Let the pupil write the following EXERCISES. If a man divided an apple into eight parts, and gave away five of these parts, how do you express the quantity he gave away, and the quantity he kept ? If a man had three apples, and cut out a fourth part of each, and gave it away, how do you express what he gave away? If a man had twelve oranges, and one sixths of each was decayed, how do you express the quantity of decayed oranges he had ? If a man had five casks of wine, and a twelfth part leak. ed out of each, how do you express what he lost? DECIMAL NUMERATION. There is another mode of writing fractions, in which the numerator only is written. The denominator, although not written, is always understood to be I, and a certain number of ciphers. These fractions are called Decimals. Thus in writing decimals, if we are to express two tenths, instead of writing it thus, to, the numerator only is written, and a comma, called a separatrix, is placed before it, thus ,2. The following is the rule, by which it is known what is the denominator. The denominator of a decimal is always one, and as many ciphers as there are figures in the numerator, or decimal. What is the denominator of this decimal, ,2 ? How many ciphers in the denominators of these decimals, ,34. ,600. ,3246. ,56945. ,3694. ? If the decimal has one figure, it expresses tenths. Thus ,2 is two tenths. If it has two figures, it expresses hundredths. Thus ,02 is two hundredths. How do decimal fractions differ from vulgar? What is the rule for as. certaining the denominator of a decimal? If it has three figures, it expresses thousandths. Thus ,002 is two thousandths. If it has four figures, it expresses tenths of thousandths. Thus ,0002 is two tenths of thousandths. If it has five figures, it expresses hundredths of thou. sandths. Thus ,00002 is two hundredths of thousandths. What does this decimal express, ,3? What does this decimal express, ,15? Ans. Fifteen hundredths. What does this decimal express, ,110 ? A decimal must always have the number of figures in the numerator, equal to the number of ciphers in the de. nominator ; therefore it is necessary to learn how many ciphers there are in each kind of denominator. If the decimal is tenths, there is one cipher in the de. nominator ; if hundredths, there are two ciphers ; if thousandths, there are three ciphers ; if tenths of thousandths, there are four ciphers; if hundredths of thousandths, there are five ciphers, &c. Of course in writing decimals, if tenths are to be ex. pressed, there must be only one figure in the numerator, or decimal; if hundredths, there must be two figures ; if thousandths, there must be three figures ; if tenths of thousandths, there must be four figures; if hundredths of thousandths, there must be five figures, &c. EXERCISES. If you are to write two tenths, how many figures must What must the number of figures in the numerator equal ? there be in the numerator or decimal, and how many ci. phers are understood to be in the denominator ? Write two tenths. (,2.) If you are to write two hundredths, how many ciphers are understood to be in the denominator, and how many figures must there be in the numerator? Write, two hundredths. In writing this last, the pupil must first write the 2, and then as there must be as many figures in the numerator, as there are ciphers in the denominator, a cipher is placed before the 2, and then the separatrix is prefixed thus, ,02. If the cipher were placed after the 2, how would it read? Ans. Twenty hundredths, instead of two hundredths. If the cipher were not placed before the 2, how would it read? Ans. Two tenths. If another cipher is placed before the ,02 thus, ,002 how does it read ? What does the denominator express, when there are three figures in the decimal. Ans. Thousandths. What does it express when there are four figures in the decimal ? Let the pupil write the following. * 1. Two tenths. Two tens. 2. Two hundredths. Two hundreds. 3. Two thousandths. Two thousands. 4. Two tenths of thousandths. Two tens of thousands. 5. Two hundredths of thousandths. Two hundreds of thousands. 6. Five tenths. Five tens. 7. Fifteen hundredths. Fifteen hundreds. 8. Fifteen thousandths. Fifteen thousands. 9. Fifteen tenths of thousandths. Fifteen tens of thousands. 10. Fifteen hundredths of thousandths. Fifteen hundreds of thousands. 11. One tenth. One ten. 13. One hundred and fifteen thousandths. One hun. dred and fifteen thousands. * The exercises in whole numbers are added, that pupils may notice the difference. 14. Five tenths. Five tens. 17. Five hundred and five thousandths. Five hundred and five thousands. 18. Fifteen thousandths. Fifteen thousands. 19. Five thousandths. Five thousands. 20. Two hundred thousandths. Two hundred thousands. 21. Twenty-nine thousandths. Twenty-nine thousands. 22. Five hundredths. Five hundreds. 23. Forty hundredths. Forty hundreds or four thousands. 24. Nine tenths of thousandths. 25. Nineteen tenths of thousandths. 26. Nine hundred tenths of thousandths. 27. Two thousand tenths of thousandths. 28. Two thousand and two tenths of thousandths. 29. Three thousand three hundred tenths of thousandths. 30. Thirty-two hundred tenths of thousandths. 31. Six tenths of thousandths. 32. Four hundredths of thousandths. 33. Fourteen hundredths of thousandths. 34. Four hundred hundredths of thousandths. 35. Two thousand and six hundredths of thousandths. 36. Sixty-four thousand hundredths of thousandths. 37. Sixteen thousand and four hundredths of thousandths. 38. Four thousand and nine hundredths of thousandths. 39. Six hundredths of thousandths. 40. Five thousand and four hundredths of thousandths. 41. Sixty-five thousand hundredths of thousandths. 42. Nine hundred and one hundredths of thousandths. 43. Twenty-nine hundred hundredths of thousandths. 44. Twelve tenths of thousandths. 45. Fifteen hundredths. 46. Sixty-four thousandths. 47. Nine hundred and one tenths of thousandths. Decimals can be read in two different ways. Thus,,21 can be read, either as two tenths, and one hun. dredth ; or as twenty-one hundredths. This can best be illustrated, by the coin of the United States. Thus, 2 dimes, 1 cent, can be read, either as twenty-one cents, or as two dimes and one cent. Thus again, 1 dollar, 3 dimes, and 2 cents, can be call. ed, either 132 cents; or 13 dimes, 2 cents; or 1 dollar, 3 dimes, and 2 cents. In like manner, decimals may be read in different ways. Thus, 234 can be read either as 234 thousandths; or 2 tenths, 3 hundredths, and 4 thousandths ; or 23 hun. dredths, and 4 thousandths; or 2 tenths, and 34 thou. sandths. Note. Let the teacher illustrate the above with coin. EXERCISES. Write two tenths. Ans. There are ten times as many hundredths as there are tenths in a thing. Therefore ,2 is ten times as many hundredths, or 20. Is there any difference in the value of ,2 and ,20 ? What is the difference between them? Ans. The ,20 has ten times more pieces, and each piece is ten times smaller than the ,2 ; but there is no difference in the value. ,3 is how many hundredths ? ,4 is how many hun. dredths ? ,30 is how many tenths ? ,40 is how many tenths ? Write two tenths, and four hundredths. In this sum how many hundredths ? Write thirty-four hundredths. In this sum how many tenths ? Write 2 tenths, 6 hundredths, or twenty-six hundredths. Write 4 tenths, 9 hundredths, and read it both ways. Write 6 tenths, 7 hundredths, 5 thousandths, or six hun. dred and seventy-five thousandths. Write 6 tenths, 4 hundredths, and 5 thousandths. Write nine tenths, six hundredths, and six thousandths, and read them both ways. Write seven tenths, six hundredths, five thousandths, and nine tenths of thousandths, and read them both ways. Write nine tenths, no hundredths, six thousandths, no |