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An Improper Fraction is one whose numerator equals or exceeds its denominator; its value is never less than the unit, 1.
Thus,, 3 15 35 50 180 are improper fractions.
39 41 8 109 90
122. A Mixed Number is a number expressed by an integer and a fraction.
Thus, 41, 171, 93 are mixed numbers.
123. Since fractions indicate division, all changes in the terms of a fraction will affect the value of that fraction according to the laws of division.
If we modify the language of the General Principles of Division (86) by substituting the words numerator, denominator, and fraction, or value of the fraction, for the words dividend, divisor, and quotient, respectively, we shall have the following principles:
GENERAL PRINCIPLES OF FRACTIONS.
124. PRINCIPLES. -I. Multiplying the numerator multiplies the fraction, and dividing the numerator divides the fraction.
II. Multiplying the denominator divides the fraction, and dividing the denominator multiplies the fraction.
III. Multiplying or dividing both terms of the fraction by the same number does not alter the value of the fraction.
These three principles may be embraced in one general law.
GENERAL LAW. A change in the NUMERATOR produces a LIKE change in the value of the fraction; but a change in the DENOMINATOR produces an OPPOSITE change in the value of the fraction.
125. Reduction of fractions is the process of changing their form without altering their value.
126. To reduce fractions to their lowest terms.
A fraction is in its lowest terms when its numerator and denominator are prime to each other; that is, when both terms have no common divisor.
1. Reduce the fraction 48 to its lowest terms.
4. Ans. 34 = 13
SOLUTION. - Dividing both terms of a fraction by the same number does not alter the value of the fraction or quotient, (124, III); hence, we divide both terms of 48, by 2, both terms of the result, 24, by 2, and both terms of this result by 3. Since the terms of are prime to each other, the lowest terms of 48 are . We have, in effect, canceled all the factors common to the numerator and denominator.
SOLUTION. - We find the greatest common divisor of 48 and 60 (103) which is 12, and divide both terms of the fraction by it, thus performing the reduction at a single division.
RULE. Cancel or reject all factors common to both
numerator and denominator.
Divide both terms by their greatest common divisor.
11. Express in its simplest form the quotient of 189 divided by 273. Ans. 13
12. Express in its simplest form the quotient of 1344 divided by 1536. Ans. 7.
127. To reduce an improper fraction to a whole or mixed number.
1. Reduce 324 to a whole or mixed number.
324 = 324 ÷ 15 = 214%
SOLUTION. Since 15 fifteenths equal 1, 324 fifteenths are equal to
as many times 1 as 15 is contained times in 324, which is 21 times. Or, since the numerator is a dividend and the denominator a divisor (118), we reduce the fraction to an equivalent whole or mixed number, by dividing the numerator, 324, by the denominator, 15. Divide the numerator by the denominator.
1. When the denominator is an exact divisor of the numerator, the result will be a whole number.
2. In all answers containing fractions reduce the fractions to their lowest terms.
2. Change 13 of a week to a mixed number.
3. In 117 of a bushel, how many bushels are there? 4. In 461 of a dollar, how many dollars are there? 5. In 872 of a pound, how many pounds are there? 6. Reduce 1258 to a mixed number.
7. Reduce 738 to a whole number.
4 184, Ans.
8. Change 1512 to a mixed number.
9. Change 7321 to a mixed number. 10. Change 237040 to a mixed number. 11. Change 2531520 to a whole number.
128. To reduce a whole number to a fraction
having a given denominator.
1. Reduce 46 yards to fourths.
SOLUTION. Since in 1 yard there are 4 fourths, in 46 yards there are 46 times 4 fourths, which are 184 fourths =184.
RULE. Multiply the whole number by the given denominator; take the product for a numerator, under which write the given denominator.
A whole number is reduced to a fractional form by writing 1 under it for a denominator; thus, 9 = f.
2. How many eighths of a bushel are there in 25 bushels? Ans. 200.
3. How many fourths of a gallon are there in 63 gallons? Ans. 252.
4. Reduce 140 pounds to sixteenths of a pound. 5. How many tenths of a dollar are there in 56 dollars? Ans. 560. 6. Reduce 94 to a fraction whose denominator is 9. 7. Reduce 180 to seventy-fifths.
8. Change 42 to the form of a fraction.
9. Change 247 to the form of a fraction. 10. Change 347 to a fraction with a denominator of 14. Ans. 4858.
129. To reduce a mixed number to an improper fraction.
1. In $53, how many eighths of a dollar are there?
8 43, Ans.
SOLUTION. Since in 1 dollar there are 8 eighth
in 5 dollars there are 5 times 8 eighths, or 40 eighths, and 40 eighths + 3 eighths = 43 eighths, or 43.
RULE. Multiply the whole number by the denominator of the fraction; to the product add the numerator, and under the sum write the denominator.
2. Change 41 dollars, to half dollars.
3. Change 714 weeks, to sevenths of a week.
6. Change 56 to an improper fraction.
PRAC. AR. -7
7. Reduce 217 to an improper fraction. Ans. 1267. 8. Reduce 2251 to an improper fraction. Ans. 5639. 9. Change 96,42% to an improper fraction. 120 10. Change 1297 to eighty-fourths. Ans. 108951, 11. What improper fraction will express 40039 ?
130. To reduce a fraction to a given denominator.
As fractions may be reduced to lower terms by division, they may also be changed to higher terms by multiplication; and all higher terms must be multiples of the lowest terms (105).
1. Reduce to a fraction whose denominator is 20.
20 ÷ 4 5
SOLUTION. First we divide 20, the required denominator, by 4, the denominator of the given fraction, to ascertain if it is a multiple of this term, 4. The division shows that it is a multiple, and that 5 is the factor which must be employed to produce this multiple of 4. We therefore multiply both terms of by 5 (124), and obtain 15, the desired result.
3 x 5 15 4 x 5 20'
RULE. Divide the required denominator by the denominator of the given fraction, and multiply both terms of the fraction by the quotient.
2. Reduce to a fraction whose denominator is 15.
Ans. f to a fraction whose denominator is 35. to a fraction whose denominator is 51. Ans. 36 to a fraction whose denominator is 150. 6. Reduce 125 to a fraction whose denominator is
Ans. 1000 7. Reduce to a fraction whose denominator is