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16. Supposing a meteor to appear so high in the heavens as to be visible at Boston, 71° 3', at the city of Washington, 77° 43', and at the Sandwich Islands, 1500 W. longitude, and that its appearance at the city of Washington be at 7 minutes past 9 o'clock in the eve
what will be the hour and minute of its appearance at Boston and at the Sandwich Islands ?
FRACTIONS. T 43. We have seen, (T 17,), that numbers expressing whole things are called integers, or whole numbers; but that, in division, it is often necessary to divide or break a whole thing into parts, and that these parts are called fractions or broken numbers.
It will be recollected, (14, ex. 11,) that when a thing or unit is divided into 3 parts, the parts or fractions are called thirds ; when into four parts, fourths ; when into six parts, sixths ; that is, the fraction takes its name or denomination from the number of parts into which the unit is divided. Thus, if the unit be divided into 16 parts, the parts are called sixteenths, and 5 of these parts would be 5 sixteenths, expressed thus, The number below the short line, (16,) as before taught, (ST 17,) 'is called the denominator, because it gives the name or denomination to the parts; the number above the line is called the numerator, because it numbers the parts.
The denominator shows how many parts it takes to make a unit, or whole thing; the numerator shows how many of these parts are expressed by the fraction.
1. If an orange be cut into 5 equal parts, by what fraction is 1 part expressed? 2 parts ? 3 parts ? 4 parts? 5 parts ? how many parts make unity or a whole orange ?
2. If a pie be cut into 8 equal pieces, and 2 of these pieces be given to Harry, what will be his fraction of the pie ? if 5 pieces be given to John, what will be his fraction ? what fraction or part of the pie will be left ?
It is important to bear in mind that fractions arise from division, (T 17,) and that the numerator may be considered a dividend, and the denominator a divisor, and the value of the fraetion is the quotient ; thus, £ is the quotient of 1 (the numerator) divided by 2, (the denominator;) is the quotient arising from 1 divided by 4, and is 3 times as much, that is, 3 divided by 4; thus, one fourth part of 3 is the same as 3 fourths of 1.
Hence, in all cases, a fraction is always expressed by the sign of division.
3 is the dividend, or numerator. expresses the quotient, of which
4 is the divisor, or denominator. 3. If 4 oranges be equally divided among 6 boys, what part of an orange is each boy's share ?
A sixth part of 1 orange is 4, and a sixth part of 4 oranges is 4 such pieces, =
Ans. of an orange. 4. If 3 apples be equally divided among 5 boys, what part of an apple is each boy's share 1 if 4 apples, what? if 2 apples, what? if 6 apples, what?
5. What is the quotient of 1 divided by 3? of 2 by 3?
of 1 by 4? of 2 by 4? of 3 by 4? of 5 by 7 ? of 6 by 8 ? of 4 by 5?
of 2 by 14 ?
6. What part of an orange is a third part of 2 oranges ? one fourth of 2 oranges
4 of 3 oranges ? oranges ?
of 2 ?
+ of 5 ?
* of 3?
of 4 ? of 'of 2?
A Proper Fraction. Since the denominator shows the number of parts necessary to make a whole thing, or 1, it is plain that, when the numerator is less than the denominator, the fraction is less than a unit, or whole thing; it is then called a proper fraction. Thus, H: , &c. are proper fractions.
An Improper Fraction. When the numerator equals or exceeds the denominator, the fraction equals or exceeds unity, or 1, and is then called an improper fraction. Thus, b, t, f, ye, are improper fractions.
A Mixed Number, as already shown, is one composed of a whole number and a fraction. Thus, 145, 131, &c. are mixed numbers.
7. A father bought 4 oranges, and cut each orange into 6 equal parts; he gave to Samuel 3 pieces, to James 5 pieces, to Mary 7 pieces, and to Nancy 9 pieces ; what was each one's fraction ?
Was James' fraction proper or improper? Why?
To change an improper fraction to a whole or mixed number. 944. It is evident that every improper fraction must contain one or more whole ones, or integers. 1. How many whole apples are there in 4 halves (1) of an apple ? in 6 ? in 8? in 10?
in 4,8 ?
in 934 1
2. How many yards in
in ? in ?
of a yard ?
in 10? in 20?
in 17 ?
in 31 ?
in 15 ? 3. How many bushels in 8 pecks ? that is, in g of a bushel ? -in 10? in u? in 43 ?
in 24? in
12e 2 ? This finding how many integers, or whole things, are contained in any improper fraction, is called reducing an improper fraction to a whole or mixed number.
4. If I give 27 children of an orange each, how many oranges will it take?
It will take 47; and it is evident that di
viding the numerator, 27, (= the number of 4) 27
parts contained in the fraction,) by the deno.
minator, 4, (= to the number of parts in 1 Ans. 6 oranges.
orange,) will give the number of whole or. anges.
Hence, To reduce an improper fraction to a whole or mixed numiber,-Rule: Divide the numerator by the denominator ; tient will be the whole or mixed number.
EXAMPLES FOR PRACTICE. 5. A man, spending of a dollar a day, in 83 days would spend one of a dollar ; how many dollars would that be? 6. In 14 of an hour, how many whole hours ?
The both part of an hour is 1 minute; therefore, the question is evidently the same as if it had been, In 1417 minutes how many hours?
Ans. 2337 hours. 7. In 8783 of a shilling, how many units or shillings ?
730 shillings. 8. Reduce 146 7 8 to whole or mixed number. Reduce jo, 2006, 177, 1786, 34606, to whole or mixed numbers.
To reduce a whole or mixed number to an improper fraction. 145. We have seen that an improper fraction may be changed 0 a whole or mixed number, and it is evident that, by reversing the operation, a whole or mixed number may be changed to the form of an improper fraction.
1. In 2 whole apples, how many halves of an apple? Ans. 4 halves; that is, . In 3 apples how many halves ? in 4 apples ? in 6 apples ? in 10 apples? in 24 ? in 60 ? in 170 ? in 492 ? • 2. Reduce 2 yards to thirds. Ans... Reduce 27 yards to thirds. Ans. . Reduce 3 yards to thirds. 3} yards. -33 yards. 5 yards.
3. Reduce 2 bushels to fourths. bushels. 64 bushels. -74 bushels. 4. In 5 dollars, how many 12
of a dollar ? * make 1 dollar; if, therefore, we multiply 16 by 12, that is, multiply the whole number by the denominator, the product will be the number of 12ths in 16 dollars; 16 X 12=192, and this, increased by the numerator of the fraction, (5,) evidently gives the whole number of 12ths ; that is 192 of a dollar, Ans. OPERATION, 16_5. dollars.
6ş yards. -2 bushels.
192 : 12ths in 16 dollars, or the whole number.
5 12ths contained in the fraction. 197 =
197, the answer. Hence, To reduce a mixed number to an improper fraction, Rule: Multiply the whole number by the denominator of the fraction, to
he product add the numerator, and write the result over the denominator.
EXAMPLES FOR PRAOTICE. 5. What is the improper fraction equivalent to 232 7 hours ?
1407 of an hour, 6. Reduce 730_8 shillings to 12ths. As
of a shilling is equal to 1 penny, the question is evidently the same as, In 730 s. 3 d., how many pence?
of a shilling; that is, 8763 pence. 7. Reduce 116, 1746, 870655, 47086, and 73 16, to improper fractions. 8. In 15617 days, how many 24ths of a day?
Ans. 3761=3761 hours. 9. In 3424 gallons, how many 4ths of a gallon?
Ans. 1371 of a gallon =1871 quarts. To reduce a fraction to its lowest or most simple terms. - 746. The numerator and the denominator, taken together, are called the terms of the fraction.
If ļ of an apple be divided into 2 equal parts, it becomes 4. The effect on the fraction is evidently the same as if we had* multiplied both of its terms by 2. In either case, the parts are made 2 times as many as they were before; but they are only HALF AS LARGE; for it will take 2 times as many fourths to make a whole one as it will take halves; and hence it is that is the same in value or quantity as t. 2
is 2 parts; and if each of these parts be again-divided into 2 equal parts, that is, if both terms of the fraction be multiplied by 2, it becomes . Hence = =, and the reverse of this is evidently true,
that It follows, therefore, by multiplying or dividing both terms of the fraction by the same number, we change its terms without altering its value.
Thus, if we reverse the above operation, and divide both terms of the fraction by 2, we obtain its equal, f ; dividing again by 2, we obtain , which is the most simple form of the fraction, because the terms are the least possible by which the fraction can be expressed.
The process of changing into its equal, t, is called reducing the fraction to its lowest terms. It consists in dividing both terms of the fraction by any number which will divide them both without a remainder, and the quotient thence arising in the same manner, and 80 on, till it appears that no number greater than I will again divide them.
A number which will divide two or more numbers without a remainaer, is called a common divisor, or common measure of those
numbers. The greatest number that will do this, is called the greatest common divisor.
1. What part of an acre are 128 rods?
Let us reduce this fraction to its lowest terms. We find, by trial, that 4 will exactly measure both 128 and 160, and, dividing, we change the fraction to its equal, 28. Again, we find that 8 is a divisor common to both terms, and, dividing, we reduce the fraction to its equal, , which is now in its lowest terms, for no greater number than ! will again measure them. The operation may be presented thus : 4) 128 8)
of an acre, Answer. 2. Reduce 958,
to their lowest terms.
Ans. , }, }, and ļ. Note. If any number ends with a cipher, it is evidently divisible by 10. If the two right hand figures are divisible by 4, ihe whole number is also. If it ends with an even number, it is divisible by 2; if with a 5 or 0, it is divisible by 5.
3. Reduce 188, 5, 195, and to their lowest terms.
147. Any fraction may evidently be reduced to its lowest terms by a single division, if we use the greatest common divisor of the two terms. The greatest common measure of any two numbers may be found by a sort of trial easily made. Let the numbers be the two terms of the fraction 14.. The common divisor cannot " exceed the less number, for it must measure it. We will try, therefore, if the less number, 128, which measures itself, will also divide or measure 160.
128 in 160 goes 1 time, and 32 remain; 128, 128 ) 160 ( 1
therefore, is not a divisor of 160. We will now 123
try whether this remainder be not the divisor
sought ; for if 32 be a divisor of 128, the for32 ) 128 ( 4 mer divisor, it must also be a divisor of 160, which consists of 128 + 32.
32 in 128 goes 4 times, without any remainder. Consequent
ly 32 is a divisor of 128 and 160. And it is evidently the greatest common divisor of these numbers; for it must be contained at least once more in 160 than in 128, and no number greater than their difference, that is, greater than 32, can do it.
Hence the rule for finding the greatest common divisor of troo numbers : Divide the greater number by the less, and that divisor by the remainder, and so on, always dividing the last divisor by the last remainder, till nothing remain. The last divisor will be the greatest common divisor required.
Note. It is evident, that, when we would find the greatest common divisor of more than two numbers, we may first find the greatest common divisor of two numbers, and then of that common divi