days? 3 days? 5. What part of a yard of cloth is 1 quarter? 6. What part of a minute is 1 second? 10 seconds? The following are to be written. qr.? Compound 7. What part of a pound is 3d. 1 numbers cannot be used in a Fraction. fore reduce 3d. 1 qr. to qrs.=13 qrs. qrs. Therefore, (§ XXVIII.) 3d. 1 qr. is 8. What part of 1 cwt. is 3 qrs. 15 lb.? 9. What part of 1 hogshead of wine, is 13 We must there- 28 This process is called bringing whole numbers of lower denom inations, to Fractions of a higher. The rule seems to be, REDUCE THE GIVEN NUMBERS TO THE LOWEST DENOMINATION MENTIONED FOR A NUMERATOR, AND AN INTEGER OF THE NEXT HIGHER DENOMINATION, TO THE SAME DENOMINATION, FOR A DENOMINATOR. NOTE. Reduce the Fraction, thus obtained to its lowest terms. 10. What part of a pound is 13s. 4d.? Ans. 18=3. 240 11. What part of 1 cwt. is 3 qrs. 15 lb. 14 oz.? Ans. 78. 12. What part of 1 bu. is 3 pks. 7 qts. 1 pt. Ans. §‡. 13. What part of 15 pipes is 25 gals.? Ans. NOTE. In this example 15 pipes must be reduced to gals.; or, which is bet ter, divide both 15 and 25 by 5, and then reduce the quotient 3 pipes to gals. Dividing both by 5 will not affect the answer, (§ XLI.) since, both numerator and denominator are equally divided. 14. What part of 2 miles is 7 fur. 11 in. 2 b. c,? 76032 4320 Ans. 33271 15. What part of 1 mo. is 22 d. 15 h. 1 m.? A. 3258 16. What part of the whole duration of the world have you lived? 17. The earth's diameter, is 7,9113 miles in diameter nearly, and the highest peak of the Himmaleh mountains, is estimated to be 27,677 ft. above the level of the ocean. What part of the whole diameter of the earth is the height of this mountain? MENTAL EXERCISES. LV. 1. What part of a penny is of a farthing? of a farthing? 2. What part of a shilling is of a penny? of a of a shilling? of a of a farthing? of a quart? It is evident, that as fractions are not in their nature different from other numbers, but only in their form, we need no other rules for reducing them from one denomination to another, than those, given for whole numbers. Hence, to reduce a Fraction from a lower to a higher denomination, DIVIDE AS IN REDUCTION ASCENDING OF WHOLE NUMBERS. 6. Reduce of a shilling to the Fraction of a £. NOTE. The results should always be reduced to their lowest terms. 7. Reduce of a dwt. to the Fraction of a lb. Troy. 8. Reduce a cwt. Ans. T 9. What part of a week is of an hour? Ans. 17. 10. Reduce 51 furlongs to the Fraction of a mile. Ans.. 11. What part of a mile is of a b. c. ? MENTAL EXERCISES. LVI. 1. of a penny is how many farthings? of a penny? 2. of a shilling is how many pence? of a shilling? 3. of a shilling is how many pence? how many pence and farthings? 4. of a lb. Troy is how many oz.? of a lb. ? The rule, of course, for the reason assigned in the last section, is the same as in whole numbers. Hence, to reduce a Fraction from a higher denomination to a lower, · MULTIPLY AS IN REDUCTION DESCENDING OF WHOLE NUMBERS. 5. Reduce of a £ to the Fraction of a penny. 6. Reduce 7. Reduce A. 240-8. of a hhd. to the Fraction of a qt. of a cwt. to the Fraction of an oz. A. =89. 8. What part of a pt. is of a bar. ? 9. What part of a minute is 10. What part of an E. E. is A. 1280=7,83. TT of a day? A. 1441. of a yd? Á. First reduce it to the Fraction of a quarter by (§ LV.) and then to the Frac LVII. 1. In of a shilling how many pence? In 3? 2. In of a shilling how many pence? In ? 3. In of a shilling how many pence? In ? 97 4. In of a peck, how many qts.? In ? In the last section, you were required to reduce Fractions of one denomination, to Fractions of another; the present case only differs, in requiring you find whole numbers. Hence, to reduce a Fraction to whole numbers of inferior denominations,. MULTIPLY AS IN REDUCTION DESCENDING OF WHOLE NUMBERS, REDUCING THE FRACTION, WHENEVER IT BECOMES IMPROPER, TO A WHOLE OR MIXED NUMBER, AND CONTINUING THE REDUCTION ONLY OF THE FRACTIONAL PART. A. 5s. 3d. 1 qrs. 5. How much is of a hhd. of wine? A. 45 gals. 6. Find of a £. 7. Find 343 of 1 cwt. A. 1 qr. 5lb. 11 oz. 1571⁄2 dr. 8. Find 486 of a week. A. 2 d. 16 h. 8 m. 17,9 sec. 9. Find of a b Apothecaries' weight. A. 23. 3 gr. 823 1273 DECIMALS. 823 1273 § LVIII. According to our system of Numeration, numbers increase from right to left in a tenfold proportion; that is, a figure becomes ten times greater than before, on being moved a single place toward the left; one hundred times, on being moved two places; one thousand times, on being moved three places, and so on. But as numbers increase from right to left in tenfold proportion, so they decrease from left to right in the same manner; that is, a figure becomes only the tenth part of its former value, on being moved a single place towards the right; the hundredth part, on being moved two places, and so on. Take for example the number 5, 5. If it be moved one place to the left, it becomes ten times as great, that is, fifty, 50. If it be moved another place, it beomes ten times as great as fifty, or a hundred times as great as at first, that is, five hundred, 500. Thus, we may go on, as far as we please, and show that each removal to the right, multiplies the number by 10, or makes it ten times as great. If this be true, then, it will, likewise, be true, that removing the other way, will divide the number by 10, or make it a tenth part as great. Let us take five hundred, the number we ended with. Remove the significant figure 5, one place to the right, and it becomes fifty, the tenth part of five hundred. 500. 50. Remove it another place, and it becomes five, the tenth part of fifty, or the hundredth part of five hundred. 5. Each of these removals, we see, divides the number by 10, or makes it the tenth part as great as before. We have now brought the number back to the units' place. But there is plainly no reason why we should stop here. If a point be made, to distinguish this place, it is evident that we may continue to remove the number in this manner to the right, cailing it at every removal, a tenth part of what it was before. Then, the first place below units would be tenths, and the five instead of being five units would be five tenths. The next place would be tenths of tenths, or hundredths; and the five would become five hundredths. .5 .05 We use a cypher here, to show that the 5 is in the second place below units. If there were no cypher, it would stand in the first place, and be .5 five tenths, as above. The next place would be tenths of hundredths or thousandths. .005 The next place, tenths of thousandths, or ten-thousandths. .0005 There is no reason to prevent our continuing this mode of writing to any distance. Thus, we see, we have a new series of orders, less than units, or whole numbers, which increase towards the left, and decrease towards the right in ten-fold proportion exactly like whole numbers. These are called DECIMALS, from the Latin word, decimus, which means tenth. We have given some examples above, which contain but one significant figure. The following contain more. The pupil should endeavour to read the figures without looking on the names, which should be covered over. If he succeeds without assistance, he will be more likely to possess a clear knowledge of the Notation. Five tenths and two hundredths, or fifty-two hundredths. .52 NOTE. We read this fifty-two hundredths, because the 5 is ten times greater than if it stood in the hundredths' place; that is, than 5 hundredths. It is therefore, ten times 5 hundredths or fifty hundredths; and two hundredths more make fifty-two hundredths. The same explanation will apply in the fol. lowing. Seven tenths and four hundredths, or seventy-four hundredths .74 Seventy-one hundredths Eighty-nine thousandths Ninety thousandths Two hundred and thirty thousandths Nine hundredths .086 .0092 .71 .089 .090 .230 .09 .23 .1 .10 .2 200 |