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-54. What part of a rod is 4 of an inch!

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Remark.-The year is estimated at 365 days in the last question and in the 59th.

56. If a person draw 3 pints from a hbd. of wine, what part of the hid. does he take away !

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58. What part of a week is 1 day and 5 hours?

Explanation.—The denominator being equal to an unit 1 of that denomination to which the fraction belongs, 1 of that denomination reduced to the lowest denomination contained in the given numbers, must be made the denominator, and the given numbers reduced to the same denomination, a numerator.

To express the whole time mentioned in the last question in hours, the one day must be reduced to hours, and 5 hours added. (See note page 119.) One day and 5 hours equal 29 hours. To know what part of a week 29 hours are, one week must be reduced to hours and made the denominator of the fraction. One week equals 168 hours, and 29 hours must be or of a week.

59. What fraction of a year is 4 minutes? * Ans. Fä Prst60. What part of a pound is 2 s. 6 d. 2 Ans. #% =}. 61. What part of a yard is 3 qi. 1 na. ? - Ans. #. 62. A merchant sells 43 yards from a piece of cloth measuring 31 yards. What part of the piece does he sell ? Ans. For. 63. If the moon pass through 13 ° 10' 30" of her orbit in one day, what part of her revolution does she perform in the same time? Ans. T$####. 64. A man owning 1 A. 3 R. 31 r. of land, laid out 56 r. of it for a garden. What part of the whole is the garden 2 Ans. #'r.

To find a common denominator.

Fractions have a common denominator when the denominators are all equal to each other. #, #, F's, are fractions having a common denominator; but #, or, and for, have not a common denominator.

RULE,

Multiply all the denominators together, the product will be a common denominator. Multiply each numerator by all the denominators except its own ; the product written over the common denominator will form a fraction expressing the same part of an unit, as the fraction whose numerator was multiplied.

1. Change #, #, # to fractions having a common denominator.

Operation.
1. 3 2.
2 4 3
— 4
2 —
4 12
3 — 2
1 3 8 - *
4 9 2 24 common denominator.
4 2 16 new numerator for 3–44.
3 —

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The same fractions reduced to a common denominator ##, ##, ##.

Illustration.—When all the denominators are multiplied together, the product is a denominator expressing an unit divided into a greater number of parts; and multiplying each numerator by all the denominators except its own, the product will be as much less than the product of the denominators, as that numerator is less than its own denominator, and will be the same part of the common denominator, as the given numerator is of the given denominator.

Let it be required to change , and # to fractions having a common denominator. To obtain the new denominator, we multiply the 3 by 4, and the product is 12. To obtain a new numerator for , we Inultiply its numerator by 4, increasing it by the same number that we did the denominator, and the result is so. To change #, we multiply both denominator and numerator by 3, and the result is #. 4 is the same part of 12 that 1 is of 3, and 3 is the same part of 12 that 1 is of 4; therefore, 1% =}, and

4.

The scholar will perceive by examining the operations of the following examples, that when more than two fractions are given, multiplying as directed by the rule, we increase the numerator and denominator of the same fractions, by the same numbers.

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mon denominator. Ans. #4, and ***. 4. Change #4 and 2011 to fractions having a common denominator. Ans. ### and 204''.

Mote 4.—Mixed numbers may be brought to improper fracto before the operation, or the whole number omitted and set o: the same fraction after the operation, as the nature of the queue may require.

5. If on and # of a dollar be expressed by parts of a dollar equally divided, what fractions will ex

press both quantities 2 Ans. #4, and #. 6. A owns 3 of a stage and Dio, ; which owns the greatest part? Ans. A.

Explanation,--After a common denominator is found, the numerators will show which is the greatest fraction.

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Illustration.—When a single thing of any kind is divided into several equal parts, or twe equal things are so divided, any number of those parts may be added together in the same manner as simple numbers. If a hogshead of wine be divided into 63 equal parts, and one man have 9 of those parts, and another 8, both would have 17 parts, the sum of 9 and 8; or, in other words, both men would possess 17 gal. of wine; for, as 63 gal. equal a há. Ho part of a hhd. is one gal. The same expressed in a fractional form is thus, os-Ho-Ho, adding the numerators only. It is necessary to write the common denominator under those parts owned by each man. to show that the parts owned by one, are just as large as those owned by the other. But if we suppose one hid. of wine divided into 63 equal parts, and one man to possess 9 of these parts, and anotherhhd. divided into 4 equal parts, and another person to possess one of those parts, we could not add the parts owned by both persons as they stand, since 1 quarter of a hind., or #, is larger than 1 gal., or go. Let the fractions expressing the parts owned by both persons, or and 4, be reduced to a common denominator, they will be os and #. After the fractions are reduced to a common denominator, each hd. is supposed di

vided into 252 equal parts, and the person who owns + of a ha. will own 63 of 252 parts, (as may be seen by dividing 252, which denotes the whole hind. by 4,) and the other 36 of the 252 equal parts. Each fractional unit in 36 and 63, expresses the same part of a hbd., 444, therefore, the sum of 36 and 63 will denote the number of parts owned by both persons.

We may have two or more fractions of equal denominators which cannot be added together until they are changed in certain respects. We cannot add + of a pound and 4 of a shilling together, for their sum #, would be neither 3 of a pound, nor 4 of a shilling. Let the 4 of a shilling, be brought to the fraction of a pound. (See Explanation to question 52d, page 152.) and it is rig of a pound; then 4 and r}w are fractions of a pound, and may be added after they are reduced to a common denominator. Several fractions of the same, or of equal units, having equal denominators, express as many fractional units of the same unit or quantity, as there are ones in all the numerators; therefore, the sum of the numerators written over the common denominators, will express the same part of an unit as is denoted by the given fractions.

RULE. After preparing the fractions as explained in the

illustration, add the numerators and write the sum over one of the denominators.”

1. How many eighths in #, ; and #! Ans. 6. 2. What is the sum of for, so and #2 Ans. #. 3. What is the sum of #, # and 63% Ans. 64. 4. What is the sum of Ps, or and #} } Ans. 21. 5. What is the sum of ; and #! Ans. }}.

* All fractions in answers after this, will be reduced to their lowest terms.

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