Reduction of improper fractions to integers or mixed numbers.

INDUCTIVE EXERCISE.

1. How many fourths of a dollar are contained in a whole dollar? Hence, how many whole dollars in of a dollar? In of a dollar? In 20 of a dollar? In 2 of a dollar? In 32 of a dollar?

2. How many eighths of a pound are contained in a whole pound? many whole pounds in of a pound? In 13 of a pound? In In 27 of a pound? In 45 of a pound? 3. How many gallons in In of a gallon? In of a gallon? In

of a gallon? In

Hence, how of a pound?

of a gallon? In 13 of a gallon? of a gallon? In 3 of a gallon?

Hence, how are improper fractions reduced to integers or mixed numbers, and why?

Reduction of fractions to higher terms of a given denominator.

INDUCTIVE EXERCISE.

1. How many sixteenths of a bushel are contained in a whole bushel? In of a bushel, and why? In of a bushel? In of a bushel? In of a bushel? In of a bushel? In of a bushel? In of a bushel ?

2. One-half of a gallon is equal to how many fourths of a gallon? To how many sixths of a gallon? To how many eighths of a gallon? To how many twelfths of a gallon?

3. Multiply both terms of $3 by 2, and what is the result? Is there any difference in value between this result and $? Why not?

[g] To find how many fractional units of lower value are equal to one fractional unit of higher value, divide the denominator which expresses the value of the lower fractional units by the denominator which expresses the value of the higher fractional units.

[h] To multiply both numerator and denominator of a fraction by the same number does not change the value of the fraction.

88. Reduce of a bushel to an equivalent fraction which will express fifty-seconds of a bushel.

89. How many sixty-fourths of a pound are equal to of a pound? 90. Reduce 185 gallons to an equivalent mixed number whose terminal fraction will express seventy-seconds of a gallon.

Reduction of fractions to lower terms of a given denominator.

INDUCTIVE EXERCISE.

1. How many halves of a dollar are contained in a whole dollar? How many fourths of a dollar? How much greater, therefore, is $ than $} ?

2. How many thirds of a bushel are contained in a whole bushel? How many twelfths of a bushel? How much greater, therefore, is } of a bushel than bushel ?

3. How many eighths of a gallon are equal to of a gallon? How many twelfths of a gallon? 4. How many sixteenths of a ton are equal to ton, therefore, are equal to of a ton? To 1 of a ton ?

of a

of a gallon? How many sixteenths
How many tenths of a gallon?
of a ton? How many eighths of a
of a ton? To g of a ton?
Το 16

[i] To divide both numerator and denominator of a fraction by the same number does not change the value of the fraction.

mixed numbers having 13 for a denominator.

107. Express of an hour, of an hour, 4 of an hour, and 4 of an hour by equivalent fractions denoting sixths of an hour.

108. Reduce 18,45 cords of wood and 1752 cords of wood to equivalent mixed numbers whose fractional units will express twentysevenths of a cord of wood.

Reduction of fractions to lowest terms.

INDUCTIVE EXERCISE.

1. In $, how many eighths of a dollar? Of the two fractions $ and $1, which has the lower terms? Are the terms of $ as low as they can be reduced? Why not? How many fourths of a dollar are contained in $ or $? Of the three fractions $, $3, and $3, which has the lowest terms? Are the terms of $ as low as they can be reduced? Why not? How many halves of a dollar are contained in $, $1, or $? Of the four fractions $1%, $1, $3, and $4, which has the lowest terms? Are the terms of $ as low as they can be reduced? Why? Hence, what is the equivalent fraction, in lowest terms, of $?

2. In the fraction $, what is the greatest common divisor of its terms 8 and 16? What is the result of dividing both terms of $ by this greatest common divisor? Is this result in its lowest terms; and why?

3. Reduce the following fractions to lowest terms: $1, $, $2, $1, $1, $1, $1, $, $1, $, $12.

[j] A fraction is in lowest terms when its numerator and denominator are prime to each other.

[k] Two numbers are made prime to each other through consecutive divisions by their common factors, or through one division by their greatest common divisor.

Reduction of two or more fractions to equivalents of a common denominator.

INDUCTIVE EXERCISE.

1. $} is equivalent to how many sixths? $ is equivalent to how many sixths? To what common denominator, therefore, may $ and $ be reduced?

2. How many twelfths of a dollar are equivalent to $? To $? $? To what common denominator, therefore, may $1, $3, and $ be reduced?

3. What is a common multiple of $6, $8, and $12? What, therefore, is a common denominator of $3, $3, and $; and why?

4. What is the least common multiple of $6, $8, and $15? What, therefore, is the least common denominator of ${, $}, and $31⁄2; and why?

5. Reduce the following to equivalent fractions having their least common denominator; $ and $3; of an apple and of an apple; of an orange and of an orange.

[7] A common denominator of two or more fractions is ANY common multiple of their several denominators.

[m] The least common denominator of two or more fractions is the LEAST common multiple of their several denominators.

129. 7 quarts, of a quart, 4 of a quart, and 6 quarts.

of a barrel, & of a barrel, of a barrel, and of a barrel.
of a cent, of a cent, and 1

130.

131.

of a cent,

132.

of a foot,

of a foot, and of a foot.

133.

of a task,

of a cent.

of a task, Ty of a task, and of a task. 134. 47 years, of a year, 1 of a year, and & of a year. 135. 142 hours, & of an hour, 3 of an hour, and 17 of an hour. 136. of a month, 1 of a month, and 23 of a month. 137. 5% days, 14 days, 351⁄2 days, and 1837 days. 138. 275 ounces, 3013 ounces, 194 ounces, and 8 139.of a journey, of a journey, & of a journey, and 140. 5 hours, 9 hours, and 8 hours.

ounces.

of a journey.

141. of an acre, 3 of an acre, 54 acres, and of an acre. 142. $17, $, $13, $18, and $13.

145. $3, $4, $3, $, $2, $8, and $3.

146. What is the least common fractional unit to which

of a mile,

§ of a mile, of a mile, of a mile, and of a mile can be reduced; and how many of these common fractional units will express the value of each fraction?

ADDITION OF FRACTIONS.

INDUCTIVE EXERCISE.

1. What part of a dollar are $1 + $}? Are $} + $? Are $3 + $? Are $$? Are $+$?

2. What part of a pound of sugar are of a pound + pound? Are of a pound + of a pound + of a pound? + of a pound + § of a pound?

of a pound + of a of a pound

Are

3. How many fractional units are expressed in of a gallon? What is the name of each fractional unit? How many fractional units are expressed in 4 of a gallon? What is the name of each fractional unit? Are thirds of a gallon and fourths of a gallon like fractional units? Can of a gallon and of a gallon be added as they stand? To what common denominator, or common fractional unit, may 3 of a gallon and of a gallon be reduced? How many twelfths of a gallon are equal to of a gallon? To of a gallon? Can of a gallon and of a gallon be added, and why? What is the sum, and what fractional units does it express ?