REDUCTION OF VULGAR FRACTIONS.

I. To find the greatest common measure or divisor of any given fraction, or of any two given numbers.

The greatest common measure of two or more numbers is the greatest number that will divide each of them without a remainder.

Any number ending with an even number or 0, may be divided by 2.

Any number ending with 5 or 0, can be divided by 5.

Any number may be divided by 3 or 9 if the sum of its digits be so. Thus, 369 can be divided by 3, for 18, the sum of 3, 6, and 9, is so. And also by 4, and 25, if the two last digits of the sum be so. And by 8 if the four last digits of the sum be so.

When the sums of the alternate digits of any number are equal, the number can be divided by 11.

RULE. Divide the greater number by the less, and the divisor by the remainder till nothing remains. The divisor which produces no remainder is the greatest common measure required.

1. What is the greatest common measure of 234?

234)540(2
468

72)234(3
216
18)72(4

72

65

2. Required the greatest common measure of 2.
3. What is the greatest common measure of
4. What is the greatest common measure of 992?
5. Find the greatest common measure of

5256

458 ?

Ans. 18

18)234(18

Ans. 12.

Ans. 5.

Ans. 8.

Ans. 18.

To find the greatest common measure of more than two numbers proceed as before, then find the greatest common measure of the third number by dividing it by the preceding common measure, &c., the greatest common measure last found will give the answer.

6. Find the greatest common measure to 432, 816, 1480. 816-432, gives 48 their greatest common measure; then 1480 48 gives 8, the last greatest common mea

sure:

}

7. Find the greatest common measure to 504, 1188, 2604.

II. To reduce a fraction to its lowest terms.

Ans. 8.

432 54 8 816 102 1480 185

Ans. 12.

RULE. Divide the numerator and the denominator by any number that will measure both without a remainder, and thus proceed with them as long as any number will divide them.-Or divide the numerator and denominator by the greatest common measure, as in the foregoing rule.

III. To reduce a whole number to a fraction of any denomination.

RULE. A whole number may become a fraction by making unity its denominator, as =3; or a whole number may have any denominator, provided it be multiplied by the given denominator.

14. Reduce 5 to a fraction whose denominator shall be 7.

5x7=35. Ans.

Ans. 28.
Ans. 5.

15. Reduce 7 to a fraction whose denominator shall be 4. 16. Reduce 9 to a fraction having 5 for its denominator. 17. Reduce 16 to a fraction, having 25 for its denominator. Ans. 100.

IV. To reduce a mixed number to an improper fraction. RULE. Multiply the whole number by the denominator of the fraction, and to the product add the numerator for a new numerator, which place over the denominator.

To express a whole number fraction-ways, set one for the denominator given. 18. Reduce 18 to an improper fraction.

18x7+3=129, new numerator.

19. Reduce 5613 to an improper fraction. 20. Reduce 183 to an improper fraction. 21. Reduce 13 to an improper fraction. 22. Reduce 273 to an improper fraction.

Ans. 129.

23. Reduce 514 to an improper fraction.

Ans. 8229.

V. To reduce an improper fraction to its proper terms.

RULE. Divide the upper term by the lower.

Ans. 18.

24. Reduce 129 to its proper terms.

129-7=183.

25. Reduce 1245 to its proper terms. 26. Reduce 318 to its proper terms. 27. Reduce to its proper terms. 28. Reduce 245 to its proper terms. 29. Reduce 8229 to its proper terms.

VI. To reduce a compound fraction to a single one.

RULE. Multiply all the numerators for a new numerator, and all

the denominators for a new denominator, reducing whole or mixed numbers to improper fractions, according to rule.

Reduce the new fraction to its lowest terms by RULE 2.

30. Reduce of of § to a single fraction.

If the same figures occur both in the numerator and denominator, they may be erased from each, or divide any two of them by their greatest common divisor. 35. Reduce of of 2 of 3 of § of } to a single fraction.

Ans.

36. Reduce of of § of of 18 of 2 to a single fraction.

Ans..

VII. To reduce a fraction to another of the same value, having the numerator given of the required fraction.

RULE. As the numerator of the given fraction is to its denominator, so is the numerator of the intended fraction to its denominator.

37. Reduce to a fraction of the same value, whose numerator shall be 12. As 23 12: 18.

Ans.

38. Reduce to a fraction of the same value, whose numerator shall

be 25.

Ans.

39. Reduce to a fraction of the same value, whose numerator shall be 47.

47

Ans.

65

VIII. To reduce a fraction to another of the same value, having the denominator given of the fraction required,

RULE. As the denominator of the given fraction is to its numerator, so is the denominator of the intended fraction to its numerator.

40. Reduce to a fraction of the same value, whose denominator shall be 18. As 32 18: 12 Ans. 1. 41. Reduce to a fraction of the same value, whose denominator shall be 35. Ans. 42. Reduce to a fraction of the same value, whose denominator shall be 65.

IX. To reduce fractions to a common denominator.

47

Ans.

65

RULE. Multiply each numerator into all the denominators, except its own, for the new numerators; and multiply all the denominators together for the common denominator.

43. Reduce and to a common denominator.

1st num. 2x7=14,

2nd num.

4x4=16, then 4×7-28 den.

44. Reduce,, and, to a common denominator. 45. Reduce,,, and, to a common denominator. Ans. 2940

Ans. and 18.

and Ans.,,.

3024 2880

33609 3340, 3960, 3360.

46. Reduce,,, and to a common denominator.

720 Ans. 864

240 720

1440 1440 1440 1440

47. Reduce 3, 3, 4, and, to a common denominator.

Ans. 672, 560 360 105

48. Reduce, 3, 3, and 3 to a common denominator.

Ans. 720 1200 540 1296

2160 2160 2150 2160

X. To reduce a mixed or complex fraction to a simple one.

RULE. Multiply the upper and lower term by the denominator of the fractional part, if there be only one, as 4 thus, 2x4+1=9; then

9

2x9=18=; Or, first by one denominator, and then by another, when more than one. Then reduce the result to its lowest terms.

XI. To reduce fractions of one denomination to another, but retaining the same value.

RULE. If from a less to a greater denomination, multiply the denominator by all the denominations from the one given to the one sought. Thus, a farthing reduced to the fraction of a £ would be of of of a £.-Or if the fraction is to be reduced from a greater to a less denomination, multiply the numerator by all the denominations, from the one given to the one sought, for a new numerator, and place it over the given denominator; then reduce to the lowest terms.

53. Reduce of a penny to the fraction of a pound.

of of 1920 Ans.

54. Reduce of a penny to the fraction of a pound. 55. Reduce of a dwt. to the fraction of a lb. Troy.

Ans. 960 Ans. 1200

1

Ans. 17800.

57. Reduce of a lb. Avoirdupois to the fraction of a cwt. Ans. 7.
of a yd. to the fraction of a mile.
of a second to the fraction of an hour.
of a pound to the fraction of a penny.

Ans.

9

39600⚫

Ans.

7×20×12=1680, 1858 reduced to its lowest term=}. Ans.
of a pound to the fraction of a penny.
of a lb. Troy to the fraction of a pennyweight.

58. Reduce 59. Reduce 60. Reduce

61. Reduce

62. Reduce

63. Reduce

64. Reduce 8 of a mile to the fraction of a yd.
17600
9
65. Reduce of an hour to the fraction of a second
39600

Ans. .

of a cwt. Avoirdupois to the fraction of a lb. Ans. 4.

Ans.

Ans.

XII. To find the value of a fraction, in the known parts of an integer. RULE. Multiply the numerator by the number of parts in the integer, and divide the product by the denominator. If a remainder, multiply it by the next inferior denomination, and divide the product by the denominator, as before.

66. Reduce of a pound sterling to its proper quantity.

3x20=60, and 4-15s. Ans.
Ans. 4 d..
Ans. £4 8s. 8d.

of two shillings. Ans. 8d.
Ans. 16s.
Ans. 6s.

67. Reduce of a shilling to its proper quantity.
68. What is the amount of of £7 15s. 2d.
69. Show that of a shilling is equal to
70. Show that of a £ is equal to of £4.
71. Show that of 14s. is equal to of 3 times 14s.
72. What is the value of 1 of £2.
Ans. £1 2s. 43d. }.
73. Reduce of a lb. Troy to its proper quantity. Ans. 7 oz. 4 dwts.
74. Reduce of a lb. Avoirdupois to its proper quantity.

75. Reduce 3 of a cwt. to its proper quantity.

Ans. 9 oz. 22 drs.

Ans. 3 qrs. 3 lb. 1 oz. 12 dr. 76. Reduce & of an ell English to its proper quantity.

Ans. 2 qrs. 3 nails. 77. Reduce of a mile to its proper quantity. Ans. 6 furl. 16 poles. 78. Reduce of an acre to its proper quantity. Ans. 2 roods, 20 poles. 79. Reduce of an hogshead of wine to its proper quantity.

80. Reduce of a barrel of beer to its proper quantity.

Ans. 54 gallons.

Ans. 12 gallons.

81. Reduce of a chaldron of coals to its proper quantity. 82. Reduce of a month to its proper time.

Ans. 15 bushels.

Ans. 2 weeks, 2 days, 19 hours.

XIII. To reduce money, weights, ana measures, to fractions.

RULE. Reduce the given quantity to the lowest term mentioned for a numerator, under which set the integral part (reduced to the same term) for a denominator, and it will give the fraction required.