Fractions Equivalent To 2/3
The equivalent fractions to ⅔
(
it reads two thirds
)
are those whose value, expressed in decimal form, is the same as that obtained by dividing 2 by 3: 0.6666… The ellipsis indicates that 6 appears infinitely many times in this division.
A fraction equivalent to 2/3 is the fraction 4/6, since r
It turns out that after explicitly solving the division between 4 and 6, the decimal 0.66666 is obtained….
Then it can be stated that 4/6 = 2/3 = 0.66666….
Fractions 2/3 and 4/6 are equivalent because dividing the numerator digit by the denominator digit gives the same periodic decimal number 0.666666…. (Source: F. Zapata)
A fraction, as its name implies, is a part or portion of the unit.
The fraction ⅔
It is obtained by dividing the unit into three equal parts and taking two of those parts.
Every fraction consists of an upper part, called the numerator , separated from the lower part or denominator , by the fraction line.
The denominator indicates how many parts the unit is divided into and the numerator indicates how many of these parts must be taken into account.
Now consider the fraction
4/6
(it reads
four sixths
). This fraction is found to be
equivalent to
23
, since for d
Divide the unit into six parts, follow these steps:

Divide the unit into three equal parts.

And then divide each of these parts in half, orb having a total of six equal parts.
Graphical verification that the fraction 2/3 is equivalent to the fraction 4/6. Source: F. Zapata.
Methods for Finding Equivalent Fractions
Note that the equivalent fraction 4/6 can be obtained from 2/3 by multiplying by 2 both the numerator and the denominator of the latter.
When the numerator and denominator of a fraction are multiplied simultaneously by the same number, an equivalent fraction is obtained.
Another way to find a fraction equivalent to another would be to divide the numerator and denominator by the same quantity, as long as the numerator and denominator are exactly divisible by the same number. But n
or it is possible to get, by dividing by the same integer, an equivalent fraction starting from 2/3, since the numbers 2 and 3
they are cousins to each other.
When the numerator and denominator of a fraction are prime numbers to each other, the fraction is said to be irreducible. And the fraction 2/3 is a good example of this kind of fractions, in fact, 2/3 represents the set of all fractions equivalent to 0.666 …
On the other hand, the fraction 4/6 is reducible and equivalent to the fraction ⅔, since the numerator 4 and the denominator 6 are even numbers, both divisible by 2.
So, the two ways to obtain fractions equivalent to a given one are:

Simultaneously amplify numerator and denominator

Reduce numerator and denominator together
Amplification of fractions
To obtain a fraction equivalent to a given one, the numerator and denominator are multiplied by the same figure. Here are some examples:
In summary, if we start from the irreducible fraction ⅔, the way to obtain any other equivalent fraction is to apply this formula:
Amplification method to obtain equivalent fractions. Source: F. Zapata
Fraction reduction
It is a method that allows obtaining an equivalent fraction, provided that the starting fraction has a numerator and denominator with one or more common divisors.
This is not the case with 2/3, which, as said before, is irreducible. But for example, the fraction 60/90 (
sixty ninety
) can be reduced to:

6/9, since both the numerator and denominator are divisible by ten.

30/45, because numerator and denominator are divisible by two.

20/30, since numerator and denominator are divisible by three.

12/18, because the numerator and denominator are divisible by five.
If you want to obtain the irreducible fraction equivalent to the original, then you need to divide both the numerator and the denominator by their
Greatest Common Divisor (GCF).
Decomposing the numerator into factors we have:
60 = 2 ^{2} ⋅ 3 ⋅ 5
And carrying out the same procedure in the denominator:
90 = 2 ⋅ 3 ^{2} ⋅ 5
The LCM are the common prime factors with their lowest exponent, that is:
LCM (60,90) = 2⋅3⋅5 = 30
So 60 times 30 gives 2, which is placed in the numerator and
Since 90 times 30 gives 3, place 3 in the denominator.
Therefore the irreducible fraction of 60/90 can be expressed as:
Ways to determine if a given fraction is equivalent to 2/3
The direct way to know if two or more fractions are equivalent, is to express the fractions directly in decimal form, and if all the digits coincide, it is certain that the fractions are equivalent. But there are other methods applicable to 2/3:
Method 1
Let us be the fraction x / y and we want to know if said fraction is equivalent to 2/3:
A question mark is placed, since it is not yet known if the values of “x” and “y” satisfy the equality.
To find out, multiply in a cross:
3x =? 2 and
Only when the equality holds, is there certainty that x / y is a fraction equivalent to 2/3.
Method 2
This method requires determining the greatest common divisor (GCF) of the numerator and denominator. Then both are divided by the GCF, and if the fraction obtained after carrying out the operation described is 2/3, then it can be said that it is a fraction equivalent to it.
Examples
Example 1
Determine if the fraction 40/60 is equivalent to ⅔.
Solution
By method 1:
The method indicates that it must be multiplied in a cross:
40 x 3 =? 60 x 2
120 =? 120
Since the equality holds, we conclude that 40/60 is equivalent to 2/3.
Example 2
Determine if the fraction 120/180 is equivalent to ⅔.
Solution
In this example, method 2 is applied. The first thing is to determine
the prime factorization of 120:
120 = 2 ^{3} ⋅ 3 ⋅ 5
And the factor decomposition of the denominator is:
180 = 2 ^{2} ⋅ 3 ^{2} ⋅ 5
To determine the GCF, the common factors with their smallest exponent are multiplied:
GCF (120,180) = 2 ^{2} ⋅ 3 ⋅ 5 = 60
Then:
120 ÷ 60 = 2
180 ÷ 60 = 3
Therefore, it is concluded that 120/180 is equivalent to 2/3, that is:
Solved exercises
Exercise 1
Are the fractions 10/15 and 12/18 equivalent?
Solution
The quickest way to check is to cross multiply, since they are not very large values:
10 x 18 =? 15 x 12
180 =? 180
An equality was obtained, then it can be stated that 10/15 = 12/18.
Exercise 2
Are the fractions
8/12 and 12/20 equivalent to ⅔?
Solution
The simplification method will be applied, which consists of simultaneously dividing the numerator and denominator by common prime factors until an irreducible expression is reached:
8/12 = 4/6 = ⅔, that is, the first fraction
is equivalent to ⅔
.
For the second fraction we have:
12/20 = 6/10 = ⅗, but ⅗ is irreducible and different from ⅔, therefore the second fraction
does not equal ⅔
.