268

CHANGING TO MIXED NUMBERS

An integer is a whole number.

For example: 6, 23, 247, 1, and 10 are integers.

A mixed number is an integer and a fraction.

For example: 61, 73, 295 are mixed numbers.

A fraction whose value is less than 1 is called a proper fraction.

3

For example: 1, 2, 3, 7, are proper fractions.

A fraction whose value is equal to 1 or greater than 1 is called an improper fraction.

For example: 4. 5, 7, 10 are improper fractions.

From the above definition it is plain that a mixed number, if reduced to a fraction, becomes an improper fraction.

It is often necessary to change a mixed number to an improper fraction. The illustration and explanation below explain how this may be done.

How many halves in 1 unit? In 2 units? In 5 units?

How many halves in 5 units? How many halves does 5 equal?

To change a mixed number to an improper fraction, multiply the whole number by the denominator of the fraction. To that product add the numerator of the fraction.

The number thus obtained will be the numerator of the improper fraction. Its denominator will be the denominator of the fraction.

For example: Reduce 2 to an improper fraction.

The illustration below shows a whole unit divided into
thirds, and also an additional of a unit. How many thirds
do you see? What does equal?

What does equal? What does
equal?

It is plain, then, that an im-

proper fraction may be changed to a mixed number.

To change an improper fraction to a mixed number,
divide the numerator of the fraction by the denominator of
the fraction.

270

ADDING AND SUBTRACTING FRACTIONS

In order to add 6 quarts, 3 gallons, and 5 pints you must first change those quantities to a common unit of measure, as shown by A below. The common unit of measure is the pint.

In like manner, to add, §, and you must first change the fractions to a common unit, and then add, as shown by B below. The common unit, called the common denominator, is eighths.

To add fractions that have different denominators, first change them to fractions that have a common denominator.

The smallest number that will contain all the denominators of several fractions is called the least common denominator.

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=

=

In changing, 7, and to frac

tions having a common denominator, it is plain that 8 is the smallest number that will contain 2, 8, and 4, the denominators of the fractions. The illustration above shows

this. Therefore 8 is the least common denominator of the fractions

21

,, and . 3, 3.

Find the smallest number that will contain the denominators

of the fractions in each of the groups below.

12. Take and

33-1

of each of the following: 36 72 48 54

81 90
90 63 24 18 15 45
45 27

13. Take and 3 of each of the following: 24 80 60 44

28 32 56 48 72 96 36 16

7

of each of the following: 24 64

144 80 56 120

17. Take 3, 4, 5, 3, and 8 of each of the following: 27 63 36 90 54 18 72 99 99 108 45 9 180

272

MULTIPLICATION OF FRACTIONS

Find inch on your ruler. How many such parts are in 1 inch. Find of a half-inch. How many such parts are in 1 inch? What is each of these parts called? How much is of ?

Find of the fourth-inch on your ruler. How many such parts are in 1 inch? What is each of these parts called? How much is of ?

In the same way find the answers to these examples.

To multiply one fraction by another, multiply the numerators together to get a new numerator, and multiply the denominators together to get a new denominator.