Lesson 22

EQUIVALENT FRACTIONS

Here is an elementary example:

fraction is one half of 1.

Every fraction has a ratio to 1. Equivlent fractions will have the same ratio to 1.

Since the numerator and denominator are natural numbers, they will have all the properties of ratios, specifically the theorem of the same multiple.

Answer.  For example,

To create them, we multiplied both 5 and 6 by the same number.  First by 2, then by 3, then by 10.

(Compare Lesson 20, Problem 2c.)

Example 2.    Write the missing numerator:

Answer.   To make 7 into 28, we have to multiply it by 4.  Therefore, we must also multiply 6 by 4:

6 7

=

24 4 × 6 28 4 × 7

In practice, to find the multiplier, mentally divide the original denominator into the new denominator, and then multiply the numerator by that quotient.  That is, say:

"7 goes into 28 four times. Four times 6 is 24."

Compare Lesson 18, Example 5.  Every property of ratios applies to fractions.

Example 3.   Write the missing numerator:

Answer.   "8 goes into 48 six times.  Six times 5 is 30."

In actual problems, we convert two (or more) fractions so that they have equal denominators.  When we do that, it is easy to compare them. Moreover, equal denominators are necessary in order to add or subtract fractions.  For we can only add or subtract quantities that have the same name, that is, that are units of the same kind; and it is the denominator of a fraction that names the unit. (Lesson 21.)

Now, since 15, for example, is a multiple of 5, we say that 5 is a divisor of 15.

5 is also a divisor of 20.  5 is a common divisor of 15 and 20.

(15 and 22 have no common divisors, except 1, which is a divisor of every number.)

Example 4.   Convert and to equivalent fractions with equal denominators.

Answer.   The denominators 3 and 8 have no common divisors (except 1).  Therefore, as a common denominator, choose 24.

To convert we said, "3 goes into 24 eight times. Eight tmes 2 is 16."

To convert we said, "8 goes into 24 three times. Three times 5 is 15."

Once we convert to a common denominator, we could then know that is greater than . Because when fractions have equal denominators, then the larger the numerator, the larger the fraction. (Lesson 20, Question 11.)

Also, we could now add those fractions:

See Lesson 21, Example 3.

We can choose the product of denominators even when the denominators have a common divisor. But their product will not then be their lowest common multiple (Lesson 23). The student should prefer the lowest common multiple because smaller numbers make for simpler calculations.

Same ratio

When fractions are equivalent, their numerators and denominators are in the same ratio.  That in fact is the best definition of equivalent fractions.

1 is half of 2.  2 is half of 4.  In fact, any fraction where the numerator is half of the denominator will be equivalent to .

1 is half of 2.  2 is half of 4.  3 is half of 6.  5 is half of 10.  And so on.  They are all at the same place on the number line.

Example 5.    and are equivalent because each numerator is a third of its denominator.

Example 6.   Write the missing numerator:

Answer.  7 is a quarter of 28.  And a quarter of 16 is 4.

 7 is to 28  as  4 is to 16.

How to simplify, or reduce, a fraction

The numerator and denominator of a fraction are called its terms. To simplify or reduce a fraction means to make the terms smaller numbers.  To do that, we divide both terms by a common divisor.

Of those three has the as the lowest terms. We cannot divide any further.

We like to express a fraction with its lowest terms because it gives a better sense of its value, and it makes for simpler calculations.

Answer.  15 and 21 have a common divisor, 3.

15 21

=

5 "3 goes into 15 five times." 7 "3 goes into 21 seven times."

Or, take a third of both 15 and 21.

"A third of 15 is 5."

"A third of 21 is 7."

Answer.  When the terms have the same number of 0's, we may ignore them.

Effectively, we have divided 200 and 1200 by 100. (Lesson 2, Question 10.)

Solution.  Divide 20 by 8.  "8 goes into 20 two (2) times (16) with 4 left over."

Or, we could reduce first.  20 and 8 have a common divisor 4:

20  8	=	5 "4 goes into 20 five times." 2 "4 goes into 8 two times."
	=	2½.

Notice that we are free to interpret the same symbol

And it indicates "the ratio of 20 to 8."

Any fraction in which the numerator and denominator are equal, is equal to 1.

Summary

At this point, please "turn" the page and do some Problems.

or

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