69. A fraction is usually expressed by writing the num bers one above the other with a short horizontal line between them. Thus, three-fourths is written; two-thirds, , etc.

70. The number below the line is called the Denominator, and shows into how many equal parts the unit is divided.

71. The number above the line is called the Numerator, and shows how many of the equal parts are taken. Thus, in the denominator, 5, shows that the unit is divided into 5 equal parts; and 4, the numerator, shows that 4 of these parts are taken.

72. The numerator and the denominator are called the Terms of the fraction.

EXERCISE.

Read-,,, 3, 1, 1, 1, 4, 3, 10.

Write one-third, two-fourths, three-ninths, six-sevenths, five-sevenths, three-eighths, five-ninths, seven-tenths, thirteen-fifteenths, sixteen-fortieths.

73. A Common Fraction is one in which the numerator and the denominator are both expressed by figures.

74. Common fractions are either proper or improper. 75. A Proper Fraction is one whose numerator is less than its denominator; as,,, etc.

76. An Improper Fraction is one whose numerator is equal to or greater than its denominator; as,,,, etc.

77. A Mixed Number is one consisting of a whole number and a fraction.

SECTION II.

REDUCTION OF FRACTIONS.

CASE I.

To Reduce Whole or Mixed Numbers to Fractions.

ORAL EXERCISE.

1. How many halves in 1 apple? How many thirds? 2. How many fourths in an orange? How many fifths? 3. How many halves in 3 apples?

SOLUTION.-In 1 apple there are 2 halves, and in 3 apples 3

SOLUTION.-In 1 there are 4; in 5 there are 5 times, or 20;

20 and are 23. Hence, in 52 there are 23.

WRITTEN PROBLEMS.

Ex. Reduce 174 to an improper fraction.

SOLUTION.

171 3

52 thirds=52.

Explanation. In 1 there are, and in 17 there

are 17 times, or 51; 51+1=52.

From the foregoing we derive the rule for reducing mixed numbers to improper fractions:

RULE.

Multiply the whole number by the denominator, and to this add the numerator, writing the product over the given denominator.

Reduce to improper fractions

SOLUTION. Since one equals, in 15 there are as many ones as

SOLUTION. Since one equals, in there are as many ones as

is contained times in §, which is 2 times and remaining, or 2}.

7. Reduce to a mixed number. 8. Reduce 27 to a mixed number. 9. Reduce 45 to a mixed number. 10. Reduce to a mixed number.

WRITTEN PROBLEMS.

Ex. Reduce to a mixed number.

SOLUTION.

8)63

77

Explanation. Since there are in 1, in there are as many ones as is contained times in 3, which is 7, and remaining, or 73.

From the foregoing we derive the

RULE.

Divide the numerator of the improper fraction by the denominator; the quotient is the whole or mixed number. Reduce to whole or mixed numbers

SOLUTION. Since there are § in 1, in } there are } of %, or 3.

2. How many fourths in ?

12.

6. How many twelfths in ? In ? In ?

7. How many twelfths in ?

SOLUTION.-In there are, and in there are 3 times, or

8. How many twelfths in ? In ?

9. How many fifteenths in ? In ? In ?

10. How many twentieths in ? In ? In ?

PRINCIPLE.—Multiplying both numerator and denominator of a fraction by the same number does not change the value of the fraction. Thus, equals, which is the same as multiplying by 3.

WRITTEN PROBLEMS.

1. Reduce to twentieths.

SOLUTION. ×8-18

Explanation. To reduce fourths to twentieths it is necessary to multiply the denominator by 5; but to preserve the value of the fraction the numerator must also be multiplied by the same number. Hence, 3=18.

15

From the foregoing we derive the rule for reduction of fractions to higher terms.

RULE.

Multiply both numerator and denominator by that number which will produce the required denominator.

SOLUTION. Since, there are as many thirds in as is contained times in, or 3. Hence, in there are 3.