Тема: Основна властивість дробу. Скорочення дробу
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T.: Today we are going to review how to reduce fractions.
Reducing Fractions to Lowest Term
Consider
the following two fractions:
1/2 and 2/4
These fractions are equivalent fractions. They
both represent the same amount. One half of an orange is equal to two quarters
of an orange. However, only one of these fractions is written in lowest terms.
A fraction
is in lowest terms when the numerator and denominator have no common
factor other than 1.
The factors
of 2 are 1 and 2.
The factors
of 4 are 1, 2, and 4.
2 and 4
share a common factor: 2.
We can reduce this fraction by dividing both
the numerator and denominator by their common factor, 2.
2 ÷ 2/4 ÷ 2
= 1/2
1 and 2 have no common factor other than 1, so
the fraction is in lowest terms.
Method #1: Common Factors
(a slow and
steady method)
Let's try another example:
30/36
Do 30 and 36 share any factors other than 1?
The factors
of 30 are 1, 2, 3, 5, 6, 10, 15, 30.
The factors
of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36.
30 and 36
have three common factors: 2, 3, and 6.
Let's see what happens if we divide the
numerator and denominator by their lowest common factor, 2. (In fact, we'd know
that they have 2 as a common factor without having to work out all their
factors, because both 30 and 36 are even numbers.)
30 ÷ 2/36 ÷
2 = 15/18
Are we done? Do 15 and 18 share any factors
other than 1?
The factors
of 15 are 1, 3, 5, 15.
The factors
of 18 are 1, 2, 3, 6, 9, 18.
15 and 18
have one common factor: 3.
Once again, we divide the numerator and
denominator by their common factor, 3.
15 ÷ 3/18 ÷
3 = 5/6
Are we done? Do 5 and 6 share any factors
other than 1?
The factors
of 5 are 1 and 5.
The factors
of 6 are 1, 2, 3, and 6.
5 and 6
have no common factors other than 1.
This method will reduce a fraction to its
lowest terms, but it can take several steps until you reach that point. What
would have happened if, instead of dividing the numerator and denominator by
their lowest common factor, we had started with their greatest common factor?
Method #2: Greatest Common Factor
(a more
efficient method)
Let's try it again:
30/36
Do 30 and 36 share any factors other than 1?
The factors
of 30 are 1, 2, 3, 5, 6, 10, 15.
The factors
of 36 are 1, 2, 3, 4, 6, 9, 12, 18.
30 and 36
have three common factors: 2, 3, and 6.
The
greatest common factor is 6.
Divide the numerator and denominator by the
greatest common factor:
30 ÷ 6/36 ÷
6 = 5/6
This time, it takes only one step to get to
the same result. To reduce a fraction to its lowest terms, divide the numerator
and denominator by the greatest common factor.
Method #3: Prime Factors
(an even
more efficient method)
Another way to reduce fractions is to break
the numerator and denominator down to their prime factors, and remove every
prime factor the two have in common. Let's do that example one more time, using
this method.
30/36
The prime
factors of 30 are 2 x 3 x 5.
The prime
factors of 36 are 2 x 2 x 3 x 3.
2 x 3 x 5/2
x 2 x 3 x 3
We remove
the 2 x 3 the numerator and denominator have in common:
5/2 x 3 =
5/6
(If you think about it, this works the same
way as the last method. The greatest common factor of two numbers is the same
as the product of the prime factors they have in common.)
T.: Let’s solve some problems to improve your knowledge.
1) Study
the table and reduce fractions to their lowest terms (card#1).
2) Reduce
fractions (card#2).
3) Reduce
fractions to their lowest terms. Change any improper fractions to mixed
numbers.
Card#1
Card#2
Reduce
fractions:
Card#3
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