In this Lesson, we will answer the following:
Here is an elementary example:
one half of 1.
As fractions of a unit of measure, equivalent fractions are equal measurements.
(Compare the theorem of the same multiple; for, a numerator has a ratio to the denominator. Every property of ratios applies to fractions. From the viewpoint of comparing fractions, however, see Problem 2 of that lesson.)
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:
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."
The student who has studied ratio and proportion
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 (see the next Lesson, Question 3), and equal denominators are necessary in order to add or subtract them (Lesson 25). 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 not a divisor of 14, because 14 is not a multiple of 5.)
5 is also a divisor of 20. 5 is a common divisor of 15 and 20.
(15 and 14 have no common divisors, except 1, which is a divisor of every number.)
Answer. The denominators 3 and 5 have no common divisors (except 1). Therefore, as a common denominator, choose 15.
Once we convert to a common denominator, we could then know
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.
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
1 is half of 2. 2 is half of 4. 3 is half of 6. 5 is half of 10. And so on. These are all at the same place on the number line.
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. To accomplish that, we divide both terms by a common divisor.
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.
Or, take a third of both 15 and 21.
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:
Notice that we are free to interpret the same symbol
"the ratio of 20 to 8."
Any fraction in which the numerator and denominator are equal, is equal to 1.
At this point, please "turn" the page and do some Problems.
Continue on to the next Section.
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